From ddeee8ed284f0e57116cf217eb82d0616d76a0c9 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Tue, 14 Oct 2025 12:35:01 +0200 Subject: [PATCH 01/50] factor out inequality of cardinalities --- src/foundation/mere-embeddings.lagda.md | 24 +- src/set-theory.lagda.md | 1 + src/set-theory/cardinalities.lagda.md | 161 +------------ .../inequality-cardinalities.lagda.md | 225 ++++++++++++++++++ .../kuratowski-finite-sets.lagda.md | 9 +- 5 files changed, 248 insertions(+), 172 deletions(-) create mode 100644 src/set-theory/inequality-cardinalities.lagda.md diff --git a/src/foundation/mere-embeddings.lagda.md b/src/foundation/mere-embeddings.lagda.md index 4fbccc13d4..931180da64 100644 --- a/src/foundation/mere-embeddings.lagda.md +++ b/src/foundation/mere-embeddings.lagda.md @@ -9,6 +9,7 @@ module foundation.mere-embeddings where ```agda open import foundation.cantor-schroder-bernstein-escardo open import foundation.embeddings +open import foundation.functoriality-propositional-truncation open import foundation.law-of-excluded-middle open import foundation.mere-equivalences open import foundation.propositional-truncations @@ -21,6 +22,12 @@ open import order-theory.large-preorders +## Idea + +A type `A` {{#concept "merely embeds" Agda=mere-emb}} into a type `B` if there +[merely exists](foundation.propositional-truncations.md) an +[embedding](foundation-core.embeddings.md) of `A` into `B`. + ## Definition ```agda @@ -46,13 +53,7 @@ refl-mere-emb X = unit-trunc-Prop id-emb transitive-mere-emb : {l1 l2 l3 : Level} {X : UU l1} {Y : UU l2} {Z : UU l3} → mere-emb Y Z → mere-emb X Y → mere-emb X Z -transitive-mere-emb g f = - apply-universal-property-trunc-Prop g - ( mere-emb-Prop _ _) - ( λ g' → - apply-universal-property-trunc-Prop f - ( mere-emb-Prop _ _) - ( λ f' → unit-trunc-Prop (comp-emb g' f'))) +transitive-mere-emb = map-binary-trunc-Prop comp-emb mere-emb-Large-Preorder : Large-Preorder lsuc (_⊔_) type-Large-Preorder mere-emb-Large-Preorder l = UU l @@ -68,11 +69,6 @@ transitive-leq-Large-Preorder mere-emb-Large-Preorder X Y Z = antisymmetric-mere-emb : {l1 l2 : Level} {X : UU l1} {Y : UU l2} → LEM (l1 ⊔ l2) → mere-emb X Y → mere-emb Y X → mere-equiv X Y -antisymmetric-mere-emb lem f g = - apply-universal-property-trunc-Prop f - ( mere-equiv-Prop _ _) - ( λ f' → - apply-universal-property-trunc-Prop g - ( mere-equiv-Prop _ _) - ( λ g' → unit-trunc-Prop (Cantor-Schröder-Bernstein-Escardó lem f' g'))) +antisymmetric-mere-emb lem = + map-binary-trunc-Prop (Cantor-Schröder-Bernstein-Escardó lem) ``` diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index 2e2b319881..bf221d479b 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -50,6 +50,7 @@ open import set-theory.cantors-diagonal-argument public open import set-theory.cardinalities public open import set-theory.countable-sets public open import set-theory.cumulative-hierarchy public +open import set-theory.inequality-cardinalities public open import set-theory.infinite-sets public open import set-theory.russells-paradox public open import set-theory.uncountable-sets public diff --git a/src/set-theory/cardinalities.lagda.md b/src/set-theory/cardinalities.lagda.md index 26956dc000..47fe3437e8 100644 --- a/src/set-theory/cardinalities.lagda.md +++ b/src/set-theory/cardinalities.lagda.md @@ -7,18 +7,10 @@ module set-theory.cardinalities where
Imports ```agda -open import foundation.binary-relations -open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.function-extensionality open import foundation.functoriality-propositional-truncation open import foundation.identity-types -open import foundation.large-binary-relations -open import foundation.law-of-excluded-middle -open import foundation.mere-embeddings open import foundation.mere-equivalences -open import foundation.propositional-extensionality -open import foundation.propositions open import foundation.set-truncations open import foundation.sets open import foundation.universe-levels @@ -42,7 +34,7 @@ element in the set truncation of the universe of all sets. ## Definition -### Cardinalities +### Cardinals ```agda cardinal-Set : (l : Level) → Set (lsuc l) @@ -51,123 +43,20 @@ cardinal-Set l = trunc-Set (Set l) cardinal : (l : Level) → UU (lsuc l) cardinal l = type-Set (cardinal-Set l) -cardinality : {l : Level} → Set l → cardinal l -cardinality A = unit-trunc-Set A +is-set-cardinal : {l : Level} → is-set (cardinal l) +is-set-cardinal {l} = is-set-type-Set (cardinal-Set l) ``` -### Inequality of cardinalities +### The cardinality of a set ```agda -leq-cardinality-Prop' : - {l1 l2 : Level} → Set l1 → cardinal l2 → Prop (l1 ⊔ l2) -leq-cardinality-Prop' {l1} {l2} X = - map-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( λ Y' → mere-emb-Prop (type-Set X) (type-Set Y')) - -compute-leq-cardinality-Prop' : - {l1 l2 : Level} (X : Set l1) (Y : Set l2) → - ( leq-cardinality-Prop' X (cardinality Y)) ＝ - ( mere-emb-Prop (type-Set X) (type-Set Y)) -compute-leq-cardinality-Prop' {l1} {l2} X = - triangle-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( λ Y' → mere-emb-Prop (type-Set X) (type-Set Y')) - -leq-cardinality-Prop : - {l1 l2 : Level} → cardinal l1 → cardinal l2 → Prop (l1 ⊔ l2) -leq-cardinality-Prop {l1} {l2} = - map-universal-property-trunc-Set - ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( leq-cardinality-Prop') - -leq-cardinality : - {l1 l2 : Level} → cardinal l1 → cardinal l2 → UU (l1 ⊔ l2) -leq-cardinality X Y = type-Prop (leq-cardinality-Prop X Y) - -is-prop-leq-cardinality : - {l1 l2 : Level} {X : cardinal l1} {Y : cardinal l2} → - is-prop (leq-cardinality X Y) -is-prop-leq-cardinality {X = X} {Y = Y} = - is-prop-type-Prop (leq-cardinality-Prop X Y) - -compute-leq-cardinality : - {l1 l2 : Level} (X : Set l1) (Y : Set l2) → - ( leq-cardinality (cardinality X) (cardinality Y)) ≃ - ( mere-emb (type-Set X) (type-Set Y)) -compute-leq-cardinality {l1} {l2} X Y = - equiv-eq-Prop - ( ( htpy-eq - ( triangle-universal-property-trunc-Set - ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( leq-cardinality-Prop') X) (cardinality Y)) ∙ - ( compute-leq-cardinality-Prop' X Y)) - -unit-leq-cardinality : - {l1 l2 : Level} (X : Set l1) (Y : Set l2) → - mere-emb (type-Set X) (type-Set Y) → - leq-cardinality (cardinality X) (cardinality Y) -unit-leq-cardinality X Y = map-inv-equiv (compute-leq-cardinality X Y) - -inv-unit-leq-cardinality : - {l1 l2 : Level} (X : Set l1) (Y : Set l2) → - leq-cardinality (cardinality X) (cardinality Y) → - mere-emb (type-Set X) (type-Set Y) -inv-unit-leq-cardinality X Y = pr1 (compute-leq-cardinality X Y) - -refl-leq-cardinality : is-reflexive-Large-Relation cardinal leq-cardinality -refl-leq-cardinality {l} = - apply-dependent-universal-property-trunc-Set' - ( λ X → set-Prop (leq-cardinality-Prop X X)) - ( λ A → unit-leq-cardinality A A (refl-mere-emb (type-Set A))) - -transitive-leq-cardinality : - {l1 l2 l3 : Level} - (X : cardinal l1) - (Y : cardinal l2) - (Z : cardinal l3) → - leq-cardinality X Y → - leq-cardinality Y Z → - leq-cardinality X Z -transitive-leq-cardinality X Y Z = - apply-dependent-universal-property-trunc-Set' - ( λ u → - set-Prop - ( function-Prop - ( leq-cardinality u Y) - ( function-Prop (leq-cardinality Y Z) - ( leq-cardinality-Prop u Z)))) - ( λ a → - apply-dependent-universal-property-trunc-Set' - ( λ v → - set-Prop - (function-Prop - (leq-cardinality (cardinality a) v) - (function-Prop (leq-cardinality v Z) - (leq-cardinality-Prop (cardinality a) Z)))) - ( λ b → - apply-dependent-universal-property-trunc-Set' - ( λ w → - set-Prop - (function-Prop - (leq-cardinality (cardinality a) (cardinality b)) - (function-Prop (leq-cardinality (cardinality b) w) - (leq-cardinality-Prop (cardinality a) w)))) - ( λ c aImports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.function-extensionality +open import foundation.identity-types +open import foundation.large-binary-relations +open import foundation.law-of-excluded-middle +open import foundation.mere-embeddings +open import foundation.propositional-extensionality +open import foundation.propositions +open import foundation.set-truncations +open import foundation.sets +open import foundation.univalence +open import foundation.universe-levels + +open import set-theory.cardinalities +``` + +
+ +## Idea + +We say a [cardinality of sets](set-theory.cardinalities.md) `X` is +{{#concept "less than or equal to" Agda=leq-cardinal}} a cardinality `Y` if any +[set](foundation-core.sets.md) in the isomorphism class represented by `X` +embeds into any set in the isomorphism class represented by `Y`. This defines +the {{#concept "standard ordering" Disambiguation="on cardinalities of sets"}} +on cardinalities of sets. + +Under the assumption of the +[law of excluded middle](foundation.law-of-excluded-middle.md), this relation is +antisymmetric and hence defines a [partial order](order-theory.posets.md) on +cardinalites. + +## Definition + +### Boundedness of the cardinality of a set + +```agda +leq-prop-cardinal' : + {l1 l2 : Level} → Set l1 → cardinal l2 → Prop (l1 ⊔ l2) +leq-prop-cardinal' {l1} {l2} X = + map-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( λ Y' → mere-emb-Prop (type-Set X) (type-Set Y')) + +compute-leq-prop-cardinal' : + {l1 l2 : Level} (X : Set l1) (Y : Set l2) → + ( leq-prop-cardinal' X (cardinality Y)) ＝ + ( mere-emb-Prop (type-Set X) (type-Set Y)) +compute-leq-prop-cardinal' {l1} {l2} X = + triangle-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( λ Y' → mere-emb-Prop (type-Set X) (type-Set Y')) +``` + +### Inequality of cardinals + +```agda +module _ + {l1 l2 : Level} + where + + leq-prop-cardinal : cardinal l1 → cardinal l2 → Prop (l1 ⊔ l2) + leq-prop-cardinal = + map-universal-property-trunc-Set + ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-prop-cardinal') + + leq-cardinal : cardinal l1 → cardinal l2 → UU (l1 ⊔ l2) + leq-cardinal X Y = type-Prop (leq-prop-cardinal X Y) + + is-prop-leq-cardinal : + {X : cardinal l1} {Y : cardinal l2} → is-prop (leq-cardinal X Y) + is-prop-leq-cardinal {X} {Y} = is-prop-type-Prop (leq-prop-cardinal X Y) +``` + +### Inequality of cardinalities + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + leq-prop-cardinality : Prop (l1 ⊔ l2) + leq-prop-cardinality = leq-prop-cardinal (cardinality X) (cardinality Y) + + leq-cardinality : UU (l1 ⊔ l2) + leq-cardinality = leq-cardinal (cardinality X) (cardinality Y) + + is-prop-leq-cardinality : is-prop leq-cardinality + is-prop-leq-cardinality = is-prop-leq-cardinal + + compute-leq-cardinality' : + leq-cardinality ＝ mere-emb (type-Set X) (type-Set Y) + compute-leq-cardinality' = + ap + ( type-Prop) + ( ( htpy-eq + ( triangle-universal-property-trunc-Set + ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-prop-cardinal') X) (cardinality Y)) ∙ + ( compute-leq-prop-cardinal' X Y)) + + compute-leq-cardinality : + leq-cardinality ≃ mere-emb (type-Set X) (type-Set Y) + compute-leq-cardinality = equiv-eq compute-leq-cardinality' + + unit-leq-cardinality : + mere-emb (type-Set X) (type-Set Y) → leq-cardinality + unit-leq-cardinality = map-inv-equiv compute-leq-cardinality + + inv-unit-leq-cardinality : + leq-cardinality → mere-emb (type-Set X) (type-Set Y) + inv-unit-leq-cardinality = pr1 compute-leq-cardinality +``` + +### Inequality on cardinals is reflexive + +```agda +refl-leq-cardinal : is-reflexive-Large-Relation cardinal leq-cardinal +refl-leq-cardinal = + apply-dependent-universal-property-trunc-Set' + ( λ X → set-Prop (leq-prop-cardinal X X)) + ( λ A → unit-leq-cardinality A A (refl-mere-emb (type-Set A))) +``` + +### Inequality on cardinals is transitive + +```agda +transitive-leq-cardinal : + {l1 l2 l3 : Level} + (X : cardinal l1) + (Y : cardinal l2) + (Z : cardinal l3) → + leq-cardinal X Y → + leq-cardinal Y Z → + leq-cardinal X Z +transitive-leq-cardinal X Y Z = + apply-dependent-universal-property-trunc-Set' + ( λ u → + set-Prop + ( function-Prop + ( leq-cardinal u Y) + ( function-Prop (leq-cardinal Y Z) + ( leq-prop-cardinal u Z)))) + ( λ a → + apply-dependent-universal-property-trunc-Set' + ( λ v → + set-Prop + (function-Prop + (leq-cardinal (cardinality a) v) + (function-Prop (leq-cardinal v Z) + (leq-prop-cardinal (cardinality a) Z)))) + ( λ b → + apply-dependent-universal-property-trunc-Set' + ( λ w → + set-Prop + (function-Prop + (leq-cardinal (cardinality a) (cardinality b)) + (function-Prop (leq-cardinal (cardinality b) w) + (leq-prop-cardinal (cardinality a) w)))) + ( λ c a Date: Tue, 14 Oct 2025 12:35:11 +0200 Subject: [PATCH 02/50] `apply-twice-dependent-universal-property-trunc-Set'` --- src/foundation/set-truncations.lagda.md | 18 ++++++++++++++++++ 1 file changed, 18 insertions(+) diff --git a/src/foundation/set-truncations.lagda.md b/src/foundation/set-truncations.lagda.md index 154cc2d2b5..abbed9793e 100644 --- a/src/foundation/set-truncations.lagda.md +++ b/src/foundation/set-truncations.lagda.md @@ -122,6 +122,24 @@ module _ (x : type-trunc-Set A) → type-Set (B x) apply-dependent-universal-property-trunc-Set' = map-inv-equiv (equiv-dependent-universal-property-trunc-Set B) f + +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : type-trunc-Set A → UU l2} + (C : (a : type-trunc-Set A) → type-trunc-Set (B a) → Set l3) + (f : + (x : A) (y : B (unit-trunc-Set x)) → + type-Set (C (unit-trunc-Set x) (unit-trunc-Set y))) + where + + apply-twice-dependent-universal-property-trunc-Set' : + (a : type-trunc-Set A) (b : type-trunc-Set (B a)) → type-Set (C a b) + apply-twice-dependent-universal-property-trunc-Set' = + apply-dependent-universal-property-trunc-Set' + ( λ x → Π-Set (trunc-Set (B x)) (C x)) + ( λ x → + apply-dependent-universal-property-trunc-Set' + ( C (unit-trunc-Set x)) + ( f x)) ``` ### The universal property of set truncations From cacf6e509b36506065ed9ce9e2f357dc5fe8e1fe Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Tue, 14 Oct 2025 13:20:41 +0200 Subject: [PATCH 03/50] fix mess --- src/foundation/set-truncations.lagda.md | 26 ++++ .../inequality-cardinalities.lagda.md | 124 ++++++++---------- 2 files changed, 80 insertions(+), 70 deletions(-) diff --git a/src/foundation/set-truncations.lagda.md b/src/foundation/set-truncations.lagda.md index abbed9793e..3059816359 100644 --- a/src/foundation/set-truncations.lagda.md +++ b/src/foundation/set-truncations.lagda.md @@ -140,6 +140,32 @@ module _ apply-dependent-universal-property-trunc-Set' ( C (unit-trunc-Set x)) ( f x)) + +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : type-trunc-Set A → UU l2} + {C : (a : type-trunc-Set A) → type-trunc-Set (B a) → UU l3} + (D : + (a : type-trunc-Set A) (b : type-trunc-Set (B a)) → type-trunc-Set (C a b) → + Set l4) + (f : + (x : A) + (y : B (unit-trunc-Set x)) + (z : C (unit-trunc-Set x) (unit-trunc-Set y)) → + type-Set (D (unit-trunc-Set x) (unit-trunc-Set y) (unit-trunc-Set z))) + where + + apply-thrice-dependent-universal-property-trunc-Set' : + (a : type-trunc-Set A) + (b : type-trunc-Set (B a)) + (c : type-trunc-Set (C a b)) → + type-Set (D a b c) + apply-thrice-dependent-universal-property-trunc-Set' = + apply-twice-dependent-universal-property-trunc-Set' + ( λ x y → Π-Set (trunc-Set (C x y)) (D x y)) + ( λ x y → + apply-dependent-universal-property-trunc-Set' + ( D (unit-trunc-Set x) (unit-trunc-Set y)) + ( f x y)) ``` ### The universal property of set truncations diff --git a/src/set-theory/inequality-cardinalities.lagda.md b/src/set-theory/inequality-cardinalities.lagda.md index 713727a567..2c69a4ff7a 100644 --- a/src/set-theory/inequality-cardinalities.lagda.md +++ b/src/set-theory/inequality-cardinalities.lagda.md @@ -29,8 +29,8 @@ open import set-theory.cardinalities ## Idea -We say a [cardinality of sets](set-theory.cardinalities.md) `X` is -{{#concept "less than or equal to" Agda=leq-cardinal}} a cardinality `Y` if any +We say a [cardinal of sets](set-theory.cardinalities.md) `X` is +{{#concept "less than or equal to" Agda=leq-cardinal}} a cardinal `Y` if any [set](foundation-core.sets.md) in the isomorphism class represented by `X` embeds into any set in the isomorphism class represented by `Y`. This defines the {{#concept "standard ordering" Disambiguation="on cardinalities of sets"}} @@ -127,58 +127,43 @@ module _ ### Inequality on cardinals is reflexive ```agda +refl-leq-cardinality : is-reflexive-Large-Relation Set leq-cardinality +refl-leq-cardinality A = unit-leq-cardinality A A (refl-mere-emb (type-Set A)) + refl-leq-cardinal : is-reflexive-Large-Relation cardinal leq-cardinal refl-leq-cardinal = apply-dependent-universal-property-trunc-Set' ( λ X → set-Prop (leq-prop-cardinal X X)) - ( λ A → unit-leq-cardinality A A (refl-mere-emb (type-Set A))) + ( refl-leq-cardinality) ``` ### Inequality on cardinals is transitive ```agda -transitive-leq-cardinal : +module _ {l1 l2 l3 : Level} - (X : cardinal l1) - (Y : cardinal l2) - (Z : cardinal l3) → - leq-cardinal X Y → - leq-cardinal Y Z → - leq-cardinal X Z -transitive-leq-cardinal X Y Z = - apply-dependent-universal-property-trunc-Set' - ( λ u → - set-Prop - ( function-Prop - ( leq-cardinal u Y) - ( function-Prop (leq-cardinal Y Z) - ( leq-prop-cardinal u Z)))) - ( λ a → - apply-dependent-universal-property-trunc-Set' - ( λ v → - set-Prop - (function-Prop - (leq-cardinal (cardinality a) v) - (function-Prop (leq-cardinal v Z) - (leq-prop-cardinal (cardinality a) Z)))) - ( λ b → - apply-dependent-universal-property-trunc-Set' - ( λ w → - set-Prop - (function-Prop - (leq-cardinal (cardinality a) (cardinality b)) - (function-Prop (leq-cardinal (cardinality b) w) - (leq-prop-cardinal (cardinality a) w)))) - ( λ c a Date: Tue, 14 Oct 2025 13:23:36 +0200 Subject: [PATCH 04/50] mere decidable emberddings --- src/foundation.lagda.md | 1 + src/foundation/decidable-embeddings.lagda.md | 4 +- .../mere-decidable-embeddings.lagda.md | 79 +++++++++++++++++++ 3 files changed, 82 insertions(+), 2 deletions(-) create mode 100644 src/foundation/mere-decidable-embeddings.lagda.md diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md index f0f52a7493..0b12e6fe16 100644 --- a/src/foundation.lagda.md +++ b/src/foundation.lagda.md @@ -300,6 +300,7 @@ open import foundation.maps-in-global-subuniverses public open import foundation.maps-in-subuniverses public open import foundation.maximum-truncation-levels public open import foundation.maybe public +open import foundation.mere-decidable-embeddings public open import foundation.mere-embeddings public open import foundation.mere-equality public open import foundation.mere-equivalences public diff --git a/src/foundation/decidable-embeddings.lagda.md b/src/foundation/decidable-embeddings.lagda.md index 051f6be3f1..63586874cb 100644 --- a/src/foundation/decidable-embeddings.lagda.md +++ b/src/foundation/decidable-embeddings.lagda.md @@ -213,8 +213,8 @@ is-decidable-emb-id : {l : Level} {A : UU l} → is-decidable-emb (id {A = A}) is-decidable-emb-id = (is-emb-id , is-decidable-map-id) -decidable-emb-id : {l : Level} {A : UU l} → A ↪ᵈ A -decidable-emb-id = (id , is-decidable-emb-id) +id-decidable-emb : {l : Level} {A : UU l} → A ↪ᵈ A +id-decidable-emb = (id , is-decidable-emb-id) is-decidable-prop-map-id : {l : Level} {A : UU l} → is-decidable-prop-map (id {A = A}) diff --git a/src/foundation/mere-decidable-embeddings.lagda.md b/src/foundation/mere-decidable-embeddings.lagda.md new file mode 100644 index 0000000000..146d48cf98 --- /dev/null +++ b/src/foundation/mere-decidable-embeddings.lagda.md @@ -0,0 +1,79 @@ +# Mere decidable embeddings + +```agda +module foundation.mere-decidable-embeddings where +``` + +
Imports + +```agda +open import foundation.decidable-embeddings +open import foundation.embeddings +open import foundation.functoriality-propositional-truncation +open import foundation.law-of-excluded-middle +open import foundation.mere-equivalences +open import foundation.propositional-truncations +open import foundation.universe-levels + +open import foundation-core.propositions + +open import order-theory.large-preorders +``` + +
+ +## Idea + +A type `A` +{{#concept "merely decidably embeds" Agda=mere-decidable-decidable-emb}} into a +type `B` if there [merely exists](foundation.propositional-truncations.md) a +[decidable embedding](logic.decidable-embeddings.md) of `A` into `B`. + +## Definition + +```agda +mere-decidable-emb-Prop : {l1 l2 : Level} → UU l1 → UU l2 → Prop (l1 ⊔ l2) +mere-decidable-emb-Prop X Y = trunc-Prop (X ↪ᵈ Y) + +mere-decidable-emb : {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2) +mere-decidable-emb X Y = type-Prop (mere-decidable-emb-Prop X Y) + +is-prop-mere-decidable-emb : + {l1 l2 : Level} (X : UU l1) (Y : UU l2) → is-prop (mere-decidable-emb X Y) +is-prop-mere-decidable-emb X Y = is-prop-type-Prop (mere-decidable-emb-Prop X Y) +``` + +## Properties + +### Types equipped with mere decidable embeddings form a preordering + +```agda +refl-mere-decidable-emb : {l1 : Level} (X : UU l1) → mere-decidable-emb X X +refl-mere-decidable-emb X = unit-trunc-Prop id-decidable-emb + +transitive-mere-decidable-emb : + {l1 l2 l3 : Level} {X : UU l1} {Y : UU l2} {Z : UU l3} → + mere-decidable-emb Y Z → mere-decidable-emb X Y → mere-decidable-emb X Z +transitive-mere-decidable-emb = map-binary-trunc-Prop comp-decidable-emb + +mere-decidable-emb-Large-Preorder : Large-Preorder lsuc (_⊔_) +type-Large-Preorder mere-decidable-emb-Large-Preorder l = UU l +leq-prop-Large-Preorder mere-decidable-emb-Large-Preorder = + mere-decidable-emb-Prop +refl-leq-Large-Preorder mere-decidable-emb-Large-Preorder = + refl-mere-decidable-emb +transitive-leq-Large-Preorder mere-decidable-emb-Large-Preorder X Y Z = + transitive-mere-decidable-emb +``` + +### Assuming WLPO, then types equipped with mere decidable embeddings form a partial ordering + +> This remains to be formalized. + +```text +antisymmetric-mere-decidable-emb : + {l1 l2 : Level} {X : UU l1} {Y : UU l2} → WLPO (l1 ⊔ l2) → + mere-decidable-emb X Y → mere-decidable-emb Y X → mere-equiv X Y +antisymmetric-mere-decidable-emb wlpo = + map-binary-trunc-Prop (decidable-emb-Cantor-Schröder-Bernstein wlpo) +``` From cfd7d6cc0f26a9527feef9976d3418d253520148 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Tue, 14 Oct 2025 13:29:06 +0200 Subject: [PATCH 05/50] complemented inequality of cardinalities --- .../mere-decidable-embeddings.lagda.md | 3 +- src/set-theory.lagda.md | 1 + ...lemented-inequality-cardinalities.lagda.md | 240 ++++++++++++++++++ 3 files changed, 242 insertions(+), 2 deletions(-) create mode 100644 src/set-theory/complemented-inequality-cardinalities.lagda.md diff --git a/src/foundation/mere-decidable-embeddings.lagda.md b/src/foundation/mere-decidable-embeddings.lagda.md index 146d48cf98..46786b4258 100644 --- a/src/foundation/mere-decidable-embeddings.lagda.md +++ b/src/foundation/mere-decidable-embeddings.lagda.md @@ -24,8 +24,7 @@ open import order-theory.large-preorders ## Idea -A type `A` -{{#concept "merely decidably embeds" Agda=mere-decidable-decidable-emb}} into a +A type `A` {{#concept "merely decidably embeds" Agda=mere-decidable-emb}} into a type `B` if there [merely exists](foundation.propositional-truncations.md) a [decidable embedding](logic.decidable-embeddings.md) of `A` into `B`. diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index bf221d479b..9ec8c17f0f 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -48,6 +48,7 @@ open import set-theory.baire-space public open import set-theory.cantor-space public open import set-theory.cantors-diagonal-argument public open import set-theory.cardinalities public +open import set-theory.complemented-inequality-cardinalities public open import set-theory.countable-sets public open import set-theory.cumulative-hierarchy public open import set-theory.inequality-cardinalities public diff --git a/src/set-theory/complemented-inequality-cardinalities.lagda.md b/src/set-theory/complemented-inequality-cardinalities.lagda.md new file mode 100644 index 0000000000..aa006de091 --- /dev/null +++ b/src/set-theory/complemented-inequality-cardinalities.lagda.md @@ -0,0 +1,240 @@ +# Complemented inequality on cardinals + +```agda +module set-theory.complemented-inequality-cardinalities where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.decidable-embeddings +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.function-extensionality +open import foundation.identity-types +open import foundation.large-binary-relations +open import foundation.mere-decidable-embeddings +open import foundation.propositional-extensionality +open import foundation.propositions +open import foundation.set-truncations +open import foundation.sets +open import foundation.univalence +open import foundation.universe-levels +open import foundation.weak-limited-principle-of-omniscience + +open import set-theory.cardinalities +``` + +
+ +## Idea + +We may say a [cardinal of sets](set-theory.cardinalities.md) `X` is +{{#concept "less than or equal to" Agda=leq-complemented-cardinality}} a +cardinal `Y` if any [set](foundation-core.sets.md) in the isomorphism class +represented by `X` [embeds](foundation-core.embeddings.md) as a +[complemented subtype](foundation.decidable-subtypes.md) into any set in the +isomorphism class represented by `Y`. In other words, if there is a +[decidable embedding](foundation.decidable-embeddings.md) from one to the other. +This defines the +{{#concept "complemented ordering" Disambiguation="on cardinalities of sets"}} +on cardinalities of sets. + +Under the assumption of the +[weak limited principle of omniscience](foundation.weak-limited-principle-of-omniscience.md), +this relation is antisymmetric and hence defines a +[partial order](order-theory.posets.md) on cardinalites. + +## Definition + +### Complemented boundedness of the cardinality of a set + +```agda +leq-complemented-prop-cardinal' : + {l1 l2 : Level} → Set l1 → cardinal l2 → Prop (l1 ⊔ l2) +leq-complemented-prop-cardinal' {l1} {l2} X = + map-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) + +compute-leq-complemented-prop-cardinal' : + {l1 l2 : Level} (X : Set l1) (Y : Set l2) → + ( leq-complemented-prop-cardinal' X (cardinality Y)) ＝ + ( mere-decidable-emb-Prop (type-Set X) (type-Set Y)) +compute-leq-complemented-prop-cardinal' {l1} {l2} X = + triangle-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) +``` + +### Complemented inequality of cardinals + +```agda +module _ + {l1 l2 : Level} + where + + leq-complemented-prop-cardinal : + cardinal l1 → cardinal l2 → Prop (l1 ⊔ l2) + leq-complemented-prop-cardinal = + map-universal-property-trunc-Set + ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-complemented-prop-cardinal') + + leq-complemented-cardinal : + cardinal l1 → cardinal l2 → UU (l1 ⊔ l2) + leq-complemented-cardinal X Y = + type-Prop (leq-complemented-prop-cardinal X Y) + + is-prop-leq-complemented-cardinal : + {X : cardinal l1} {Y : cardinal l2} → + is-prop (leq-complemented-cardinal X Y) + is-prop-leq-complemented-cardinal {X} {Y} = + is-prop-type-Prop (leq-complemented-prop-cardinal X Y) +``` + +### Complemented inequality of cardinalities + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + leq-complemented-prop-cardinality : Prop (l1 ⊔ l2) + leq-complemented-prop-cardinality = + leq-complemented-prop-cardinal (cardinality X) (cardinality Y) + + leq-complemented-cardinality : UU (l1 ⊔ l2) + leq-complemented-cardinality = + leq-complemented-cardinal (cardinality X) (cardinality Y) + + is-prop-leq-complemented-cardinality : + is-prop leq-complemented-cardinality + is-prop-leq-complemented-cardinality = + is-prop-leq-complemented-cardinal + + compute-leq-complemented-cardinality' : + leq-complemented-cardinality ＝ + mere-decidable-emb (type-Set X) (type-Set Y) + compute-leq-complemented-cardinality' = + ap + ( type-Prop) + ( ( htpy-eq + ( triangle-universal-property-trunc-Set + ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-complemented-prop-cardinal') X) (cardinality Y)) ∙ + ( compute-leq-complemented-prop-cardinal' X Y)) + + compute-leq-complemented-cardinality : + leq-complemented-cardinality ≃ + mere-decidable-emb (type-Set X) (type-Set Y) + compute-leq-complemented-cardinality = + equiv-eq compute-leq-complemented-cardinality' + + unit-leq-complemented-cardinality : + mere-decidable-emb (type-Set X) (type-Set Y) → + leq-complemented-cardinality + unit-leq-complemented-cardinality = + map-inv-equiv compute-leq-complemented-cardinality + + inv-unit-leq-complemented-cardinality : + leq-complemented-cardinality → + mere-decidable-emb (type-Set X) (type-Set Y) + inv-unit-leq-complemented-cardinality = + pr1 compute-leq-complemented-cardinality +``` + +### Complemented inequality on cardinals is reflexive + +```agda +refl-leq-complemented-cardinality : + is-reflexive-Large-Relation Set leq-complemented-cardinality +refl-leq-complemented-cardinality A = + unit-leq-complemented-cardinality A A + ( refl-mere-decidable-emb (type-Set A)) + +refl-leq-complemented-cardinal : + is-reflexive-Large-Relation cardinal leq-complemented-cardinal +refl-leq-complemented-cardinal = + apply-dependent-universal-property-trunc-Set' + ( λ X → set-Prop (leq-complemented-prop-cardinal X X)) + ( refl-leq-complemented-cardinality) +``` + +### Complemented inequality on cardinals is transitive + +```agda +module _ + {l1 l2 l3 : Level} + where + + transitive-leq-complemented-cardinality : + (X : Set l1) (Y : Set l2) (Z : Set l3) → + leq-complemented-cardinality Y Z → + leq-complemented-cardinality X Y → + leq-complemented-cardinality X Z + transitive-leq-complemented-cardinality X Y Z Y≤Z X≤Y = + unit-leq-complemented-cardinality X Z + ( transitive-mere-decidable-emb + ( inv-unit-leq-complemented-cardinality Y Z Y≤Z) + ( inv-unit-leq-complemented-cardinality X Y X≤Y)) + + transitive-leq-complemented-cardinal : + (X : cardinal l1) (Y : cardinal l2) (Z : cardinal l3) → + leq-complemented-cardinal Y Z → + leq-complemented-cardinal X Y → + leq-complemented-cardinal X Z + transitive-leq-complemented-cardinal = + apply-thrice-dependent-universal-property-trunc-Set' + ( λ X Y Z → + ( leq-complemented-cardinal Y Z → + leq-complemented-cardinal X Y → + leq-complemented-cardinal X Z) , + ( is-set-function-type + ( is-set-function-type + ( is-set-is-prop is-prop-leq-complemented-cardinal)))) + ( transitive-leq-complemented-cardinality) +``` + +## Properties + +### Assuming the weak limited principle of omniscience, then complemented inequality is antisymmetric + +Using the previous result and assuming the weak limited principle of +omniscience, we can conclude `leq-complemented-cardinal` is a partial order by +showing that it is antisymmetric. + +> This remains to be formalized. + +```text +module _ + {l : Level} (wlpo : WLPO l) + where + + antisymmetric-leq-complemented-cardinality : + (X Y : Set l) → + leq-complemented-cardinality X Y → + leq-complemented-cardinality Y X → + cardinality X ＝ cardinality Y + antisymmetric-leq-complemented-cardinality X Y X≤Y Y≤X = + eq-mere-equiv-cardinality X Y + ( antisymmetric-mere-decidable-emb + ( wlpo) + ( inv-unit-leq-complemented-cardinality X Y X≤Y) + ( inv-unit-leq-complemented-cardinality Y X Y≤X)) + + antisymmetric-leq-complemented-cardinal : + (X Y : cardinal l) → + leq-complemented-cardinal X Y → leq-complemented-cardinal Y X → X ＝ Y + antisymmetric-leq-complemented-cardinal = + apply-twice-dependent-universal-property-trunc-Set' + ( λ X Y → + set-Prop + ( function-Prop + ( leq-complemented-cardinal X Y) + ( function-Prop + ( leq-complemented-cardinal Y X) + ( Id-Prop (cardinal-Set l) X Y)))) + ( antisymmetric-leq-complemented-cardinality) +``` From d3cb84eb03f285708d833fc65a764fed49b1a89a Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Tue, 14 Oct 2025 13:40:10 +0200 Subject: [PATCH 06/50] imports and fix --- src/foundation/mere-decidable-embeddings.lagda.md | 5 +---- .../complemented-inequality-cardinalities.lagda.md | 2 -- 2 files changed, 1 insertion(+), 6 deletions(-) diff --git a/src/foundation/mere-decidable-embeddings.lagda.md b/src/foundation/mere-decidable-embeddings.lagda.md index 46786b4258..2412f4d2cf 100644 --- a/src/foundation/mere-decidable-embeddings.lagda.md +++ b/src/foundation/mere-decidable-embeddings.lagda.md @@ -8,10 +8,7 @@ module foundation.mere-decidable-embeddings where ```agda open import foundation.decidable-embeddings -open import foundation.embeddings open import foundation.functoriality-propositional-truncation -open import foundation.law-of-excluded-middle -open import foundation.mere-equivalences open import foundation.propositional-truncations open import foundation.universe-levels @@ -26,7 +23,7 @@ open import order-theory.large-preorders A type `A` {{#concept "merely decidably embeds" Agda=mere-decidable-emb}} into a type `B` if there [merely exists](foundation.propositional-truncations.md) a -[decidable embedding](logic.decidable-embeddings.md) of `A` into `B`. +[decidable embedding](foundation.decidable-embeddings.md) of `A` into `B`. ## Definition diff --git a/src/set-theory/complemented-inequality-cardinalities.lagda.md b/src/set-theory/complemented-inequality-cardinalities.lagda.md index aa006de091..9ede149bc2 100644 --- a/src/set-theory/complemented-inequality-cardinalities.lagda.md +++ b/src/set-theory/complemented-inequality-cardinalities.lagda.md @@ -8,7 +8,6 @@ module set-theory.complemented-inequality-cardinalities where ```agda open import foundation.action-on-identifications-functions -open import foundation.decidable-embeddings open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality @@ -21,7 +20,6 @@ open import foundation.set-truncations open import foundation.sets open import foundation.univalence open import foundation.universe-levels -open import foundation.weak-limited-principle-of-omniscience open import set-theory.cardinalities ``` From 68d3fe25d73d19a1994245818ccc630d86fc7986 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Thu, 16 Oct 2025 00:38:51 +0200 Subject: [PATCH 07/50] more edits --- ...lemented-inequality-cardinalities.lagda.md | 51 +++++++++-------- .../inequality-cardinalities.lagda.md | 56 ++++++++++--------- 2 files changed, 60 insertions(+), 47 deletions(-) diff --git a/src/set-theory/complemented-inequality-cardinalities.lagda.md b/src/set-theory/complemented-inequality-cardinalities.lagda.md index 9ede149bc2..690f8ad6d7 100644 --- a/src/set-theory/complemented-inequality-cardinalities.lagda.md +++ b/src/set-theory/complemented-inequality-cardinalities.lagda.md @@ -49,21 +49,24 @@ this relation is antisymmetric and hence defines a ### Complemented boundedness of the cardinality of a set ```agda -leq-complemented-prop-cardinal' : - {l1 l2 : Level} → Set l1 → cardinal l2 → Prop (l1 ⊔ l2) -leq-complemented-prop-cardinal' {l1} {l2} X = - map-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) - -compute-leq-complemented-prop-cardinal' : - {l1 l2 : Level} (X : Set l1) (Y : Set l2) → - ( leq-complemented-prop-cardinal' X (cardinality Y)) ＝ - ( mere-decidable-emb-Prop (type-Set X) (type-Set Y)) -compute-leq-complemented-prop-cardinal' {l1} {l2} X = - triangle-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) +module _ + {l1 l2 : Level} (X : Set l1) + where + + leq-complemented-prop-cardinal' : cardinal l2 → Prop (l1 ⊔ l2) + leq-complemented-prop-cardinal' = + map-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) + + compute-leq-complemented-prop-cardinal' : + (Y : Set l2) → + leq-complemented-prop-cardinal' (cardinality Y) ＝ + mere-decidable-emb-Prop (type-Set X) (type-Set Y) + compute-leq-complemented-prop-cardinal' = + triangle-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) ``` ### Complemented inequality of cardinals @@ -112,17 +115,21 @@ module _ is-prop-leq-complemented-cardinality = is-prop-leq-complemented-cardinal + compute-leq-complemented-prop-cardinality' : + leq-complemented-prop-cardinality ＝ + mere-decidable-emb-Prop (type-Set X) (type-Set Y) + compute-leq-complemented-prop-cardinality' = + ( htpy-eq + ( triangle-universal-property-trunc-Set + ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-complemented-prop-cardinal') X) (cardinality Y)) ∙ + ( compute-leq-complemented-prop-cardinal' X Y) + compute-leq-complemented-cardinality' : leq-complemented-cardinality ＝ mere-decidable-emb (type-Set X) (type-Set Y) compute-leq-complemented-cardinality' = - ap - ( type-Prop) - ( ( htpy-eq - ( triangle-universal-property-trunc-Set - ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( leq-complemented-prop-cardinal') X) (cardinality Y)) ∙ - ( compute-leq-complemented-prop-cardinal' X Y)) + ap type-Prop compute-leq-complemented-prop-cardinality' compute-leq-complemented-cardinality : leq-complemented-cardinality ≃ diff --git a/src/set-theory/inequality-cardinalities.lagda.md b/src/set-theory/inequality-cardinalities.lagda.md index 2c69a4ff7a..371c58f439 100644 --- a/src/set-theory/inequality-cardinalities.lagda.md +++ b/src/set-theory/inequality-cardinalities.lagda.md @@ -46,21 +46,24 @@ cardinalites. ### Boundedness of the cardinality of a set ```agda -leq-prop-cardinal' : - {l1 l2 : Level} → Set l1 → cardinal l2 → Prop (l1 ⊔ l2) -leq-prop-cardinal' {l1} {l2} X = - map-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( λ Y' → mere-emb-Prop (type-Set X) (type-Set Y')) - -compute-leq-prop-cardinal' : - {l1 l2 : Level} (X : Set l1) (Y : Set l2) → - ( leq-prop-cardinal' X (cardinality Y)) ＝ - ( mere-emb-Prop (type-Set X) (type-Set Y)) -compute-leq-prop-cardinal' {l1} {l2} X = - triangle-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( λ Y' → mere-emb-Prop (type-Set X) (type-Set Y')) +module _ + {l1 l2 : Level} (X : Set l1) + where + + leq-prop-cardinal' : cardinal l2 → Prop (l1 ⊔ l2) + leq-prop-cardinal' = + map-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( λ Y' → mere-emb-Prop (type-Set X) (type-Set Y')) + + compute-leq-prop-cardinal' : + (Y : Set l2) → + leq-prop-cardinal' (cardinality Y) ＝ + mere-emb-Prop (type-Set X) (type-Set Y) + compute-leq-prop-cardinal' = + triangle-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( λ Y' → mere-emb-Prop (type-Set X) (type-Set Y')) ``` ### Inequality of cardinals @@ -100,20 +103,23 @@ module _ is-prop-leq-cardinality : is-prop leq-cardinality is-prop-leq-cardinality = is-prop-leq-cardinal - compute-leq-cardinality' : + eq-compute-leq-prop-cardinality : + leq-prop-cardinality ＝ mere-emb-Prop (type-Set X) (type-Set Y) + eq-compute-leq-prop-cardinality = + ( htpy-eq + ( triangle-universal-property-trunc-Set + ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-prop-cardinal') X) (cardinality Y)) ∙ + ( compute-leq-prop-cardinal' X Y) + + eq-compute-leq-cardinality : leq-cardinality ＝ mere-emb (type-Set X) (type-Set Y) - compute-leq-cardinality' = - ap - ( type-Prop) - ( ( htpy-eq - ( triangle-universal-property-trunc-Set - ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( leq-prop-cardinal') X) (cardinality Y)) ∙ - ( compute-leq-prop-cardinal' X Y)) + eq-compute-leq-cardinality = + ap type-Prop eq-compute-leq-prop-cardinality compute-leq-cardinality : leq-cardinality ≃ mere-emb (type-Set X) (type-Set Y) - compute-leq-cardinality = equiv-eq compute-leq-cardinality' + compute-leq-cardinality = equiv-eq eq-compute-leq-cardinality unit-leq-cardinality : mere-emb (type-Set X) (type-Set Y) → leq-cardinality From b8c8bfdfa923c806e30d6ebcae8d2f352ffa12e4 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Thu, 16 Oct 2025 00:43:40 +0200 Subject: [PATCH 08/50] capitalize `Cardinal` --- src/set-theory/cardinalities.lagda.md | 14 +-- ...lemented-inequality-cardinalities.lagda.md | 98 +++++++++---------- .../inequality-cardinalities.lagda.md | 74 +++++++------- .../kuratowski-finite-sets.lagda.md | 2 +- 4 files changed, 94 insertions(+), 94 deletions(-) diff --git a/src/set-theory/cardinalities.lagda.md b/src/set-theory/cardinalities.lagda.md index 47fe3437e8..c6f67d636b 100644 --- a/src/set-theory/cardinalities.lagda.md +++ b/src/set-theory/cardinalities.lagda.md @@ -37,20 +37,20 @@ element in the set truncation of the universe of all sets. ### Cardinals ```agda -cardinal-Set : (l : Level) → Set (lsuc l) -cardinal-Set l = trunc-Set (Set l) +Cardinal-Set : (l : Level) → Set (lsuc l) +Cardinal-Set l = trunc-Set (Set l) -cardinal : (l : Level) → UU (lsuc l) -cardinal l = type-Set (cardinal-Set l) +Cardinal : (l : Level) → UU (lsuc l) +Cardinal l = type-Set (Cardinal-Set l) -is-set-cardinal : {l : Level} → is-set (cardinal l) -is-set-cardinal {l} = is-set-type-Set (cardinal-Set l) +is-set-Cardinal : {l : Level} → is-set (Cardinal l) +is-set-Cardinal {l} = is-set-type-Set (Cardinal-Set l) ``` ### The cardinality of a set ```agda -cardinality : {l : Level} → Set l → cardinal l +cardinality : {l : Level} → Set l → Cardinal l cardinality A = unit-trunc-Set A ``` diff --git a/src/set-theory/complemented-inequality-cardinalities.lagda.md b/src/set-theory/complemented-inequality-cardinalities.lagda.md index 690f8ad6d7..b3845df6e9 100644 --- a/src/set-theory/complemented-inequality-cardinalities.lagda.md +++ b/src/set-theory/complemented-inequality-cardinalities.lagda.md @@ -53,17 +53,17 @@ module _ {l1 l2 : Level} (X : Set l1) where - leq-complemented-prop-cardinal' : cardinal l2 → Prop (l1 ⊔ l2) - leq-complemented-prop-cardinal' = + leq-complemented-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) + leq-complemented-prop-Cardinal' = map-universal-property-trunc-Set ( Prop-Set (l1 ⊔ l2)) ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) - compute-leq-complemented-prop-cardinal' : + compute-leq-complemented-prop-Cardinal' : (Y : Set l2) → - leq-complemented-prop-cardinal' (cardinality Y) ＝ + leq-complemented-prop-Cardinal' (cardinality Y) ＝ mere-decidable-emb-Prop (type-Set X) (type-Set Y) - compute-leq-complemented-prop-cardinal' = + compute-leq-complemented-prop-Cardinal' = triangle-universal-property-trunc-Set ( Prop-Set (l1 ⊔ l2)) ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) @@ -76,23 +76,23 @@ module _ {l1 l2 : Level} where - leq-complemented-prop-cardinal : - cardinal l1 → cardinal l2 → Prop (l1 ⊔ l2) - leq-complemented-prop-cardinal = + leq-complemented-prop-Cardinal : + Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) + leq-complemented-prop-Cardinal = map-universal-property-trunc-Set - ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( leq-complemented-prop-cardinal') - - leq-complemented-cardinal : - cardinal l1 → cardinal l2 → UU (l1 ⊔ l2) - leq-complemented-cardinal X Y = - type-Prop (leq-complemented-prop-cardinal X Y) - - is-prop-leq-complemented-cardinal : - {X : cardinal l1} {Y : cardinal l2} → - is-prop (leq-complemented-cardinal X Y) - is-prop-leq-complemented-cardinal {X} {Y} = - is-prop-type-Prop (leq-complemented-prop-cardinal X Y) + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-complemented-prop-Cardinal') + + leq-complemented-Cardinal : + Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) + leq-complemented-Cardinal X Y = + type-Prop (leq-complemented-prop-Cardinal X Y) + + is-prop-leq-complemented-Cardinal : + {X : Cardinal l1} {Y : Cardinal l2} → + is-prop (leq-complemented-Cardinal X Y) + is-prop-leq-complemented-Cardinal {X} {Y} = + is-prop-type-Prop (leq-complemented-prop-Cardinal X Y) ``` ### Complemented inequality of cardinalities @@ -104,16 +104,16 @@ module _ leq-complemented-prop-cardinality : Prop (l1 ⊔ l2) leq-complemented-prop-cardinality = - leq-complemented-prop-cardinal (cardinality X) (cardinality Y) + leq-complemented-prop-Cardinal (cardinality X) (cardinality Y) leq-complemented-cardinality : UU (l1 ⊔ l2) leq-complemented-cardinality = - leq-complemented-cardinal (cardinality X) (cardinality Y) + leq-complemented-Cardinal (cardinality X) (cardinality Y) is-prop-leq-complemented-cardinality : is-prop leq-complemented-cardinality is-prop-leq-complemented-cardinality = - is-prop-leq-complemented-cardinal + is-prop-leq-complemented-Cardinal compute-leq-complemented-prop-cardinality' : leq-complemented-prop-cardinality ＝ @@ -121,9 +121,9 @@ module _ compute-leq-complemented-prop-cardinality' = ( htpy-eq ( triangle-universal-property-trunc-Set - ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( leq-complemented-prop-cardinal') X) (cardinality Y)) ∙ - ( compute-leq-complemented-prop-cardinal' X Y) + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-complemented-prop-Cardinal') X) (cardinality Y)) ∙ + ( compute-leq-complemented-prop-Cardinal' X Y) compute-leq-complemented-cardinality' : leq-complemented-cardinality ＝ @@ -159,11 +159,11 @@ refl-leq-complemented-cardinality A = unit-leq-complemented-cardinality A A ( refl-mere-decidable-emb (type-Set A)) -refl-leq-complemented-cardinal : - is-reflexive-Large-Relation cardinal leq-complemented-cardinal -refl-leq-complemented-cardinal = +refl-leq-complemented-Cardinal : + is-reflexive-Large-Relation Cardinal leq-complemented-Cardinal +refl-leq-complemented-Cardinal = apply-dependent-universal-property-trunc-Set' - ( λ X → set-Prop (leq-complemented-prop-cardinal X X)) + ( λ X → set-Prop (leq-complemented-prop-Cardinal X X)) ( refl-leq-complemented-cardinality) ``` @@ -185,20 +185,20 @@ module _ ( inv-unit-leq-complemented-cardinality Y Z Y≤Z) ( inv-unit-leq-complemented-cardinality X Y X≤Y)) - transitive-leq-complemented-cardinal : - (X : cardinal l1) (Y : cardinal l2) (Z : cardinal l3) → - leq-complemented-cardinal Y Z → - leq-complemented-cardinal X Y → - leq-complemented-cardinal X Z - transitive-leq-complemented-cardinal = + transitive-leq-complemented-Cardinal : + (X : Cardinal l1) (Y : Cardinal l2) (Z : Cardinal l3) → + leq-complemented-Cardinal Y Z → + leq-complemented-Cardinal X Y → + leq-complemented-Cardinal X Z + transitive-leq-complemented-Cardinal = apply-thrice-dependent-universal-property-trunc-Set' ( λ X Y Z → - ( leq-complemented-cardinal Y Z → - leq-complemented-cardinal X Y → - leq-complemented-cardinal X Z) , + ( leq-complemented-Cardinal Y Z → + leq-complemented-Cardinal X Y → + leq-complemented-Cardinal X Z) , ( is-set-function-type ( is-set-function-type - ( is-set-is-prop is-prop-leq-complemented-cardinal)))) + ( is-set-is-prop is-prop-leq-complemented-Cardinal)))) ( transitive-leq-complemented-cardinality) ``` @@ -207,7 +207,7 @@ module _ ### Assuming the weak limited principle of omniscience, then complemented inequality is antisymmetric Using the previous result and assuming the weak limited principle of -omniscience, we can conclude `leq-complemented-cardinal` is a partial order by +omniscience, we can conclude `leq-complemented-Cardinal` is a partial order by showing that it is antisymmetric. > This remains to be formalized. @@ -229,17 +229,17 @@ module _ ( inv-unit-leq-complemented-cardinality X Y X≤Y) ( inv-unit-leq-complemented-cardinality Y X Y≤X)) - antisymmetric-leq-complemented-cardinal : - (X Y : cardinal l) → - leq-complemented-cardinal X Y → leq-complemented-cardinal Y X → X ＝ Y - antisymmetric-leq-complemented-cardinal = + antisymmetric-leq-complemented-Cardinal : + (X Y : Cardinal l) → + leq-complemented-Cardinal X Y → leq-complemented-Cardinal Y X → X ＝ Y + antisymmetric-leq-complemented-Cardinal = apply-twice-dependent-universal-property-trunc-Set' ( λ X Y → set-Prop ( function-Prop - ( leq-complemented-cardinal X Y) + ( leq-complemented-Cardinal X Y) ( function-Prop - ( leq-complemented-cardinal Y X) - ( Id-Prop (cardinal-Set l) X Y)))) + ( leq-complemented-Cardinal Y X) + ( Id-Prop (Cardinal-Set l) X Y)))) ( antisymmetric-leq-complemented-cardinality) ``` diff --git a/src/set-theory/inequality-cardinalities.lagda.md b/src/set-theory/inequality-cardinalities.lagda.md index 371c58f439..15e858df87 100644 --- a/src/set-theory/inequality-cardinalities.lagda.md +++ b/src/set-theory/inequality-cardinalities.lagda.md @@ -30,7 +30,7 @@ open import set-theory.cardinalities ## Idea We say a [cardinal of sets](set-theory.cardinalities.md) `X` is -{{#concept "less than or equal to" Agda=leq-cardinal}} a cardinal `Y` if any +{{#concept "less than or equal to" Agda=leq-Cardinal}} a Cardinal `Y` if any [set](foundation-core.sets.md) in the isomorphism class represented by `X` embeds into any set in the isomorphism class represented by `Y`. This defines the {{#concept "standard ordering" Disambiguation="on cardinalities of sets"}} @@ -50,17 +50,17 @@ module _ {l1 l2 : Level} (X : Set l1) where - leq-prop-cardinal' : cardinal l2 → Prop (l1 ⊔ l2) - leq-prop-cardinal' = + leq-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) + leq-prop-Cardinal' = map-universal-property-trunc-Set ( Prop-Set (l1 ⊔ l2)) ( λ Y' → mere-emb-Prop (type-Set X) (type-Set Y')) - compute-leq-prop-cardinal' : + compute-leq-prop-Cardinal' : (Y : Set l2) → - leq-prop-cardinal' (cardinality Y) ＝ + leq-prop-Cardinal' (cardinality Y) ＝ mere-emb-Prop (type-Set X) (type-Set Y) - compute-leq-prop-cardinal' = + compute-leq-prop-Cardinal' = triangle-universal-property-trunc-Set ( Prop-Set (l1 ⊔ l2)) ( λ Y' → mere-emb-Prop (type-Set X) (type-Set Y')) @@ -73,18 +73,18 @@ module _ {l1 l2 : Level} where - leq-prop-cardinal : cardinal l1 → cardinal l2 → Prop (l1 ⊔ l2) - leq-prop-cardinal = + leq-prop-Cardinal : Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) + leq-prop-Cardinal = map-universal-property-trunc-Set - ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( leq-prop-cardinal') + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-prop-Cardinal') - leq-cardinal : cardinal l1 → cardinal l2 → UU (l1 ⊔ l2) - leq-cardinal X Y = type-Prop (leq-prop-cardinal X Y) + leq-Cardinal : Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) + leq-Cardinal X Y = type-Prop (leq-prop-Cardinal X Y) - is-prop-leq-cardinal : - {X : cardinal l1} {Y : cardinal l2} → is-prop (leq-cardinal X Y) - is-prop-leq-cardinal {X} {Y} = is-prop-type-Prop (leq-prop-cardinal X Y) + is-prop-leq-Cardinal : + {X : Cardinal l1} {Y : Cardinal l2} → is-prop (leq-Cardinal X Y) + is-prop-leq-Cardinal {X} {Y} = is-prop-type-Prop (leq-prop-Cardinal X Y) ``` ### Inequality of cardinalities @@ -95,22 +95,22 @@ module _ where leq-prop-cardinality : Prop (l1 ⊔ l2) - leq-prop-cardinality = leq-prop-cardinal (cardinality X) (cardinality Y) + leq-prop-cardinality = leq-prop-Cardinal (cardinality X) (cardinality Y) leq-cardinality : UU (l1 ⊔ l2) - leq-cardinality = leq-cardinal (cardinality X) (cardinality Y) + leq-cardinality = leq-Cardinal (cardinality X) (cardinality Y) is-prop-leq-cardinality : is-prop leq-cardinality - is-prop-leq-cardinality = is-prop-leq-cardinal + is-prop-leq-cardinality = is-prop-leq-Cardinal eq-compute-leq-prop-cardinality : leq-prop-cardinality ＝ mere-emb-Prop (type-Set X) (type-Set Y) eq-compute-leq-prop-cardinality = ( htpy-eq ( triangle-universal-property-trunc-Set - ( hom-set-Set (cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( leq-prop-cardinal') X) (cardinality Y)) ∙ - ( compute-leq-prop-cardinal' X Y) + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-prop-Cardinal') X) (cardinality Y)) ∙ + ( compute-leq-prop-Cardinal' X Y) eq-compute-leq-cardinality : leq-cardinality ＝ mere-emb (type-Set X) (type-Set Y) @@ -136,10 +136,10 @@ module _ refl-leq-cardinality : is-reflexive-Large-Relation Set leq-cardinality refl-leq-cardinality A = unit-leq-cardinality A A (refl-mere-emb (type-Set A)) -refl-leq-cardinal : is-reflexive-Large-Relation cardinal leq-cardinal -refl-leq-cardinal = +refl-leq-Cardinal : is-reflexive-Large-Relation Cardinal leq-Cardinal +refl-leq-Cardinal = apply-dependent-universal-property-trunc-Set' - ( λ X → set-Prop (leq-prop-cardinal X X)) + ( λ X → set-Prop (leq-prop-Cardinal X X)) ( refl-leq-cardinality) ``` @@ -159,16 +159,16 @@ module _ ( inv-unit-leq-cardinality Y Z Y≤Z) ( inv-unit-leq-cardinality X Y X≤Y)) - transitive-leq-cardinal : - (X : cardinal l1) (Y : cardinal l2) (Z : cardinal l3) → - leq-cardinal Y Z → leq-cardinal X Y → leq-cardinal X Z - transitive-leq-cardinal = + transitive-leq-Cardinal : + (X : Cardinal l1) (Y : Cardinal l2) (Z : Cardinal l3) → + leq-Cardinal Y Z → leq-Cardinal X Y → leq-Cardinal X Z + transitive-leq-Cardinal = apply-thrice-dependent-universal-property-trunc-Set' ( λ X Y Z → - ( leq-cardinal Y Z → leq-cardinal X Y → leq-cardinal X Z) , + ( leq-Cardinal Y Z → leq-Cardinal X Y → leq-Cardinal X Z) , ( is-set-function-type ( is-set-function-type - ( is-set-is-prop is-prop-leq-cardinal)))) + ( is-set-is-prop is-prop-leq-Cardinal)))) ( transitive-leq-cardinality) ``` @@ -177,7 +177,7 @@ module _ ### Assuming excluded middle, then inequality is antisymmetric Using the previous result and assuming excluded middle, we can conclude -`leq-cardinal` is a partial order by showing that it is antisymmetric. +`leq-Cardinal` is a partial order by showing that it is antisymmetric. ```agda module _ @@ -196,16 +196,16 @@ module _ ( inv-unit-leq-cardinality X Y X≤Y) ( inv-unit-leq-cardinality Y X Y≤X)) - antisymmetric-leq-cardinal : - (X Y : cardinal l) → - leq-cardinal X Y → leq-cardinal Y X → X ＝ Y - antisymmetric-leq-cardinal = + antisymmetric-leq-Cardinal : + (X Y : Cardinal l) → + leq-Cardinal X Y → leq-Cardinal Y X → X ＝ Y + antisymmetric-leq-Cardinal = apply-twice-dependent-universal-property-trunc-Set' ( λ X Y → set-Prop ( function-Prop - ( leq-cardinal X Y) - ( function-Prop (leq-cardinal Y X) (Id-Prop (cardinal-Set l) X Y)))) + ( leq-Cardinal X Y) + ( function-Prop (leq-Cardinal Y X) (Id-Prop (Cardinal-Set l) X Y)))) ( antisymmetric-leq-cardinality) ``` diff --git a/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md b/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md index bd7a99df27..4303035237 100644 --- a/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md +++ b/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md @@ -168,7 +168,7 @@ module _ ```agda cardinality-Kuratowski-Finite-Set : - {l : Level} → Kuratowski-Finite-Set l → cardinal l + {l : Level} → Kuratowski-Finite-Set l → Cardinal l cardinality-Kuratowski-Finite-Set X = cardinality (set-Kuratowski-Finite-Set X) From 1780e5e1c880f2dfd9d15ad1589b49d2faacd601 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Thu, 16 Oct 2025 13:53:42 +0200 Subject: [PATCH 09/50] similarity of cardinalities --- src/set-theory.lagda.md | 1 + src/set-theory/cardinalities.lagda.md | 18 -- .../equality-cardinalities.lagda.md | 294 ++++++++++++++++++ 3 files changed, 295 insertions(+), 18 deletions(-) create mode 100644 src/set-theory/equality-cardinalities.lagda.md diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index 9ec8c17f0f..6d0d018636 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -51,6 +51,7 @@ open import set-theory.cardinalities public open import set-theory.complemented-inequality-cardinalities public open import set-theory.countable-sets public open import set-theory.cumulative-hierarchy public +open import set-theory.equality-cardinalities public open import set-theory.inequality-cardinalities public open import set-theory.infinite-sets public open import set-theory.russells-paradox public diff --git a/src/set-theory/cardinalities.lagda.md b/src/set-theory/cardinalities.lagda.md index c6f67d636b..0c1ab3306c 100644 --- a/src/set-theory/cardinalities.lagda.md +++ b/src/set-theory/cardinalities.lagda.md @@ -54,24 +54,6 @@ cardinality : {l : Level} → Set l → Cardinal l cardinality A = unit-trunc-Set A ``` -## Properties - -### Equality of cardinalities is equivalent to mere equivalence - -```agda -is-effective-cardinality : - {l : Level} (X Y : Set l) → - (cardinality X ＝ cardinality Y) ≃ mere-equiv (type-Set X) (type-Set Y) -is-effective-cardinality X Y = - ( equiv-trunc-Prop (extensionality-Set X Y)) ∘e - ( is-effective-unit-trunc-Set (Set _) X Y) - -eq-mere-equiv-cardinality : - {l : Level} (X Y : Set l) → - mere-equiv (type-Set X) (type-Set Y) → cardinality X ＝ cardinality Y -eq-mere-equiv-cardinality X Y = map-inv-equiv (is-effective-cardinality X Y) -``` - ## External links - [Cardinality](https://en.wikipedia.org/wiki/Cardinality) at Wikipedia diff --git a/src/set-theory/equality-cardinalities.lagda.md b/src/set-theory/equality-cardinalities.lagda.md new file mode 100644 index 0000000000..ac772032af --- /dev/null +++ b/src/set-theory/equality-cardinalities.lagda.md @@ -0,0 +1,294 @@ +# Equality of cardinals of sets + +```agda +module set-theory.equality-cardinalities where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.function-extensionality +open import foundation.functoriality-propositional-truncation +open import foundation.identity-types +open import foundation.large-equivalence-relations +open import foundation.large-similarity-relations +open import foundation.mere-equivalences +open import foundation.propositional-extensionality +open import foundation.propositions +open import foundation.set-truncations +open import foundation.sets +open import foundation.univalence +open import foundation.universe-levels + +open import set-theory.cardinalities +``` + +
+ +## Idea + +Two [cardinals of sets](set-theory.cardinalities.md) `X` and `Y` are +{{#concept "similar" Disambiguation="cardinals" Agda=sim-Cardinal}} if there +[merely](foundation.inhabited-types.md) is an +[equivalence](foundation-core.equivalences.md) between any two representing two +types. + +## Definition + +### The underlying similarity of cardinalities + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + sim-prop-cardinality' : Prop (l1 ⊔ l2) + sim-prop-cardinality' = mere-equiv-Prop (type-Set X) (type-Set Y) + + sim-cardinality' : UU (l1 ⊔ l2) + sim-cardinality' = mere-equiv (type-Set X) (type-Set Y) + + is-prop-sim-sim-cardinality' : is-prop sim-cardinality' + is-prop-sim-sim-cardinality' = is-prop-mere-equiv (type-Set X) (type-Set Y) + +refl-sim-cardinality' : {l : Level} (X : Set l) → sim-cardinality' X X +refl-sim-cardinality' X = refl-mere-equiv (type-Set X) + +transitive-sim-cardinality' : + {l1 l2 l3 : Level} (X : Set l1) (Y : Set l2) (Z : Set l3) → + sim-cardinality' Y Z → sim-cardinality' X Y → sim-cardinality' X Z +transitive-sim-cardinality' X Y Z = + transitive-mere-equiv (type-Set X) (type-Set Y) (type-Set Z) + +symmetric-sim-cardinality' : + {l1 l2 : Level} (X : Set l1) (Y : Set l2) → + sim-cardinality' X Y → sim-cardinality' Y X +symmetric-sim-cardinality' X Y = + symmetric-mere-equiv (type-Set X) (type-Set Y) +``` + +### Similarity of cardinalities with a cardinal + +```agda +module _ + {l1 l2 : Level} (X : Set l1) + where + + sim-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) + sim-prop-Cardinal' = + map-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( sim-prop-cardinality' X) + + sim-Cardinal' : Cardinal l2 → UU (l1 ⊔ l2) + sim-Cardinal' Y = type-Prop (sim-prop-Cardinal' Y) + + is-prop-sim-sim-Cardinal' : (Y : Cardinal l2) → is-prop (sim-Cardinal' Y) + is-prop-sim-sim-Cardinal' Y = is-prop-type-Prop (sim-prop-Cardinal' Y) + + eq-compute-sim-prop-Cardinal' : + (Y : Set l2) → + sim-prop-Cardinal' (cardinality Y) ＝ sim-prop-cardinality' X Y + eq-compute-sim-prop-Cardinal' = + triangle-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( sim-prop-cardinality' X) + + eq-compute-sim-Cardinal' : + (Y : Set l2) → + sim-Cardinal' (cardinality Y) ＝ sim-cardinality' X Y + eq-compute-sim-Cardinal' Y = + ap type-Prop (eq-compute-sim-prop-Cardinal' Y) +``` + +### Similarity of cardinals + +```agda +module _ + {l1 l2 : Level} (X : Cardinal l1) (Y : Cardinal l2) + where + + sim-prop-Cardinal : Prop (l1 ⊔ l2) + sim-prop-Cardinal = + map-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( sim-prop-Cardinal') + ( X) + ( Y) + + sim-Cardinal : UU (l1 ⊔ l2) + sim-Cardinal = type-Prop sim-prop-Cardinal + + is-prop-sim-sim-Cardinal : is-prop sim-Cardinal + is-prop-sim-sim-Cardinal = is-prop-type-Prop sim-prop-Cardinal + + sim-set-Cardinal : Set (l1 ⊔ l2) + sim-set-Cardinal = set-Prop sim-prop-Cardinal +``` + +### Similarity of cardinalities + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + sim-prop-cardinality : Prop (l1 ⊔ l2) + sim-prop-cardinality = sim-prop-Cardinal (cardinality X) (cardinality Y) + + sim-cardinality : UU (l1 ⊔ l2) + sim-cardinality = type-Prop sim-prop-cardinality + + is-prop-sim-sim-cardinality : is-prop sim-cardinality + is-prop-sim-sim-cardinality = is-prop-type-Prop sim-prop-cardinality + + eq-compute-sim-prop-cardinality : + sim-prop-cardinality ＝ mere-equiv-Prop (type-Set X) (type-Set Y) + eq-compute-sim-prop-cardinality = + ( htpy-eq + ( triangle-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( sim-prop-Cardinal') + ( X)) + ( cardinality Y)) ∙ + ( eq-compute-sim-prop-Cardinal' X Y) + + eq-compute-sim-cardinality : + sim-cardinality ＝ mere-equiv (type-Set X) (type-Set Y) + eq-compute-sim-cardinality = + ap type-Prop eq-compute-sim-prop-cardinality + + compute-sim-cardinality : + sim-cardinality ≃ mere-equiv (type-Set X) (type-Set Y) + compute-sim-cardinality = equiv-eq eq-compute-sim-cardinality + + unit-sim-cardinality : + mere-equiv (type-Set X) (type-Set Y) → sim-cardinality + unit-sim-cardinality = map-inv-equiv compute-sim-cardinality + + inv-unit-sim-cardinality : + sim-cardinality → mere-equiv (type-Set X) (type-Set Y) + inv-unit-sim-cardinality = pr1 compute-sim-cardinality +``` + +## Properties + +### Equality of cardinalities is equivalent to mere equivalence + +```agda +is-effective-cardinality : + {l : Level} (X Y : Set l) → + (cardinality X ＝ cardinality Y) ≃ mere-equiv (type-Set X) (type-Set Y) +is-effective-cardinality X Y = + ( equiv-trunc-Prop (extensionality-Set X Y)) ∘e + ( is-effective-unit-trunc-Set (Set _) X Y) + +eq-mere-equiv-cardinality : + {l : Level} (X Y : Set l) → + mere-equiv (type-Set X) (type-Set Y) → cardinality X ＝ cardinality Y +eq-mere-equiv-cardinality X Y = map-inv-equiv (is-effective-cardinality X Y) +``` + +### Similarity of cardinals is reflexive + +```agda +refl-sim-Cardinal : {l : Level} (X : Cardinal l) → sim-Cardinal X X +refl-sim-Cardinal = + apply-dependent-universal-property-trunc-Set' + ( λ X → sim-set-Cardinal X X) + ( λ X → unit-sim-cardinality X X (refl-mere-equiv (type-Set X))) +``` + +### Similarity of cardinals is symmetric + +```agda +abstract + symmetric-sim-Cardinal : + {l1 l2 : Level} + (X : Cardinal l1) (Y : Cardinal l2) → + sim-Cardinal X Y → sim-Cardinal Y X + symmetric-sim-Cardinal = + apply-twice-dependent-universal-property-trunc-Set' + ( λ X Y → hom-set-Set (sim-set-Cardinal X Y) (sim-set-Cardinal Y X)) + ( λ X Y p → + unit-sim-cardinality Y X + ( symmetric-sim-cardinality' X Y (inv-unit-sim-cardinality X Y p))) +``` + +### Similarity of cardinals is transitive + +```agda +abstract + transitive-sim-Cardinal : + {l1 l2 l3 : Level} + (X : Cardinal l1) (Y : Cardinal l2) (Z : Cardinal l3) → + sim-Cardinal Y Z → sim-Cardinal X Y → sim-Cardinal X Z + transitive-sim-Cardinal = + apply-thrice-dependent-universal-property-trunc-Set' + ( λ X Y Z → + hom-set-Set + ( sim-set-Cardinal Y Z) + ( hom-set-Set (sim-set-Cardinal X Y) (sim-set-Cardinal X Z))) + ( λ X Y Z p q → + unit-sim-cardinality X Z + ( transitive-sim-cardinality' X Y Z + ( inv-unit-sim-cardinality Y Z p) + ( inv-unit-sim-cardinality X Y q))) +``` + +### Similarity of cardinals is extensional + +```agda +module _ + {l : Level} + where + + sim-eq-Cardinal : + (X Y : Cardinal l) → X ＝ Y → sim-Cardinal X Y + sim-eq-Cardinal X .X refl = refl-sim-Cardinal X + + abstract + eq-sim-Cardinal : + (X Y : Cardinal l) → sim-Cardinal X Y → X ＝ Y + eq-sim-Cardinal = + apply-twice-dependent-universal-property-trunc-Set' + ( λ X Y → + hom-set-Set + ( sim-set-Cardinal X Y) + ( set-Prop (Id-Prop (Cardinal-Set l) X Y))) + ( λ X Y p → + eq-mere-equiv-cardinality X Y (inv-unit-sim-cardinality X Y p)) + + abstract + antisymmetric-sim-Cardinal : + (X Y : Cardinal l) → sim-Cardinal X Y → sim-Cardinal Y X → X ＝ Y + antisymmetric-sim-Cardinal X Y p _ = eq-sim-Cardinal X Y p +``` + +### Similarity of cardinals is a large similarity relation + +```agda +large-equivalence-relation-Cardinal : Large-Equivalence-Relation (_⊔_) Cardinal +large-equivalence-relation-Cardinal = + λ where + .sim-prop-Large-Equivalence-Relation → sim-prop-Cardinal + .refl-sim-Large-Equivalence-Relation → refl-sim-Cardinal + .symmetric-sim-Large-Equivalence-Relation → symmetric-sim-Cardinal + .transitive-sim-Large-Equivalence-Relation → transitive-sim-Cardinal + +large-similarity-relation-Cardinal : Large-Similarity-Relation (_⊔_) Cardinal +large-similarity-relation-Cardinal = + λ where + .large-equivalence-relation-Large-Similarity-Relation → + large-equivalence-relation-Cardinal + .eq-sim-Large-Similarity-Relation → + eq-sim-Cardinal +``` + +## External links + +- [Cardinality](https://en.wikipedia.org/wiki/Cardinality) at Wikipedia +- [cardinal number](https://ncatlab.org/nlab/show/cardinal+number) at $n$Lab From d30a83338eaa69d5478879f231f071cbedb6aae9 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Thu, 16 Oct 2025 14:02:47 +0200 Subject: [PATCH 10/50] cleanup --- .../equality-cardinalities.lagda.md | 48 ++++++++++++++----- .../inequality-cardinalities.lagda.md | 1 + 2 files changed, 37 insertions(+), 12 deletions(-) diff --git a/src/set-theory/equality-cardinalities.lagda.md b/src/set-theory/equality-cardinalities.lagda.md index ac772032af..da5d178d65 100644 --- a/src/set-theory/equality-cardinalities.lagda.md +++ b/src/set-theory/equality-cardinalities.lagda.md @@ -34,7 +34,8 @@ Two [cardinals of sets](set-theory.cardinalities.md) `X` and `Y` are {{#concept "similar" Disambiguation="cardinals" Agda=sim-Cardinal}} if there [merely](foundation.inhabited-types.md) is an [equivalence](foundation-core.equivalences.md) between any two representing two -types. +types. This characterizes [equality](foundation-core.identity-types.md) of +cardinals. ## Definition @@ -195,16 +196,27 @@ eq-mere-equiv-cardinality X Y = map-inv-equiv (is-effective-cardinality X Y) ### Similarity of cardinals is reflexive ```agda +refl-sim-cardinality : {l : Level} (X : Set l) → sim-cardinality X X +refl-sim-cardinality X = unit-sim-cardinality X X (refl-mere-equiv (type-Set X)) + refl-sim-Cardinal : {l : Level} (X : Cardinal l) → sim-Cardinal X X refl-sim-Cardinal = apply-dependent-universal-property-trunc-Set' ( λ X → sim-set-Cardinal X X) - ( λ X → unit-sim-cardinality X X (refl-mere-equiv (type-Set X))) + ( refl-sim-cardinality) ``` ### Similarity of cardinals is symmetric ```agda +symmetric-sim-cardinality : + {l1 l2 : Level} + (X : Set l1) (Y : Set l2) → + sim-cardinality X Y → sim-cardinality Y X +symmetric-sim-cardinality X Y p = + unit-sim-cardinality Y X + ( symmetric-sim-cardinality' X Y (inv-unit-sim-cardinality X Y p)) + abstract symmetric-sim-Cardinal : {l1 l2 : Level} @@ -213,14 +225,22 @@ abstract symmetric-sim-Cardinal = apply-twice-dependent-universal-property-trunc-Set' ( λ X Y → hom-set-Set (sim-set-Cardinal X Y) (sim-set-Cardinal Y X)) - ( λ X Y p → - unit-sim-cardinality Y X - ( symmetric-sim-cardinality' X Y (inv-unit-sim-cardinality X Y p))) + ( symmetric-sim-cardinality) ``` ### Similarity of cardinals is transitive ```agda +transitive-sim-cardinality : + {l1 l2 l3 : Level} + (X : Set l1) (Y : Set l2) (Z : Set l3) → + sim-cardinality Y Z → sim-cardinality X Y → sim-cardinality X Z +transitive-sim-cardinality X Y Z p q = + unit-sim-cardinality X Z + ( transitive-sim-cardinality' X Y Z + ( inv-unit-sim-cardinality Y Z p) + ( inv-unit-sim-cardinality X Y q)) + abstract transitive-sim-Cardinal : {l1 l2 l3 : Level} @@ -232,11 +252,7 @@ abstract hom-set-Set ( sim-set-Cardinal Y Z) ( hom-set-Set (sim-set-Cardinal X Y) (sim-set-Cardinal X Z))) - ( λ X Y Z p q → - unit-sim-cardinality X Z - ( transitive-sim-cardinality' X Y Z - ( inv-unit-sim-cardinality Y Z p) - ( inv-unit-sim-cardinality X Y q))) + ( transitive-sim-cardinality) ``` ### Similarity of cardinals is extensional @@ -250,6 +266,15 @@ module _ (X Y : Cardinal l) → X ＝ Y → sim-Cardinal X Y sim-eq-Cardinal X .X refl = refl-sim-Cardinal X + sim-eq-cardinality : + (X Y : Set l) → cardinality X ＝ cardinality Y → sim-cardinality X Y + sim-eq-cardinality X Y = sim-eq-Cardinal (cardinality X) (cardinality Y) + + eq-sim-cardinality : + (X Y : Set l) → sim-cardinality X Y → cardinality X ＝ cardinality Y + eq-sim-cardinality X Y p = + eq-mere-equiv-cardinality X Y (inv-unit-sim-cardinality X Y p) + abstract eq-sim-Cardinal : (X Y : Cardinal l) → sim-Cardinal X Y → X ＝ Y @@ -259,8 +284,7 @@ module _ hom-set-Set ( sim-set-Cardinal X Y) ( set-Prop (Id-Prop (Cardinal-Set l) X Y))) - ( λ X Y p → - eq-mere-equiv-cardinality X Y (inv-unit-sim-cardinality X Y p)) + ( eq-sim-cardinality) abstract antisymmetric-sim-Cardinal : diff --git a/src/set-theory/inequality-cardinalities.lagda.md b/src/set-theory/inequality-cardinalities.lagda.md index 15e858df87..08f86f1195 100644 --- a/src/set-theory/inequality-cardinalities.lagda.md +++ b/src/set-theory/inequality-cardinalities.lagda.md @@ -23,6 +23,7 @@ open import foundation.univalence open import foundation.universe-levels open import set-theory.cardinalities +open import set-theory.equality-cardinalities ``` From 2a22efbd5d75ba6a81343655100f22111ad8b5a2 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Thu, 16 Oct 2025 14:08:37 +0200 Subject: [PATCH 11/50] fix import --- src/univalent-combinatorics/kuratowski-finite-sets.lagda.md | 1 + 1 file changed, 1 insertion(+) diff --git a/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md b/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md index 4303035237..e9ab2f70bb 100644 --- a/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md +++ b/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md @@ -23,6 +23,7 @@ open import foundation.surjective-maps open import foundation.universe-levels open import set-theory.cardinalities +open import set-theory.equality-cardinalities open import set-theory.inequality-cardinalities open import univalent-combinatorics.dedekind-finite-sets From aae3c7b7b2a95eaf7832204ca7177810f237ada0 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Thu, 16 Oct 2025 18:44:07 +0200 Subject: [PATCH 12/50] wip dependent sums and products of cardinals --- src/foundation-core/fibers-of-maps.lagda.md | 3 + src/foundation.lagda.md | 1 + .../functoriality-truncation.lagda.md | 24 +++ src/foundation/nonsurjective-maps.lagda.md | 155 ++++++++++++++ src/foundation/projective-types.lagda.md | 10 +- src/set-theory.lagda.md | 7 + .../addition-cardinalities.lagda.md | 112 ++++++++++ .../cardinality-projective-sets.lagda.md | 141 +++++++++++++ .../dependent-products-cardinals.lagda.md | 89 ++++++++ .../dependent-sums-cardinals.lagda.md | 106 ++++++++++ ...emented-boundedness-cardinalities.lagda.md | 129 ++++++++++++ ...lemented-inequality-cardinalities.lagda.md | 198 ++++++++++++++++++ src/set-theory/zero-cardinal.lagda.md | 67 ++++++ 13 files changed, 1037 insertions(+), 5 deletions(-) create mode 100644 src/foundation/nonsurjective-maps.lagda.md create mode 100644 src/set-theory/addition-cardinalities.lagda.md create mode 100644 src/set-theory/cardinality-projective-sets.lagda.md create mode 100644 src/set-theory/dependent-products-cardinals.lagda.md create mode 100644 src/set-theory/dependent-sums-cardinals.lagda.md create mode 100644 src/set-theory/strict-complemented-boundedness-cardinalities.lagda.md create mode 100644 src/set-theory/strict-complemented-inequality-cardinalities.lagda.md create mode 100644 src/set-theory/zero-cardinal.lagda.md diff --git a/src/foundation-core/fibers-of-maps.lagda.md b/src/foundation-core/fibers-of-maps.lagda.md index 178e7d1397..f2fad80631 100644 --- a/src/foundation-core/fibers-of-maps.lagda.md +++ b/src/foundation-core/fibers-of-maps.lagda.md @@ -386,6 +386,9 @@ module _ pr1 (map-inv-compute-fiber-comp t) = pr1 (pr2 t) pr2 (map-inv-compute-fiber-comp t) = ap g (pr2 (pr2 t)) ∙ pr2 (pr1 t) + inclusion-fiber-comp : (t : fiber g x) → fiber h (pr1 t) → fiber (g ∘ h) x + inclusion-fiber-comp t s = map-inv-compute-fiber-comp (t , s) + is-section-map-inv-compute-fiber-comp : is-section map-compute-fiber-comp map-inv-compute-fiber-comp is-section-map-inv-compute-fiber-comp ((.(h a) , refl) , (a , refl)) = refl diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md index 3023e76790..2f5d9e4a47 100644 --- a/src/foundation.lagda.md +++ b/src/foundation.lagda.md @@ -333,6 +333,7 @@ open import foundation.negated-equality public open import foundation.negation public open import foundation.noncontractible-types public open import foundation.noninjective-maps public +open import foundation.nonsurjective-maps public open import foundation.null-homotopic-maps public open import foundation.operations-span-diagrams public open import foundation.operations-spans public diff --git a/src/foundation/functoriality-truncation.lagda.md b/src/foundation/functoriality-truncation.lagda.md index 92e1760616..1a1ca29dcf 100644 --- a/src/foundation/functoriality-truncation.lagda.md +++ b/src/foundation/functoriality-truncation.lagda.md @@ -9,6 +9,7 @@ module foundation.functoriality-truncation where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types +open import foundation.evaluation-functions open import foundation.function-extensionality open import foundation.truncations open import foundation.universe-levels @@ -180,3 +181,26 @@ module _ ( inclusion-retract R) ( retraction-retract R) ``` + +### The truncation of the evaluation map + +```agda +module _ + {l1 l2 : Level} (k : 𝕋) + where + + map-trunc-ev : + {X : UU l1} {Y : X → UU l2} → + (x : X) → type-trunc k ((x : X) → Y x) → type-trunc k (Y x) + map-trunc-ev x = map-trunc k (ev x) + + map-distributive-trunc-Π : + {X : UU l1} (Y : X → UU l2) → + type-trunc k ((x : X) → Y x) → (x : X) → type-trunc k (Y x) + map-distributive-trunc-Π Y f x = map-trunc-ev x f + + map-distributive-trunc-function-type : + (X : UU l1) (Y : UU l2) → type-trunc k (X → Y) → X → type-trunc k Y + map-distributive-trunc-function-type X Y = + map-distributive-trunc-Π (λ _ → Y) +``` diff --git a/src/foundation/nonsurjective-maps.lagda.md b/src/foundation/nonsurjective-maps.lagda.md new file mode 100644 index 0000000000..08b5601af6 --- /dev/null +++ b/src/foundation/nonsurjective-maps.lagda.md @@ -0,0 +1,155 @@ +# Nonsurjective maps + +```agda +module foundation.nonsurjective-maps where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.connected-maps +open import foundation.contractible-types +open import foundation.coproduct-types +open import foundation.decidable-maps +open import foundation.dependent-pair-types +open import foundation.diagonal-maps-of-types +open import foundation.disjunction +open import foundation.embeddings +open import foundation.empty-types +open import foundation.equality-cartesian-product-types +open import foundation.existential-quantification +open import foundation.functoriality-cartesian-product-types +open import foundation.functoriality-coproduct-types +open import foundation.functoriality-propositional-truncation +open import foundation.fundamental-theorem-of-identity-types +open import foundation.homotopy-induction +open import foundation.identity-types +open import foundation.inhabited-types +open import foundation.negation +open import foundation.postcomposition-dependent-functions +open import foundation.propositional-truncations +open import foundation.split-surjective-maps +open import foundation.structure-identity-principle +open import foundation.subtype-identity-principle +open import foundation.truncated-types +open import foundation.univalence +open import foundation.universal-property-family-of-fibers-of-maps +open import foundation.universal-property-propositional-truncation +open import foundation.universe-levels + +open import foundation-core.cartesian-product-types +open import foundation-core.constant-maps +open import foundation-core.contractible-maps +open import foundation-core.equivalences +open import foundation-core.fibers-of-maps +open import foundation-core.function-types +open import foundation-core.functoriality-dependent-function-types +open import foundation-core.homotopies +open import foundation-core.precomposition-dependent-functions +open import foundation-core.propositional-maps +open import foundation-core.propositions +open import foundation-core.retracts-of-types +open import foundation-core.sections +open import foundation-core.sets +open import foundation-core.subtypes +open import foundation-core.torsorial-type-families +open import foundation-core.truncated-maps +open import foundation-core.truncation-levels + +open import logic.propositionally-decidable-maps + +open import orthogonal-factorization-systems.extensions-maps +``` + +
+ +## Idea + +A map `f : A → B` is {{#concept "nonsurjective"}} if there +[exists](foundation.existential-quantification.md) a +[fiber](foundation-core.fibers-of-maps.md) that is [not](foundation.negation.md) +[inhabited](foundation.inhabited-types.md). + +## Definitions + +### The nonimage + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) + where + + nonim : UU (l1 ⊔ l2) + nonim = Σ B (λ y → ¬ fiber f y) +``` + +### Nonsurjectivity + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) + where + + is-nonsurjective : UU (l1 ⊔ l2) + is-nonsurjective = ║ nonim f ║₋₁ + + is-prop-is-nonsurjective : is-prop is-nonsurjective + is-prop-is-nonsurjective = is-prop-exists-structure B (λ y → ¬ fiber f y) + + is-nonsurjective-Prop : Prop (l1 ⊔ l2) + is-nonsurjective-Prop = exists-structure-Prop B (λ y → ¬ fiber f y) +``` + +## Properties + +### If `g ∘ f` is nonsurjective and `g` is decidable then `g` or `f` is nonsurjective + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {f : A → B} {g : B → C} + where + + decide-nonim-comp : + is-decidable-map g → + nonim (g ∘ f) → nonim f + nonim g + decide-nonim-comp H (c , np) = + map-coproduct + ( λ (y , q) → y , map-neg (inclusion-fiber-comp g f c (y , q)) np) + ( c ,_) + ( H c) + + decide-is-nonsurjective-nonim-comp' : + is-decidable-map g → + nonim (g ∘ f) → is-nonsurjective f + is-nonsurjective g + decide-is-nonsurjective-nonim-comp' H (c , np) = + map-coproduct + ( λ (y , q) → + unit-trunc-Prop (y , map-neg (inclusion-fiber-comp g f c (y , q)) np)) + (λ p → unit-trunc-Prop (c , p)) + ( H c) + + is-nonsurjective-is-nonsurjective-comp' : + is-decidable-map g → + is-nonsurjective (g ∘ f) → + disjunction-type (is-nonsurjective f) (is-nonsurjective g) + is-nonsurjective-is-nonsurjective-comp' H = + map-trunc-Prop (decide-is-nonsurjective-nonim-comp' H) + + decide-is-nonsurjective-nonim-comp : + is-inhabited-or-empty-map g → + nonim (g ∘ f) → is-nonsurjective f + is-nonsurjective g + decide-is-nonsurjective-nonim-comp H (c , np) = + map-coproduct + ( map-trunc-Prop + ( λ (y , q) → y , map-neg (inclusion-fiber-comp g f c (y , q)) np)) + ( λ p → unit-trunc-Prop (c , p)) + ( H c) + + is-nonsurjective-is-nonsurjective-comp : + is-inhabited-or-empty-map g → + is-nonsurjective (g ∘ f) → + disjunction-type (is-nonsurjective f) (is-nonsurjective g) + is-nonsurjective-is-nonsurjective-comp H = + map-trunc-Prop (decide-is-nonsurjective-nonim-comp H) +``` diff --git a/src/foundation/projective-types.lagda.md b/src/foundation/projective-types.lagda.md index 07ee95a086..7e0578ef97 100644 --- a/src/foundation/projective-types.lagda.md +++ b/src/foundation/projective-types.lagda.md @@ -26,17 +26,17 @@ open import foundation-core.truncated-types A type `X` is said to be {{#concept "set-projective" Agda=is-set-projective}} if for every [surjective map](foundation.surjective-maps.md) `f : A ↠ B` onto a -[set](foundation-core.sets.md) `B` the [postcomposition -function[(foundation-core.postcomposition-functions.md) +[set](foundation-core.sets.md) `B` the +[postcomposition function](foundation-core.postcomposition-functions.md) ```text (X → A) → (X → B) ``` is surjective. This is [equivalent](foundation.logical-equivalences.md) to the -condition that for every [equivalence -relation[(foundation-core.equivalence-relations.md) `R` on a type `A` the -natural map +condition that for every +[equivalence relation](foundation-core.equivalence-relations.md) `R` on a type +`A` the natural map ```text (X → A)/~ → (X → A/R) diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index 6d0d018636..8e4bf784cb 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -44,18 +44,25 @@ theory, and [Russell's paradox](set-theory.russells-paradox.md). ```agda module set-theory where +open import set-theory.addition-cardinalities public open import set-theory.baire-space public open import set-theory.cantor-space public open import set-theory.cantors-diagonal-argument public open import set-theory.cardinalities public +open import set-theory.cardinality-projective-sets public open import set-theory.complemented-inequality-cardinalities public open import set-theory.countable-sets public open import set-theory.cumulative-hierarchy public +open import set-theory.dependent-products-cardinals public +open import set-theory.dependent-sums-cardinals public open import set-theory.equality-cardinalities public open import set-theory.inequality-cardinalities public open import set-theory.infinite-sets public open import set-theory.russells-paradox public +open import set-theory.strict-complemented-boundedness-cardinalities public +open import set-theory.strict-complemented-inequality-cardinalities public open import set-theory.uncountable-sets public +open import set-theory.zero-cardinal public ``` ## References diff --git a/src/set-theory/addition-cardinalities.lagda.md b/src/set-theory/addition-cardinalities.lagda.md new file mode 100644 index 0000000000..8d0d8db340 --- /dev/null +++ b/src/set-theory/addition-cardinalities.lagda.md @@ -0,0 +1,112 @@ +# Coproducts of cardinalities + +```agda +module set-theory.addition-cardinalities where +``` + +
Imports + +```agda +open import elementary-number-theory.natural-numbers + +open import foundation.action-on-identifications-functions +open import foundation.coproduct-types +open import foundation.decidable-propositions +open import foundation.decidable-subtypes +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.discrete-types +open import foundation.double-negation +open import foundation.equality-coproduct-types +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.functoriality-set-truncation +open import foundation.identity-types +open import foundation.isolated-elements +open import foundation.logical-equivalences +open import foundation.negated-equality +open import foundation.negation +open import foundation.powersets +open import foundation.propositional-truncations +open import foundation.sections +open import foundation.set-truncations +open import foundation.sets +open import foundation.surjective-maps +open import foundation.universe-levels + +open import foundation-core.empty-types +open import foundation-core.fibers-of-maps +open import foundation-core.propositions + +open import logic.propositionally-decidable-types + +open import set-theory.cardinalities +open import set-theory.cardinality-projective-sets +open import set-theory.equality-cardinalities +open import set-theory.inequality-cardinalities +open import set-theory.zero-cardinal +``` + +
+ +## Idea + +Given two [cardinals](set-theory.cardinalities.lagda.md) `A` and `B`, we may +form their {{#concept "sum" Disambiguation="cardinal" Agda=add-Cardinal}}, or +**coproduct** cardinal, `A + B`. It is defined on isomorphism classes as +[coproducts](foundation-core.coproduct-types.md) of +[sets](foundation-core.sets.md). + +## Definitions + +### The underlying addition of cardinalities + +```agda +module _ + {l1 l2 : Level} + where + + add-cardinality' : Set l1 → Set l2 → Cardinal (l1 ⊔ l2) + add-cardinality' X Y = cardinality (coproduct-Set X Y) +``` + +### Addition of cardinals + +```agda +module _ + {l1 l2 : Level} + where + + add-Cardinal : Cardinal l1 → Cardinal l2 → Cardinal (l1 ⊔ l2) + add-Cardinal = binary-map-trunc-Set coproduct-Set +``` + +### Addition of cardinalities + +```agda +module _ + {l1 l2 : Level} + where + + add-cardinality : Set l1 → Set l2 → Cardinal (l1 ⊔ l2) + add-cardinality X Y = add-Cardinal (cardinality X) (cardinality Y) + + compute-add-cardinality : + (X : Set l1) (Y : Set l2) → + add-Cardinal (cardinality X) (cardinality Y) ＝ add-cardinality' X Y + compute-add-cardinality = {! !} +``` + +## Properties + +### The zero cardinal is the additive unit + +```agda +module _ + {l : Level} + where + + left-unit-law-add-Cardinal : + {l : Level} (X : Cardinal l) → sim-Cardinal (add-Cardinal zero-Cardinal X) X + left-unit-law-add-Cardinal X = {! !} +``` diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md new file mode 100644 index 0000000000..0a545534d7 --- /dev/null +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -0,0 +1,141 @@ +# Cardinality-projective sets + +```agda +module set-theory.cardinality-projective-sets where +``` + +
Imports + +```agda +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.function-types +open import foundation.functoriality-truncation +open import foundation.identity-types +open import foundation.mere-equivalences +open import foundation.sections +open import foundation.set-truncations +open import foundation.sets +open import foundation.truncation-levels +open import foundation.universe-levels + +open import set-theory.cardinalities +``` + +
+ +## Idea + +For every type $X$ there is a map $║ X → Set ║₀ → (X → Cardinal)$. We call sets +$X$ for which this map has a section +{{#concept "cardinality-projective" Disamibguation="sets"}}. Over such types we +may form [dependent sum](set-theory.dependent-sums-cardinalities.md) and +[dependent product](set-theory.dependent-products-cardinalities.md) cardinals. + +Note that classically, the universe of sets is itself a set, and so trivially +$║ X → Set ║₀ → (X → ║ Set ║₀)$ is an equivalence. However, with univalence, +meaning that $║ Set ║₀$ is itself the type of cardinalities, this is no longer +the case. + +**Terminology.** This is nonstandard terminology and is subject to change. + +## Definition + +```agda +is-cardinality-projective-set-Level : + {l1 : Level} (l2 : Level) (X : Set l1) → UU (l1 ⊔ lsuc l2) +is-cardinality-projective-set-Level l2 X = + section (map-distributive-trunc-function-type zero-𝕋 (type-Set X) (Set l2)) +``` + +### The universe of cardinality-projective sets at a universe level + +```agda +Cardinality-Projective-Set : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) +Cardinality-Projective-Set l1 l2 = + Σ (Set l1) (is-cardinality-projective-set-Level l2) + +module _ + {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + where + + set-Cardinality-Projective-Set : Set l1 + set-Cardinality-Projective-Set = pr1 X + + type-Cardinality-Projective-Set : UU l1 + type-Cardinality-Projective-Set = type-Set set-Cardinality-Projective-Set + + is-set-type-Cardinality-Projective-Set : + is-set type-Cardinality-Projective-Set + is-set-type-Cardinality-Projective-Set = + is-set-type-Set set-Cardinality-Projective-Set + + is-cardinality-projective-Cardinality-Projective-Set : + is-cardinality-projective-set-Level l2 set-Cardinality-Projective-Set + is-cardinality-projective-Cardinality-Projective-Set = pr2 X + + map-section-Cardinality-Projective-Set : + ( type-Cardinality-Projective-Set → Cardinal l2) → + ║ (type-Cardinality-Projective-Set → Set l2) ║₀ + map-section-Cardinality-Projective-Set = + map-section + ( map-distributive-trunc-function-type zero-𝕋 + ( type-Cardinality-Projective-Set) + ( Set l2)) + ( is-cardinality-projective-Cardinality-Projective-Set) + + is-section-map-section-Cardinality-Projective-Set : + is-section + ( map-distributive-trunc-function-type zero-𝕋 + ( type-Cardinality-Projective-Set) + ( Set l2)) + ( map-section-Cardinality-Projective-Set) + is-section-map-section-Cardinality-Projective-Set = + is-section-map-section + ( map-distributive-trunc-function-type zero-𝕋 + ( type-Cardinality-Projective-Set) + ( Set l2)) + ( is-cardinality-projective-Cardinality-Projective-Set) +``` + +```text + (X → Set) ↖ + | \ \ + | \ \ + ∨ ∨ \ + (X → Card) -> ║ X → Set ║₀ + + - ∘ η + (X → Set) + | \ + η | \ - ∘ η + ∨ ↘ + ║X → Set║₀ ---> (X → Card) + <--- + 𝓈 +``` + +Naturality says + +```text + ev x + (X → Set) -----> Set + | | + η | | η + ∨ ║ev x║₀ ∨ + ║X → Set║₀ ----> Card +``` + +```agda + compute-map-section-at-cardinality-Cardinality-Projective-Set : + (K : type-Cardinality-Projective-Set → Set l2) → + map-section-Cardinality-Projective-Set (cardinality ∘ K) ＝ unit-trunc-Set K + compute-map-section-at-cardinality-Cardinality-Projective-Set = TODO + where postulate TODO : _ +``` + +## See also + +- In + [Distributivity of set truncation over finite products](univalent-combinatorics.distributivity-of-set-truncation-over-finite-products.md) + it is demonstrated that finite types are cardinality-projective. diff --git a/src/set-theory/dependent-products-cardinals.lagda.md b/src/set-theory/dependent-products-cardinals.lagda.md new file mode 100644 index 0000000000..fc35200aee --- /dev/null +++ b/src/set-theory/dependent-products-cardinals.lagda.md @@ -0,0 +1,89 @@ +# Dependent products of cardinals + +```agda +module set-theory.dependent-products-cardinals where +``` + +
Imports + +```agda +open import elementary-number-theory.natural-numbers + +open import foundation.action-on-identifications-functions +open import foundation.coproduct-types +open import foundation.decidable-propositions +open import foundation.decidable-subtypes +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.discrete-types +open import foundation.double-negation +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.functoriality-set-truncation +open import foundation.identity-types +open import foundation.isolated-elements +open import foundation.logical-equivalences +open import foundation.negated-equality +open import foundation.negation +open import foundation.powersets +open import foundation.propositional-truncations +open import foundation.sections +open import foundation.set-truncations +open import foundation.sets +open import foundation.surjective-maps +open import foundation.universe-levels + +open import foundation-core.empty-types +open import foundation-core.fibers-of-maps +open import foundation-core.propositions + +open import logic.propositionally-decidable-types + +open import set-theory.cardinalities +open import set-theory.cardinality-projective-sets +``` + +
+ +## Idea + +Given a family of cardinals $κ : I → \mathrm{Cardinal}$ over a +[cardinality-projective set](set-theory.cardinality-projective-sets.md) $I$, +then we may define the {{#concept "dependent product cardinal" Agda=Π-Cardinal}} +$Π_{i∈I}κᵢ$, as the cardinality of the +[dependent product](foundation.dependent-function-types.md) of any family of +representing sets $Kᵢ$. + +## Definitions + +```agda +module _ + {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + where + + Π-Cardinal : + (type-Cardinality-Projective-Set X → Cardinal l2) → Cardinal (l1 ⊔ l2) + Π-Cardinal Y = + map-trunc-Set + ( Π-Set (set-Cardinality-Projective-Set X)) + ( map-section-Cardinality-Projective-Set X Y) + + compute-Π-Cardinal : + (K : type-Cardinality-Projective-Set X → Set l2) → + Π-Cardinal (cardinality ∘ K) ＝ + cardinality (Π-Set (set-Cardinality-Projective-Set X) K) + compute-Π-Cardinal K = + equational-reasoning + map-trunc-Set + ( Π-Set (set-Cardinality-Projective-Set X)) + ( map-section-Cardinality-Projective-Set X (cardinality ∘ K)) + ＝ map-trunc-Set (Π-Set (pr1 X)) (unit-trunc-Set K) + by + ap + ( map-trunc-Set (Π-Set (set-Cardinality-Projective-Set X))) + ( compute-map-section-at-cardinality-Cardinality-Projective-Set X K) + ＝ cardinality + ( Π-Set (set-Cardinality-Projective-Set X) K) + by + naturality-unit-trunc-Set (Π-Set (set-Cardinality-Projective-Set X)) K +``` diff --git a/src/set-theory/dependent-sums-cardinals.lagda.md b/src/set-theory/dependent-sums-cardinals.lagda.md new file mode 100644 index 0000000000..0da9746fd5 --- /dev/null +++ b/src/set-theory/dependent-sums-cardinals.lagda.md @@ -0,0 +1,106 @@ +# Dependent sums of cardinals + +```agda +module set-theory.dependent-sums-cardinals where +``` + +
Imports + +```agda +open import elementary-number-theory.natural-numbers + +open import foundation.action-on-identifications-functions +open import foundation.coproduct-types +open import foundation.decidable-propositions +open import foundation.decidable-subtypes +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.discrete-types +open import foundation.double-negation +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.functoriality-set-truncation +open import foundation.identity-types +open import foundation.isolated-elements +open import foundation.logical-equivalences +open import foundation.negated-equality +open import foundation.negation +open import foundation.powersets +open import foundation.propositional-truncations +open import foundation.sections +open import foundation.set-truncations +open import foundation.sets +open import foundation.surjective-maps +open import foundation.universe-levels + +open import foundation-core.empty-types +open import foundation-core.fibers-of-maps +open import foundation-core.propositions + +open import logic.propositionally-decidable-types + +open import set-theory.cardinalities +open import set-theory.cardinality-projective-sets +open import set-theory.inequality-cardinalities +``` + +
+ +## Idea + +Given a family of cardinals $κ : I → \mathrm{Cardinal}$ over a +[cardinality-projective set](set-theory.cardinality-projective-sets.md) $I$, +then we may define the {{#concept "dependent sum cardinal"}} $Σ_{i∈I}κᵢ$, as the +cardinality of the [dependent sum](foundation.dependent-pair-types.md) of any +family of representing sets $Kᵢ$. + +## Definitions + +```agda +module _ + {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + where + + Σ-Cardinal : + (type-Cardinality-Projective-Set X → Cardinal l2) → Cardinal (l1 ⊔ l2) + Σ-Cardinal Y = + map-trunc-Set + ( Σ-Set (set-Cardinality-Projective-Set X)) + ( map-section-Cardinality-Projective-Set X Y) + + compute-Σ-Cardinal : + (K : type-Cardinality-Projective-Set X → Set l2) → + Σ-Cardinal (cardinality ∘ K) ＝ + cardinality (Σ-Set (set-Cardinality-Projective-Set X) K) + compute-Σ-Cardinal K = + equational-reasoning + map-trunc-Set + ( Σ-Set (set-Cardinality-Projective-Set X)) + ( map-section-Cardinality-Projective-Set X (cardinality ∘ K)) + ＝ map-trunc-Set (Σ-Set (pr1 X)) (unit-trunc-Set K) + by + ap + ( map-trunc-Set (Σ-Set (set-Cardinality-Projective-Set X))) + ( compute-map-section-at-cardinality-Cardinality-Projective-Set X K) + ＝ cardinality + ( Σ-Set (set-Cardinality-Projective-Set X) K) + by + naturality-unit-trunc-Set (Σ-Set (set-Cardinality-Projective-Set X)) K +``` + +## Properties + +### Inequality is preserved under dependent sums + +```agda +module _ + {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + where + + leq-Σ-Cardinal : + (K P : type-Cardinality-Projective-Set X → Cardinal l2) → + ((i : type-Cardinality-Projective-Set X) → leq-Cardinal (K i) (P i)) → + leq-Cardinal (Σ-Cardinal X K) (Σ-Cardinal X P) + leq-Σ-Cardinal K P f = {! !} + -- proof somehow proceeds by using that since `X` is cardinality-projective, it suffices to show this for families of sets, and then it's just an easy fact of dependent sums. +``` diff --git a/src/set-theory/strict-complemented-boundedness-cardinalities.lagda.md b/src/set-theory/strict-complemented-boundedness-cardinalities.lagda.md new file mode 100644 index 0000000000..d8e0d7bd17 --- /dev/null +++ b/src/set-theory/strict-complemented-boundedness-cardinalities.lagda.md @@ -0,0 +1,129 @@ +# Strict complemented boundedness on cardinals + +```agda +module set-theory.strict-complemented-boundedness-cardinalities where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.decidable-embeddings +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.identity-types +open import foundation.large-binary-relations +open import foundation.law-of-excluded-middle +open import foundation.mere-embeddings +open import foundation.nonsurjective-maps +open import foundation.propositional-extensionality +open import foundation.propositions +open import foundation.set-truncations +open import foundation.sets +open import foundation.univalence +open import foundation.universe-levels + +open import set-theory.cardinalities +open import set-theory.inequality-cardinalities +``` + +
+ +## Idea + +We say a [cardinal of sets](set-theory.cardinalities.md) `X` is +{{#concept "strictly complemented bounded" Agda=strictly-complemented-bounded-Cardinal}} +by a cardinal `Y` if every decidable embedding of a +[set](foundation-core.sets.md) in the isomorphism class represented by `X` into +a set in the isomorphism class represented by `Y` is +[nonsurjective](foundation.nonsurjective-maps.md) in the sense that there is an +element in `Y` it does not hit. + +## Definition + +### Strict complemented boundedness of cardinality of a set with respect to a set + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + strictly-complemented-bounded-cardinality : UU (l1 ⊔ l2) + strictly-complemented-bounded-cardinality = + (f : type-Set X ↪ᵈ type-Set Y) → is-nonsurjective (map-decidable-emb f) + + strictly-complemented-bounded-prop-cardinality : Prop (l1 ⊔ l2) + strictly-complemented-bounded-prop-cardinality = + Π-Prop + ( type-Set X ↪ᵈ type-Set Y) + ( is-nonsurjective-Prop ∘ map-decidable-emb) + + is-prop-strictly-complemented-bounded-cardinality : + is-prop strictly-complemented-bounded-cardinality + is-prop-strictly-complemented-bounded-cardinality = + is-prop-type-Prop strictly-complemented-bounded-prop-cardinality +``` + +### Strict complemented boundedness of a cardinal with respect to a set + +```agda +module _ + {l1 l2 : Level} (X : Set l1) + where + + strictly-complemented-bounded-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) + strictly-complemented-bounded-prop-Cardinal' = + map-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( strictly-complemented-bounded-prop-cardinality X) + + strictly-complemented-bounded-Cardinal' : Cardinal l2 → UU (l1 ⊔ l2) + strictly-complemented-bounded-Cardinal' Y = + type-Prop (strictly-complemented-bounded-prop-Cardinal' Y) + + compute-strictly-complemented-bounded-prop-Cardinal' : + (Y : Set l2) → + strictly-complemented-bounded-prop-Cardinal' (cardinality Y) ＝ + strictly-complemented-bounded-prop-cardinality X Y + compute-strictly-complemented-bounded-prop-Cardinal' = + triangle-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( strictly-complemented-bounded-prop-cardinality X) +``` + +### Strict complemented boundedness of cardinals + +```agda +module _ + {l1 l2 : Level} + where + + strictly-complemented-bounded-prop-Cardinal : + Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) + strictly-complemented-bounded-prop-Cardinal = + map-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( strictly-complemented-bounded-prop-Cardinal') + + strictly-complemented-bounded-Cardinal : + Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) + strictly-complemented-bounded-Cardinal X Y = + type-Prop (strictly-complemented-bounded-prop-Cardinal X Y) + + compute-strictly-complemented-bounded-prop-Cardinal : + (X : Set l1) (Y : Set l2) → + strictly-complemented-bounded-prop-Cardinal + ( cardinality X) + ( cardinality Y) ＝ + strictly-complemented-bounded-prop-cardinality X Y + compute-strictly-complemented-bounded-prop-Cardinal X Y = + ( htpy-eq + ( triangle-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( strictly-complemented-bounded-prop-Cardinal') + ( X)) + ( cardinality Y)) ∙ + ( compute-strictly-complemented-bounded-prop-Cardinal' X Y) +``` diff --git a/src/set-theory/strict-complemented-inequality-cardinalities.lagda.md b/src/set-theory/strict-complemented-inequality-cardinalities.lagda.md new file mode 100644 index 0000000000..117261b853 --- /dev/null +++ b/src/set-theory/strict-complemented-inequality-cardinalities.lagda.md @@ -0,0 +1,198 @@ +# Strict complemented inequality on cardinals + +```agda +module set-theory.strict-complemented-inequality-cardinalities where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.conjunction +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.function-extensionality +open import foundation.identity-types +open import foundation.large-binary-relations +open import foundation.law-of-excluded-middle +open import foundation.mere-embeddings +open import foundation.nonsurjective-maps +open import foundation.propositional-extensionality +open import foundation.propositions +open import foundation.set-truncations +open import foundation.sets +open import foundation.univalence +open import foundation.universe-levels + +open import set-theory.cardinalities +open import set-theory.complemented-inequality-cardinalities +open import set-theory.inequality-cardinalities +open import set-theory.strict-complemented-boundedness-cardinalities +``` + +
+ +## Idea + +We may say a [cardinal of sets](set-theory.cardinalities.md) `X` is +{{#concept "less than" Agda=le-complemented-Cardinal}} a cardinal `Y` if any +[set](foundation-core.sets.md) in the isomorphism class represented by `X` +embeds as a complemented subset into any set in the isomorphism class +represented by `Y`, and every decidable embedding `X ↪ Y` is +[nonsurjective](foundation.nonsurjective-maps.md) in the sense that there is an +element in `Y` they do not hit. This is a positive way of saying that `X` and +`Y` are different. This defines a +{{#concept "strict ordering" Disambiguation="on cardinalities of sets"}} on +cardinalities of sets. + +## Definition + +### The underlying strict complemented inequality of cardinalitiess + +```agda +module _ + {l1 l2 : Level} + where + + le-complemented-prop-cardinality' : Set l1 → Set l2 → Prop (l1 ⊔ l2) + le-complemented-prop-cardinality' X Y = + ( leq-complemented-prop-cardinality X Y) ∧ + ( strictly-complemented-bounded-prop-cardinality X Y) + + le-complemented-cardinality' : Set l1 → Set l2 → UU (l1 ⊔ l2) + le-complemented-cardinality' X Y = + type-Prop (le-complemented-prop-cardinality' X Y) + + is-prop-le-complemented-cardinality' : + (X : Set l1) (Y : Set l2) → is-prop (le-complemented-cardinality' X Y) + is-prop-le-complemented-cardinality' X Y = + is-prop-type-Prop (le-complemented-prop-cardinality' X Y) +``` + +### Strict complemented inequality of cardinals + +```agda +module _ + {l1 l2 : Level} + where + + le-complemented-prop-Cardinal : Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) + le-complemented-prop-Cardinal X Y = + ( leq-complemented-prop-Cardinal X Y) ∧ + ( strictly-complemented-bounded-prop-Cardinal X Y) + + le-complemented-Cardinal : Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) + le-complemented-Cardinal X Y = type-Prop (le-complemented-prop-Cardinal X Y) + + is-prop-le-complemented-Cardinal : + {X : Cardinal l1} {Y : Cardinal l2} → is-prop (le-complemented-Cardinal X Y) + is-prop-le-complemented-Cardinal {X} {Y} = + is-prop-type-Prop (le-complemented-prop-Cardinal X Y) +``` + +### Strict complemented inequality of cardinalitiess + +```agda +module _ + {l1 l2 : Level} + where + + le-complemented-prop-cardinality : Set l1 → Set l2 → Prop (l1 ⊔ l2) + le-complemented-prop-cardinality X Y = + ( leq-complemented-prop-cardinality X Y) ∧ + ( strictly-complemented-bounded-prop-cardinality X Y) + + le-complemented-cardinality' : Set l1 → Set l2 → UU (l1 ⊔ l2) + le-complemented-cardinality' X Y = + type-Prop (le-complemented-prop-cardinality' X Y) + + is-prop-le-complemented-cardinality' : + (X : Set l1) (Y : Set l2) → is-prop (le-complemented-cardinality' X Y) + is-prop-le-complemented-cardinality' X Y = + is-prop-type-Prop (le-complemented-prop-cardinality' X Y) +``` + +### Inequality on cardinals is reflexive + +```text +refl-le-complemented-cardinality : is-reflexive-Large-Relation Set le-complemented-cardinality +refl-le-complemented-cardinality A = unit-le-complemented-cardinality A A (refl-mere-emb (type-Set A)) + +refl-le-complemented-Cardinal : is-reflexive-Large-Relation Cardinal le-complemented-Cardinal +refl-le-complemented-Cardinal = + apply-dependent-universal-property-trunc-Set' + ( λ X → set-Prop (le-complemented-prop-Cardinal X X)) + ( refl-le-complemented-cardinality) +``` + +### Inequality on cardinals is transitive + +```text +module _ + {l1 l2 l3 : Level} + where + + transitive-le-complemented-cardinality : + (X : Set l1) (Y : Set l2) (Z : Set l3) → + le-complemented-cardinality Y Z → le-complemented-cardinality X Y → le-complemented-cardinality X Z + transitive-le-complemented-cardinality X Y Z Y≤Z X≤Y = + unit-le-complemented-cardinality X Z + ( transitive-mere-emb + ( inv-unit-le-complemented-cardinality Y Z Y≤Z) + ( inv-unit-le-complemented-cardinality X Y X≤Y)) + + transitive-le-complemented-Cardinal : + (X : Cardinal l1) (Y : Cardinal l2) (Z : Cardinal l3) → + le-complemented-Cardinal Y Z → le-complemented-Cardinal X Y → le-complemented-Cardinal X Z + transitive-le-complemented-Cardinal = + apply-thrice-dependent-universal-property-trunc-Set' + ( λ X Y Z → + ( le-complemented-Cardinal Y Z → le-complemented-Cardinal X Y → le-complemented-Cardinal X Z) , + ( is-set-function-type + ( is-set-function-type + ( is-set-is-prop is-prop-le-complemented-Cardinal)))) + ( transitive-le-complemented-cardinality) +``` + +## Properties + +### Assuming excluded middle, then inequality is antisymmetric + +Using the previous result and assuming excluded middle, we can conclude +`le-complemented-Cardinal` is a partial order by showing that it is +antisymmetric. + +```text +module _ + {l : Level} (lem : LEM l) + where + + antisymmetric-le-complemented-cardinality : + (X Y : Set l) → + le-complemented-cardinality X Y → + le-complemented-cardinality Y X → + cardinality X ＝ cardinality Y + antisymmetric-le-complemented-cardinality X Y X≤Y Y≤X = + eq-mere-equiv-cardinality X Y + ( antisymmetric-mere-emb + ( lem) + ( inv-unit-le-complemented-cardinality X Y X≤Y) + ( inv-unit-le-complemented-cardinality Y X Y≤X)) + + antisymmetric-le-complemented-Cardinal : + (X Y : Cardinal l) → + le-complemented-Cardinal X Y → le-complemented-Cardinal Y X → X ＝ Y + antisymmetric-le-complemented-Cardinal = + apply-twice-dependent-universal-property-trunc-Set' + ( λ X Y → + set-Prop + ( function-Prop + ( le-complemented-Cardinal X Y) + ( function-Prop (le-complemented-Cardinal Y X) (Id-Prop (Cardinal-Set l) X Y)))) + ( antisymmetric-le-complemented-cardinality) +``` + +## External links + +- [Cardinality](https://en.wikipedia.org/wiki/Cardinality) at Wikipedia +- [cardinal number](https://ncatlab.org/nlab/show/cardinal+number) at $n$Lab diff --git a/src/set-theory/zero-cardinal.lagda.md b/src/set-theory/zero-cardinal.lagda.md new file mode 100644 index 0000000000..e482ccbba9 --- /dev/null +++ b/src/set-theory/zero-cardinal.lagda.md @@ -0,0 +1,67 @@ +# The zero cardinal + +```agda +module set-theory.zero-cardinal where +``` + +
Imports + +```agda +open import elementary-number-theory.natural-numbers + +open import foundation.action-on-identifications-functions +open import foundation.coproduct-types +open import foundation.decidable-propositions +open import foundation.decidable-subtypes +open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.discrete-types +open import foundation.double-negation +open import foundation.equality-coproduct-types +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.functoriality-set-truncation +open import foundation.identity-types +open import foundation.isolated-elements +open import foundation.logical-equivalences +open import foundation.negated-equality +open import foundation.negation +open import foundation.powersets +open import foundation.propositional-truncations +open import foundation.sections +open import foundation.set-truncations +open import foundation.sets +open import foundation.surjective-maps +open import foundation.universe-levels + +open import foundation-core.empty-types +open import foundation-core.fibers-of-maps +open import foundation-core.propositions + +open import logic.propositionally-decidable-types + +open import set-theory.cardinalities +open import set-theory.cardinality-projective-sets +open import set-theory.inequality-cardinalities +``` + +
+ +## Idea + +The {{#concept "zero cardinal" Agda=zero-Cardinal}}, or **empty cardinal**, is +the isomorphism class of [empty](foundation.empty-types.md) +[sets](foundation-core.sets.md). + +## Definitions + +```agda +zero-Cardinal : Cardinal lzero +zero-Cardinal = cardinality empty-Set +``` + +## Properties + +### Any cardinal less than or equal to the zero cardinal is the zero cardinal + +> This remains to be formalized. From 553d88c3263d04214b5b5fe36ee543efed402b8b Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Thu, 16 Oct 2025 19:08:59 +0200 Subject: [PATCH 13/50] Fix capitalization in inequality-cardinalities.md --- src/set-theory/inequality-cardinalities.lagda.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/set-theory/inequality-cardinalities.lagda.md b/src/set-theory/inequality-cardinalities.lagda.md index 08f86f1195..451f303dc1 100644 --- a/src/set-theory/inequality-cardinalities.lagda.md +++ b/src/set-theory/inequality-cardinalities.lagda.md @@ -31,7 +31,7 @@ open import set-theory.equality-cardinalities ## Idea We say a [cardinal of sets](set-theory.cardinalities.md) `X` is -{{#concept "less than or equal to" Agda=leq-Cardinal}} a Cardinal `Y` if any +{{#concept "less than or equal to" Agda=leq-Cardinal}} a cardinal `Y` if any [set](foundation-core.sets.md) in the isomorphism class represented by `X` embeds into any set in the isomorphism class represented by `Y`. This defines the {{#concept "standard ordering" Disambiguation="on cardinalities of sets"}} From 0d7365d55930cc75225cb63295d3e7eddba166d1 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Thu, 16 Oct 2025 19:11:40 +0200 Subject: [PATCH 14/50] Clarify antisymmetry of leq-Cardinal under LEM MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Clarified the conclusion about `leq-Cardinal` being antisymmetric and a partial order, referencing the Cantor–Schröder–Bernstein theorem and the law of excluded middle. --- src/set-theory/inequality-cardinalities.lagda.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/src/set-theory/inequality-cardinalities.lagda.md b/src/set-theory/inequality-cardinalities.lagda.md index 451f303dc1..74eb9eef15 100644 --- a/src/set-theory/inequality-cardinalities.lagda.md +++ b/src/set-theory/inequality-cardinalities.lagda.md @@ -177,8 +177,9 @@ module _ ### Assuming excluded middle, then inequality is antisymmetric -Using the previous result and assuming excluded middle, we can conclude -`leq-Cardinal` is a partial order by showing that it is antisymmetric. +Using that mere equivalence characterizes equality of cardinals we can conclude +by the Cantor–Schröder–Bernstein theorem, assuming the law of excluded middle, +that `leq-Cardinal` is antisymmetric and hence a partial order. ```agda module _ From 95ddf58027be70ea66c113b19d968ba48cc883d2 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Fri, 17 Oct 2025 00:23:06 +0200 Subject: [PATCH 15/50] additions nonsurjective maps --- src/foundation-core/fibers-of-maps.lagda.md | 29 ++++-- src/foundation/nonsurjective-maps.lagda.md | 108 ++++++++++++++++++-- 2 files changed, 122 insertions(+), 15 deletions(-) diff --git a/src/foundation-core/fibers-of-maps.lagda.md b/src/foundation-core/fibers-of-maps.lagda.md index f2fad80631..8b570f95e9 100644 --- a/src/foundation-core/fibers-of-maps.lagda.md +++ b/src/foundation-core/fibers-of-maps.lagda.md @@ -63,6 +63,24 @@ module _ ## Properties +### Fibers of compositions + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + (g : B → X) (h : A → B) {x : X} + where + + inclusion-fiber-comp : (t : fiber g x) → fiber h (pr1 t) → fiber (g ∘ h) x + inclusion-fiber-comp (b , q) (a , p) = (a , ap g p ∙ q) + + left-fiber-comp : fiber (g ∘ h) x → fiber g x + left-fiber-comp (a , r) = (h a , r) + + right-fiber-comp : (q : fiber (g ∘ h) x) → fiber h (pr1 (left-fiber-comp q)) + right-fiber-comp (a , r) = (a , refl) +``` + ### Characterization of the identity types of the fibers of a map #### The case of `fiber` @@ -376,18 +394,11 @@ module _ map-compute-fiber-comp : fiber (g ∘ h) x → Σ (fiber g x) (λ t → fiber h (pr1 t)) - pr1 (pr1 (map-compute-fiber-comp (a , p))) = h a - pr2 (pr1 (map-compute-fiber-comp (a , p))) = p - pr1 (pr2 (map-compute-fiber-comp (a , p))) = a - pr2 (pr2 (map-compute-fiber-comp (a , p))) = refl + map-compute-fiber-comp t = (left-fiber-comp g h t , right-fiber-comp g h t) map-inv-compute-fiber-comp : Σ (fiber g x) (λ t → fiber h (pr1 t)) → fiber (g ∘ h) x - pr1 (map-inv-compute-fiber-comp t) = pr1 (pr2 t) - pr2 (map-inv-compute-fiber-comp t) = ap g (pr2 (pr2 t)) ∙ pr2 (pr1 t) - - inclusion-fiber-comp : (t : fiber g x) → fiber h (pr1 t) → fiber (g ∘ h) x - inclusion-fiber-comp t s = map-inv-compute-fiber-comp (t , s) + map-inv-compute-fiber-comp (t , s) = inclusion-fiber-comp g h t s is-section-map-inv-compute-fiber-comp : is-section map-compute-fiber-comp map-inv-compute-fiber-comp diff --git a/src/foundation/nonsurjective-maps.lagda.md b/src/foundation/nonsurjective-maps.lagda.md index 08b5601af6..79e022788a 100644 --- a/src/foundation/nonsurjective-maps.lagda.md +++ b/src/foundation/nonsurjective-maps.lagda.md @@ -15,23 +15,27 @@ open import foundation.decidable-maps open import foundation.dependent-pair-types open import foundation.diagonal-maps-of-types open import foundation.disjunction +open import foundation.double-negation open import foundation.embeddings open import foundation.empty-types open import foundation.equality-cartesian-product-types open import foundation.existential-quantification open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-coproduct-types +open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-propositional-truncation open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopy-induction open import foundation.identity-types open import foundation.inhabited-types +open import foundation.injective-maps open import foundation.negation open import foundation.postcomposition-dependent-functions open import foundation.propositional-truncations open import foundation.split-surjective-maps open import foundation.structure-identity-principle open import foundation.subtype-identity-principle +open import foundation.surjective-maps open import foundation.truncated-types open import foundation.univalence open import foundation.universal-property-family-of-fibers-of-maps @@ -69,11 +73,11 @@ open import orthogonal-factorization-systems.extensions-maps A map `f : A → B` is {{#concept "nonsurjective"}} if there [exists](foundation.existential-quantification.md) a [fiber](foundation-core.fibers-of-maps.md) that is [not](foundation.negation.md) -[inhabited](foundation.inhabited-types.md). +inhabited. ## Definitions -### The nonimage +### The nonimage of a map ```agda module _ @@ -84,7 +88,7 @@ module _ nonim = Σ B (λ y → ¬ fiber f y) ``` -### Nonsurjectivity +### Nonsurjectivity of a map ```agda module _ @@ -101,8 +105,65 @@ module _ is-nonsurjective-Prop = exists-structure-Prop B (λ y → ¬ fiber f y) ``` +### Nonsurjective maps between types + +```agda +nonsurjective-map : {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2) +nonsurjective-map A B = Σ (A → B) is-nonsurjective + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : nonsurjective-map A B) + where + + map-nonsurjective-map : A → B + map-nonsurjective-map = pr1 f + + is-nonsurjective-map-nonsurjective-map : + is-nonsurjective map-nonsurjective-map + is-nonsurjective-map-nonsurjective-map = pr2 f +``` + ## Properties +### Nonsurjective maps are not surjective + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + where + + is-not-surjective-is-nonsurjective' : ¬¬ nonim f → ¬ is-surjective f + is-not-surjective-is-nonsurjective' F H = + F (λ (x , np) → rec-trunc-Prop empty-Prop np (H x)) + + is-not-surjective-is-nonsurjective : is-nonsurjective f → ¬ is-surjective f + is-not-surjective-is-nonsurjective F = + is-not-surjective-is-nonsurjective' + ( intro-double-negation-type-trunc-Prop F) +``` + +### If `g ∘ f` is nonsurjective and `g` is surjective then `f` is nonsurjective + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {f : A → B} {g : B → C} + where + + is-nonsurjective-right-nonim-comp-is-surjective-left : + is-surjective g → nonim (g ∘ f) → is-nonsurjective f + is-nonsurjective-right-nonim-comp-is-surjective-left G (c , np) = + map-trunc-Prop + ( λ (y , q) → (y , map-neg (inclusion-fiber-comp g f (y , q)) np)) + ( G c) + + is-nonsurjective-right-is-surjective-left : + is-surjective g → is-nonsurjective (g ∘ f) → is-nonsurjective f + is-nonsurjective-right-is-surjective-left G = + rec-trunc-Prop + ( is-nonsurjective-Prop f) + ( is-nonsurjective-right-nonim-comp-is-surjective-left G) +``` + ### If `g ∘ f` is nonsurjective and `g` is decidable then `g` or `f` is nonsurjective ```agda @@ -115,7 +176,7 @@ module _ nonim (g ∘ f) → nonim f + nonim g decide-nonim-comp H (c , np) = map-coproduct - ( λ (y , q) → y , map-neg (inclusion-fiber-comp g f c (y , q)) np) + ( λ (y , q) → y , map-neg (inclusion-fiber-comp g f (y , q)) np) ( c ,_) ( H c) @@ -125,7 +186,7 @@ module _ decide-is-nonsurjective-nonim-comp' H (c , np) = map-coproduct ( λ (y , q) → - unit-trunc-Prop (y , map-neg (inclusion-fiber-comp g f c (y , q)) np)) + unit-trunc-Prop (y , map-neg (inclusion-fiber-comp g f (y , q)) np)) (λ p → unit-trunc-Prop (c , p)) ( H c) @@ -135,14 +196,18 @@ module _ disjunction-type (is-nonsurjective f) (is-nonsurjective g) is-nonsurjective-is-nonsurjective-comp' H = map-trunc-Prop (decide-is-nonsurjective-nonim-comp' H) +``` +In fact, it suffices that `g` is propositionally decidable. + +```agda decide-is-nonsurjective-nonim-comp : is-inhabited-or-empty-map g → nonim (g ∘ f) → is-nonsurjective f + is-nonsurjective g decide-is-nonsurjective-nonim-comp H (c , np) = map-coproduct ( map-trunc-Prop - ( λ (y , q) → y , map-neg (inclusion-fiber-comp g f c (y , q)) np)) + ( λ (y , q) → y , map-neg (inclusion-fiber-comp g f (y , q)) np)) ( λ p → unit-trunc-Prop (c , p)) ( H c) @@ -153,3 +218,34 @@ module _ is-nonsurjective-is-nonsurjective-comp H = map-trunc-Prop (decide-is-nonsurjective-nonim-comp H) ``` + +### If `g` is nonsurjective then so is `g ∘ f` + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {f : A → B} {g : B → C} + where + + nonim-comp : nonim g → nonim (g ∘ f) + nonim-comp (c , np) = (c , map-neg (left-fiber-comp g f) np) + + is-nonsurjective-comp : is-nonsurjective g → is-nonsurjective (g ∘ f) + is-nonsurjective-comp = map-trunc-Prop nonim-comp +``` + +### If `g` is injective and `f` is nonsurjective then so is `g ∘ f` + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {f : A → B} {g : B → C} + where + + nonim-comp-is-injective-left : is-injective g → nonim f → nonim (g ∘ f) + nonim-comp-is-injective-left G (c , np) = + ( g c , map-neg (tot (λ _ → G)) np) + + is-nonsurjective-comp-is-injective-left : + is-injective g → is-nonsurjective f → is-nonsurjective (g ∘ f) + is-nonsurjective-comp-is-injective-left G = + map-trunc-Prop (nonim-comp-is-injective-left G) +``` From bd2211f4ef7fcf7924ae7a2f2a59b701978cf1f4 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Fri, 17 Oct 2025 01:05:55 +0200 Subject: [PATCH 16/50] inhabited cardinals --- src/set-theory.lagda.md | 1 + .../addition-cardinalities.lagda.md | 6 +- .../cardinality-projective-sets.lagda.md | 4 +- src/set-theory/inhabited-cardinals.lagda.md | 120 ++++++++++++++++++ 4 files changed, 126 insertions(+), 5 deletions(-) create mode 100644 src/set-theory/inhabited-cardinals.lagda.md diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index 8e4bf784cb..0739d9ec71 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -58,6 +58,7 @@ open import set-theory.dependent-sums-cardinals public open import set-theory.equality-cardinalities public open import set-theory.inequality-cardinalities public open import set-theory.infinite-sets public +open import set-theory.inhabited-cardinals public open import set-theory.russells-paradox public open import set-theory.strict-complemented-boundedness-cardinalities public open import set-theory.strict-complemented-inequality-cardinalities public diff --git a/src/set-theory/addition-cardinalities.lagda.md b/src/set-theory/addition-cardinalities.lagda.md index 8d0d8db340..a1e141b35a 100644 --- a/src/set-theory/addition-cardinalities.lagda.md +++ b/src/set-theory/addition-cardinalities.lagda.md @@ -1,4 +1,4 @@ -# Coproducts of cardinalities +# Addition of cardinalities ```agda module set-theory.addition-cardinalities where @@ -51,8 +51,8 @@ open import set-theory.zero-cardinal ## Idea -Given two [cardinals](set-theory.cardinalities.lagda.md) `A` and `B`, we may -form their {{#concept "sum" Disambiguation="cardinal" Agda=add-Cardinal}}, or +Given two [cardinals](set-theory.cardinalities.md) `A` and `B`, we may form +their {{#concept "sum" Disambiguation="cardinal" Agda=add-Cardinal}}, or **coproduct** cardinal, `A + B`. It is defined on isomorphism classes as [coproducts](foundation-core.coproduct-types.md) of [sets](foundation-core.sets.md). diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index 0a545534d7..53f1894fe6 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -29,8 +29,8 @@ open import set-theory.cardinalities For every type $X$ there is a map $║ X → Set ║₀ → (X → Cardinal)$. We call sets $X$ for which this map has a section {{#concept "cardinality-projective" Disamibguation="sets"}}. Over such types we -may form [dependent sum](set-theory.dependent-sums-cardinalities.md) and -[dependent product](set-theory.dependent-products-cardinalities.md) cardinals. +may form [dependent sum](set-theory.dependent-sums-cardinals.md) and +[dependent product](set-theory.dependent-products-cardinals.md) cardinals. Note that classically, the universe of sets is itself a set, and so trivially $║ X → Set ║₀ → (X → ║ Set ║₀)$ is an equivalence. However, with univalence, diff --git a/src/set-theory/inhabited-cardinals.lagda.md b/src/set-theory/inhabited-cardinals.lagda.md new file mode 100644 index 0000000000..7c7ab49152 --- /dev/null +++ b/src/set-theory/inhabited-cardinals.lagda.md @@ -0,0 +1,120 @@ +# Inhabited cardinals + +```agda +module set-theory.inhabited-cardinals where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.function-types +open import foundation.identity-types +open import foundation.inhabited-types +open import foundation.propositional-extensionality +open import foundation.propositions +open import foundation.set-truncations +open import foundation.sets +open import foundation.univalence +open import foundation.universe-levels + +open import set-theory.cardinalities +``` + +
+ +## Idea + +A [cardinal of sets](set-theory.cardinalities.lagda.md) `κ` is +{{#concept "inhabited" Disambiguation="cardinal" Agda=is-inhabited-Cardinal}}, +if any [set](foundation-core.sets.md) in its isomorphism class is +[inhabited](foundation-core.inhabited.md). + +## Definitions + +### The predicate on cardinals of being inhabited + +```agda +module _ + {l : Level} (κ : Cardinal l) + where + + is-inhabited-prop-Cardinal : Prop l + is-inhabited-prop-Cardinal = + apply-universal-property-trunc-Set' κ + ( Prop-Set l) + ( is-inhabited-Prop ∘ type-Set) + + is-inhabited-Cardinal : UU l + is-inhabited-Cardinal = type-Prop is-inhabited-prop-Cardinal + + is-prop-is-inhabited-Cardinal : is-prop is-inhabited-Cardinal + is-prop-is-inhabited-Cardinal = + is-prop-type-Prop is-inhabited-prop-Cardinal +``` + +### Inhabited cardinalities + +```agda +module _ + {l : Level} (X : Set l) + where + + is-inhabited-prop-cardinality : Prop l + is-inhabited-prop-cardinality = is-inhabited-prop-Cardinal (cardinality X) + + is-inhabited-cardinality : UU l + is-inhabited-cardinality = is-inhabited-Cardinal (cardinality X) + + is-prop-is-inhabited-cardinality : is-prop is-inhabited-cardinality + is-prop-is-inhabited-cardinality = + is-prop-is-inhabited-Cardinal (cardinality X) + + eq-compute-is-inhabited-prop-cardinality : + is-inhabited-prop-cardinality ＝ is-inhabited-Prop (type-Set X) + eq-compute-is-inhabited-prop-cardinality = + triangle-universal-property-trunc-Set + ( Prop-Set l) + ( is-inhabited-Prop ∘ type-Set) + ( X) + + eq-compute-is-inhabited-cardinality : + is-inhabited-cardinality ＝ is-inhabited (type-Set X) + eq-compute-is-inhabited-cardinality = + ap type-Prop eq-compute-is-inhabited-prop-cardinality + + compute-is-inhabited-cardinality : + is-inhabited-cardinality ≃ is-inhabited (type-Set X) + compute-is-inhabited-cardinality = + equiv-eq eq-compute-is-inhabited-cardinality + + unit-is-inhabited-cardinality : + is-inhabited (type-Set X) → is-inhabited-cardinality + unit-is-inhabited-cardinality = + map-inv-equiv compute-is-inhabited-cardinality + + inv-unit-is-inhabited-cardinality : + is-inhabited-cardinality → is-inhabited (type-Set X) + inv-unit-is-inhabited-cardinality = + map-equiv compute-is-inhabited-cardinality +``` + +### The universe of inhabited cardinals + +```agda +Inhabited-Cardinal : (l : Level) → UU (lsuc l) +Inhabited-Cardinal l = Σ (Cardinal l) is-inhabited-Cardinal + +module _ + {l : Level} (κ : Inhabited-Cardinal l) + where + + cardinal-Inhabited-Cardinal : Cardinal l + cardinal-Inhabited-Cardinal = pr1 κ + + is-inhabited-Inhabited-Cardinal : + is-inhabited-Cardinal cardinal-Inhabited-Cardinal + is-inhabited-Inhabited-Cardinal = pr2 κ +``` From ded3d726834c8574c3a4189d645bae62965eba4d Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Fri, 17 Oct 2025 12:33:45 +0200 Subject: [PATCH 17/50] file names: use `cardinals` instead of `cardinalities` --- src/set-theory.lagda.md | 9 +- ...dinalities.lagda.md => cardinals.lagda.md} | 4 +- ...omplemented-inequality-cardinals.lagda.md} | 6 +- .../equality-cardinalities.lagda.md | 318 ------------------ ...lagda.md => inequality-cardinals.lagda.md} | 8 +- src/set-theory/infinite-sets.lagda.md | 2 +- .../complements-isolated-elements.lagda.md | 2 +- .../ferrers-diagrams.lagda.md | 2 +- .../kuratowski-finite-sets.lagda.md | 6 +- 9 files changed, 19 insertions(+), 338 deletions(-) rename src/set-theory/{cardinalities.lagda.md => cardinals.lagda.md} (96%) rename src/set-theory/{complemented-inequality-cardinalities.lagda.md => complemented-inequality-cardinals.lagda.md} (97%) delete mode 100644 src/set-theory/equality-cardinalities.lagda.md rename src/set-theory/{inequality-cardinalities.lagda.md => inequality-cardinals.lagda.md} (97%) diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index 6d0d018636..b10e1a3556 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -34,7 +34,7 @@ structure. In fact, with univalence it is a proper In this module, we consider ideas historically related to the study of set theories both as foundations of set-level mathematics, but also as a study of hierarchies in mathematics. This includes ideas such as -[cardinality](set-theory.cardinalities.md) and +[cardinality](set-theory.cardinals.md) and [infinity](set-theory.infinite-sets.md), the [cumulative hierarchy](set-theory.cumulative-hierarchy.md) as a model of set theory, and [Russell's paradox](set-theory.russells-paradox.md). @@ -47,12 +47,11 @@ module set-theory where open import set-theory.baire-space public open import set-theory.cantor-space public open import set-theory.cantors-diagonal-argument public -open import set-theory.cardinalities public -open import set-theory.complemented-inequality-cardinalities public +open import set-theory.cardinals public +open import set-theory.complemented-inequality-cardinals public open import set-theory.countable-sets public open import set-theory.cumulative-hierarchy public -open import set-theory.equality-cardinalities public -open import set-theory.inequality-cardinalities public +open import set-theory.inequality-cardinals public open import set-theory.infinite-sets public open import set-theory.russells-paradox public open import set-theory.uncountable-sets public diff --git a/src/set-theory/cardinalities.lagda.md b/src/set-theory/cardinals.lagda.md similarity index 96% rename from src/set-theory/cardinalities.lagda.md rename to src/set-theory/cardinals.lagda.md index 0c1ab3306c..8fbe61cb9a 100644 --- a/src/set-theory/cardinalities.lagda.md +++ b/src/set-theory/cardinals.lagda.md @@ -1,7 +1,7 @@ -# Cardinalities of sets +# Cardinals ```agda -module set-theory.cardinalities where +module set-theory.cardinals where ```
Imports diff --git a/src/set-theory/complemented-inequality-cardinalities.lagda.md b/src/set-theory/complemented-inequality-cardinals.lagda.md similarity index 97% rename from src/set-theory/complemented-inequality-cardinalities.lagda.md rename to src/set-theory/complemented-inequality-cardinals.lagda.md index b3845df6e9..f8f3b7c85f 100644 --- a/src/set-theory/complemented-inequality-cardinalities.lagda.md +++ b/src/set-theory/complemented-inequality-cardinals.lagda.md @@ -1,7 +1,7 @@ # Complemented inequality on cardinals ```agda -module set-theory.complemented-inequality-cardinalities where +module set-theory.complemented-inequality-cardinals where ```
Imports @@ -21,14 +21,14 @@ open import foundation.sets open import foundation.univalence open import foundation.universe-levels -open import set-theory.cardinalities +open import set-theory.cardinals ```
## Idea -We may say a [cardinal of sets](set-theory.cardinalities.md) `X` is +We may say a [cardinal of sets](set-theory.cardinals.md) `X` is {{#concept "less than or equal to" Agda=leq-complemented-cardinality}} a cardinal `Y` if any [set](foundation-core.sets.md) in the isomorphism class represented by `X` [embeds](foundation-core.embeddings.md) as a diff --git a/src/set-theory/equality-cardinalities.lagda.md b/src/set-theory/equality-cardinalities.lagda.md deleted file mode 100644 index da5d178d65..0000000000 --- a/src/set-theory/equality-cardinalities.lagda.md +++ /dev/null @@ -1,318 +0,0 @@ -# Equality of cardinals of sets - -```agda -module set-theory.equality-cardinalities where -``` - -
Imports - -```agda -open import foundation.action-on-identifications-functions -open import foundation.dependent-pair-types -open import foundation.equivalences -open import foundation.function-extensionality -open import foundation.functoriality-propositional-truncation -open import foundation.identity-types -open import foundation.large-equivalence-relations -open import foundation.large-similarity-relations -open import foundation.mere-equivalences -open import foundation.propositional-extensionality -open import foundation.propositions -open import foundation.set-truncations -open import foundation.sets -open import foundation.univalence -open import foundation.universe-levels - -open import set-theory.cardinalities -``` - -
- -## Idea - -Two [cardinals of sets](set-theory.cardinalities.md) `X` and `Y` are -{{#concept "similar" Disambiguation="cardinals" Agda=sim-Cardinal}} if there -[merely](foundation.inhabited-types.md) is an -[equivalence](foundation-core.equivalences.md) between any two representing two -types. This characterizes [equality](foundation-core.identity-types.md) of -cardinals. - -## Definition - -### The underlying similarity of cardinalities - -```agda -module _ - {l1 l2 : Level} (X : Set l1) (Y : Set l2) - where - - sim-prop-cardinality' : Prop (l1 ⊔ l2) - sim-prop-cardinality' = mere-equiv-Prop (type-Set X) (type-Set Y) - - sim-cardinality' : UU (l1 ⊔ l2) - sim-cardinality' = mere-equiv (type-Set X) (type-Set Y) - - is-prop-sim-sim-cardinality' : is-prop sim-cardinality' - is-prop-sim-sim-cardinality' = is-prop-mere-equiv (type-Set X) (type-Set Y) - -refl-sim-cardinality' : {l : Level} (X : Set l) → sim-cardinality' X X -refl-sim-cardinality' X = refl-mere-equiv (type-Set X) - -transitive-sim-cardinality' : - {l1 l2 l3 : Level} (X : Set l1) (Y : Set l2) (Z : Set l3) → - sim-cardinality' Y Z → sim-cardinality' X Y → sim-cardinality' X Z -transitive-sim-cardinality' X Y Z = - transitive-mere-equiv (type-Set X) (type-Set Y) (type-Set Z) - -symmetric-sim-cardinality' : - {l1 l2 : Level} (X : Set l1) (Y : Set l2) → - sim-cardinality' X Y → sim-cardinality' Y X -symmetric-sim-cardinality' X Y = - symmetric-mere-equiv (type-Set X) (type-Set Y) -``` - -### Similarity of cardinalities with a cardinal - -```agda -module _ - {l1 l2 : Level} (X : Set l1) - where - - sim-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) - sim-prop-Cardinal' = - map-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( sim-prop-cardinality' X) - - sim-Cardinal' : Cardinal l2 → UU (l1 ⊔ l2) - sim-Cardinal' Y = type-Prop (sim-prop-Cardinal' Y) - - is-prop-sim-sim-Cardinal' : (Y : Cardinal l2) → is-prop (sim-Cardinal' Y) - is-prop-sim-sim-Cardinal' Y = is-prop-type-Prop (sim-prop-Cardinal' Y) - - eq-compute-sim-prop-Cardinal' : - (Y : Set l2) → - sim-prop-Cardinal' (cardinality Y) ＝ sim-prop-cardinality' X Y - eq-compute-sim-prop-Cardinal' = - triangle-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( sim-prop-cardinality' X) - - eq-compute-sim-Cardinal' : - (Y : Set l2) → - sim-Cardinal' (cardinality Y) ＝ sim-cardinality' X Y - eq-compute-sim-Cardinal' Y = - ap type-Prop (eq-compute-sim-prop-Cardinal' Y) -``` - -### Similarity of cardinals - -```agda -module _ - {l1 l2 : Level} (X : Cardinal l1) (Y : Cardinal l2) - where - - sim-prop-Cardinal : Prop (l1 ⊔ l2) - sim-prop-Cardinal = - map-universal-property-trunc-Set - ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( sim-prop-Cardinal') - ( X) - ( Y) - - sim-Cardinal : UU (l1 ⊔ l2) - sim-Cardinal = type-Prop sim-prop-Cardinal - - is-prop-sim-sim-Cardinal : is-prop sim-Cardinal - is-prop-sim-sim-Cardinal = is-prop-type-Prop sim-prop-Cardinal - - sim-set-Cardinal : Set (l1 ⊔ l2) - sim-set-Cardinal = set-Prop sim-prop-Cardinal -``` - -### Similarity of cardinalities - -```agda -module _ - {l1 l2 : Level} (X : Set l1) (Y : Set l2) - where - - sim-prop-cardinality : Prop (l1 ⊔ l2) - sim-prop-cardinality = sim-prop-Cardinal (cardinality X) (cardinality Y) - - sim-cardinality : UU (l1 ⊔ l2) - sim-cardinality = type-Prop sim-prop-cardinality - - is-prop-sim-sim-cardinality : is-prop sim-cardinality - is-prop-sim-sim-cardinality = is-prop-type-Prop sim-prop-cardinality - - eq-compute-sim-prop-cardinality : - sim-prop-cardinality ＝ mere-equiv-Prop (type-Set X) (type-Set Y) - eq-compute-sim-prop-cardinality = - ( htpy-eq - ( triangle-universal-property-trunc-Set - ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( sim-prop-Cardinal') - ( X)) - ( cardinality Y)) ∙ - ( eq-compute-sim-prop-Cardinal' X Y) - - eq-compute-sim-cardinality : - sim-cardinality ＝ mere-equiv (type-Set X) (type-Set Y) - eq-compute-sim-cardinality = - ap type-Prop eq-compute-sim-prop-cardinality - - compute-sim-cardinality : - sim-cardinality ≃ mere-equiv (type-Set X) (type-Set Y) - compute-sim-cardinality = equiv-eq eq-compute-sim-cardinality - - unit-sim-cardinality : - mere-equiv (type-Set X) (type-Set Y) → sim-cardinality - unit-sim-cardinality = map-inv-equiv compute-sim-cardinality - - inv-unit-sim-cardinality : - sim-cardinality → mere-equiv (type-Set X) (type-Set Y) - inv-unit-sim-cardinality = pr1 compute-sim-cardinality -``` - -## Properties - -### Equality of cardinalities is equivalent to mere equivalence - -```agda -is-effective-cardinality : - {l : Level} (X Y : Set l) → - (cardinality X ＝ cardinality Y) ≃ mere-equiv (type-Set X) (type-Set Y) -is-effective-cardinality X Y = - ( equiv-trunc-Prop (extensionality-Set X Y)) ∘e - ( is-effective-unit-trunc-Set (Set _) X Y) - -eq-mere-equiv-cardinality : - {l : Level} (X Y : Set l) → - mere-equiv (type-Set X) (type-Set Y) → cardinality X ＝ cardinality Y -eq-mere-equiv-cardinality X Y = map-inv-equiv (is-effective-cardinality X Y) -``` - -### Similarity of cardinals is reflexive - -```agda -refl-sim-cardinality : {l : Level} (X : Set l) → sim-cardinality X X -refl-sim-cardinality X = unit-sim-cardinality X X (refl-mere-equiv (type-Set X)) - -refl-sim-Cardinal : {l : Level} (X : Cardinal l) → sim-Cardinal X X -refl-sim-Cardinal = - apply-dependent-universal-property-trunc-Set' - ( λ X → sim-set-Cardinal X X) - ( refl-sim-cardinality) -``` - -### Similarity of cardinals is symmetric - -```agda -symmetric-sim-cardinality : - {l1 l2 : Level} - (X : Set l1) (Y : Set l2) → - sim-cardinality X Y → sim-cardinality Y X -symmetric-sim-cardinality X Y p = - unit-sim-cardinality Y X - ( symmetric-sim-cardinality' X Y (inv-unit-sim-cardinality X Y p)) - -abstract - symmetric-sim-Cardinal : - {l1 l2 : Level} - (X : Cardinal l1) (Y : Cardinal l2) → - sim-Cardinal X Y → sim-Cardinal Y X - symmetric-sim-Cardinal = - apply-twice-dependent-universal-property-trunc-Set' - ( λ X Y → hom-set-Set (sim-set-Cardinal X Y) (sim-set-Cardinal Y X)) - ( symmetric-sim-cardinality) -``` - -### Similarity of cardinals is transitive - -```agda -transitive-sim-cardinality : - {l1 l2 l3 : Level} - (X : Set l1) (Y : Set l2) (Z : Set l3) → - sim-cardinality Y Z → sim-cardinality X Y → sim-cardinality X Z -transitive-sim-cardinality X Y Z p q = - unit-sim-cardinality X Z - ( transitive-sim-cardinality' X Y Z - ( inv-unit-sim-cardinality Y Z p) - ( inv-unit-sim-cardinality X Y q)) - -abstract - transitive-sim-Cardinal : - {l1 l2 l3 : Level} - (X : Cardinal l1) (Y : Cardinal l2) (Z : Cardinal l3) → - sim-Cardinal Y Z → sim-Cardinal X Y → sim-Cardinal X Z - transitive-sim-Cardinal = - apply-thrice-dependent-universal-property-trunc-Set' - ( λ X Y Z → - hom-set-Set - ( sim-set-Cardinal Y Z) - ( hom-set-Set (sim-set-Cardinal X Y) (sim-set-Cardinal X Z))) - ( transitive-sim-cardinality) -``` - -### Similarity of cardinals is extensional - -```agda -module _ - {l : Level} - where - - sim-eq-Cardinal : - (X Y : Cardinal l) → X ＝ Y → sim-Cardinal X Y - sim-eq-Cardinal X .X refl = refl-sim-Cardinal X - - sim-eq-cardinality : - (X Y : Set l) → cardinality X ＝ cardinality Y → sim-cardinality X Y - sim-eq-cardinality X Y = sim-eq-Cardinal (cardinality X) (cardinality Y) - - eq-sim-cardinality : - (X Y : Set l) → sim-cardinality X Y → cardinality X ＝ cardinality Y - eq-sim-cardinality X Y p = - eq-mere-equiv-cardinality X Y (inv-unit-sim-cardinality X Y p) - - abstract - eq-sim-Cardinal : - (X Y : Cardinal l) → sim-Cardinal X Y → X ＝ Y - eq-sim-Cardinal = - apply-twice-dependent-universal-property-trunc-Set' - ( λ X Y → - hom-set-Set - ( sim-set-Cardinal X Y) - ( set-Prop (Id-Prop (Cardinal-Set l) X Y))) - ( eq-sim-cardinality) - - abstract - antisymmetric-sim-Cardinal : - (X Y : Cardinal l) → sim-Cardinal X Y → sim-Cardinal Y X → X ＝ Y - antisymmetric-sim-Cardinal X Y p _ = eq-sim-Cardinal X Y p -``` - -### Similarity of cardinals is a large similarity relation - -```agda -large-equivalence-relation-Cardinal : Large-Equivalence-Relation (_⊔_) Cardinal -large-equivalence-relation-Cardinal = - λ where - .sim-prop-Large-Equivalence-Relation → sim-prop-Cardinal - .refl-sim-Large-Equivalence-Relation → refl-sim-Cardinal - .symmetric-sim-Large-Equivalence-Relation → symmetric-sim-Cardinal - .transitive-sim-Large-Equivalence-Relation → transitive-sim-Cardinal - -large-similarity-relation-Cardinal : Large-Similarity-Relation (_⊔_) Cardinal -large-similarity-relation-Cardinal = - λ where - .large-equivalence-relation-Large-Similarity-Relation → - large-equivalence-relation-Cardinal - .eq-sim-Large-Similarity-Relation → - eq-sim-Cardinal -``` - -## External links - -- [Cardinality](https://en.wikipedia.org/wiki/Cardinality) at Wikipedia -- [cardinal number](https://ncatlab.org/nlab/show/cardinal+number) at $n$Lab diff --git a/src/set-theory/inequality-cardinalities.lagda.md b/src/set-theory/inequality-cardinals.lagda.md similarity index 97% rename from src/set-theory/inequality-cardinalities.lagda.md rename to src/set-theory/inequality-cardinals.lagda.md index 74eb9eef15..a76559edbe 100644 --- a/src/set-theory/inequality-cardinalities.lagda.md +++ b/src/set-theory/inequality-cardinals.lagda.md @@ -1,7 +1,7 @@ # Inequality on cardinals ```agda -module set-theory.inequality-cardinalities where +module set-theory.inequality-cardinals where ```
Imports @@ -22,15 +22,15 @@ open import foundation.sets open import foundation.univalence open import foundation.universe-levels -open import set-theory.cardinalities -open import set-theory.equality-cardinalities +open import set-theory.cardinals +open import set-theory.equality-cardinals ```
## Idea -We say a [cardinal of sets](set-theory.cardinalities.md) `X` is +We say a [cardinal of sets](set-theory.cardinals.md) `X` is {{#concept "less than or equal to" Agda=leq-Cardinal}} a cardinal `Y` if any [set](foundation-core.sets.md) in the isomorphism class represented by `X` embeds into any set in the isomorphism class represented by `Y`. This defines diff --git a/src/set-theory/infinite-sets.lagda.md b/src/set-theory/infinite-sets.lagda.md index d94e6be82f..0016b64538 100644 --- a/src/set-theory/infinite-sets.lagda.md +++ b/src/set-theory/infinite-sets.lagda.md @@ -24,7 +24,7 @@ open import univalent-combinatorics.standard-finite-types A [set](foundation-core.sets.md) `A` is said to be {{#concept "infinite" Disambiguation="set" WD="infinite set" WDID=Q205140}} if -it contains arbitrarily [large](set-theory.cardinalities.md) +it contains arbitrarily [large](set-theory.cardinals.md) [finite](univalent-combinatorics.finite-types.md) [subsets](foundation-core.subtypes.md). diff --git a/src/univalent-combinatorics/complements-isolated-elements.lagda.md b/src/univalent-combinatorics/complements-isolated-elements.lagda.md index add3d8ff3f..74b9e4ee67 100644 --- a/src/univalent-combinatorics/complements-isolated-elements.lagda.md +++ b/src/univalent-combinatorics/complements-isolated-elements.lagda.md @@ -27,7 +27,7 @@ open import univalent-combinatorics.finite-types ## Idea For any element `x` in a [finite type](univalent-combinatorics.finite-types.md) -`X` of [cardinality](set-theory.cardinalities.md) `n+1`, we can construct its +`X` of [cardinality](set-theory.cardinals.md) `n+1`, we can construct its **complement**, which is a type of cardinality `n`. ## Definition diff --git a/src/univalent-combinatorics/ferrers-diagrams.lagda.md b/src/univalent-combinatorics/ferrers-diagrams.lagda.md index 034b26e3ad..ecc31bf35f 100644 --- a/src/univalent-combinatorics/ferrers-diagrams.lagda.md +++ b/src/univalent-combinatorics/ferrers-diagrams.lagda.md @@ -40,7 +40,7 @@ ferrers diagram of a [finite type](univalent-combinatorics.finite-types.md) `A` consists of a finite type `X` and a family `Y` of inhabited finite types over `X` such that `Σ X Y` is merely equivalent to `A`. The number of finite Ferrers diagrams of `A` is the [partition number](univalent-combinatorics.partitions.md) -of the [cardinality](set-theory.cardinalities.md) of `A`. +of the [cardinality](set-theory.cardinals.md) of `A`. ## Definition diff --git a/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md b/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md index e9ab2f70bb..d8e33ac8d4 100644 --- a/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md +++ b/src/univalent-combinatorics/kuratowski-finite-sets.lagda.md @@ -22,9 +22,9 @@ open import foundation.sets open import foundation.surjective-maps open import foundation.universe-levels -open import set-theory.cardinalities -open import set-theory.equality-cardinalities -open import set-theory.inequality-cardinalities +open import set-theory.cardinals +open import set-theory.equality-cardinals +open import set-theory.inequality-cardinals open import univalent-combinatorics.dedekind-finite-sets open import univalent-combinatorics.dedekind-finite-types From 3e96ca6fd23d30a746e130af418ee6a77c59dee6 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Fri, 17 Oct 2025 12:43:29 +0200 Subject: [PATCH 18/50] restore equality of cardinals --- src/set-theory/equality-cardinals.lagda.md | 318 +++++++++++++++++++++ 1 file changed, 318 insertions(+) create mode 100644 src/set-theory/equality-cardinals.lagda.md diff --git a/src/set-theory/equality-cardinals.lagda.md b/src/set-theory/equality-cardinals.lagda.md new file mode 100644 index 0000000000..3787e99b51 --- /dev/null +++ b/src/set-theory/equality-cardinals.lagda.md @@ -0,0 +1,318 @@ +# Equality of cardinals + +```agda +module set-theory.equality-cardinals where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.function-extensionality +open import foundation.functoriality-propositional-truncation +open import foundation.identity-types +open import foundation.large-equivalence-relations +open import foundation.large-similarity-relations +open import foundation.mere-equivalences +open import foundation.propositional-extensionality +open import foundation.propositions +open import foundation.set-truncations +open import foundation.sets +open import foundation.univalence +open import foundation.universe-levels + +open import set-theory.cardinalities +``` + +
+ +## Idea + +Two [cardinals of sets](set-theory.cardinalities.md) `X` and `Y` are +{{#concept "similar" Disambiguation="cardinals" Agda=sim-Cardinal}} if there +[merely](foundation.inhabited-types.md) is an +[equivalence](foundation-core.equivalences.md) between any two representing two +types. This characterizes [equality](foundation-core.identity-types.md) of +cardinals. + +## Definition + +### The underlying similarity of cardinalities + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + sim-prop-cardinality' : Prop (l1 ⊔ l2) + sim-prop-cardinality' = mere-equiv-Prop (type-Set X) (type-Set Y) + + sim-cardinality' : UU (l1 ⊔ l2) + sim-cardinality' = mere-equiv (type-Set X) (type-Set Y) + + is-prop-sim-sim-cardinality' : is-prop sim-cardinality' + is-prop-sim-sim-cardinality' = is-prop-mere-equiv (type-Set X) (type-Set Y) + +refl-sim-cardinality' : {l : Level} (X : Set l) → sim-cardinality' X X +refl-sim-cardinality' X = refl-mere-equiv (type-Set X) + +transitive-sim-cardinality' : + {l1 l2 l3 : Level} (X : Set l1) (Y : Set l2) (Z : Set l3) → + sim-cardinality' Y Z → sim-cardinality' X Y → sim-cardinality' X Z +transitive-sim-cardinality' X Y Z = + transitive-mere-equiv (type-Set X) (type-Set Y) (type-Set Z) + +symmetric-sim-cardinality' : + {l1 l2 : Level} (X : Set l1) (Y : Set l2) → + sim-cardinality' X Y → sim-cardinality' Y X +symmetric-sim-cardinality' X Y = + symmetric-mere-equiv (type-Set X) (type-Set Y) +``` + +### Similarity of cardinalities with a cardinal + +```agda +module _ + {l1 l2 : Level} (X : Set l1) + where + + sim-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) + sim-prop-Cardinal' = + map-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( sim-prop-cardinality' X) + + sim-Cardinal' : Cardinal l2 → UU (l1 ⊔ l2) + sim-Cardinal' Y = type-Prop (sim-prop-Cardinal' Y) + + is-prop-sim-sim-Cardinal' : (Y : Cardinal l2) → is-prop (sim-Cardinal' Y) + is-prop-sim-sim-Cardinal' Y = is-prop-type-Prop (sim-prop-Cardinal' Y) + + eq-compute-sim-prop-Cardinal' : + (Y : Set l2) → + sim-prop-Cardinal' (cardinality Y) ＝ sim-prop-cardinality' X Y + eq-compute-sim-prop-Cardinal' = + triangle-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( sim-prop-cardinality' X) + + eq-compute-sim-Cardinal' : + (Y : Set l2) → + sim-Cardinal' (cardinality Y) ＝ sim-cardinality' X Y + eq-compute-sim-Cardinal' Y = + ap type-Prop (eq-compute-sim-prop-Cardinal' Y) +``` + +### Similarity of cardinals + +```agda +module _ + {l1 l2 : Level} (X : Cardinal l1) (Y : Cardinal l2) + where + + sim-prop-Cardinal : Prop (l1 ⊔ l2) + sim-prop-Cardinal = + map-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( sim-prop-Cardinal') + ( X) + ( Y) + + sim-Cardinal : UU (l1 ⊔ l2) + sim-Cardinal = type-Prop sim-prop-Cardinal + + is-prop-sim-sim-Cardinal : is-prop sim-Cardinal + is-prop-sim-sim-Cardinal = is-prop-type-Prop sim-prop-Cardinal + + sim-set-Cardinal : Set (l1 ⊔ l2) + sim-set-Cardinal = set-Prop sim-prop-Cardinal +``` + +### Similarity of cardinalities + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + sim-prop-cardinality : Prop (l1 ⊔ l2) + sim-prop-cardinality = sim-prop-Cardinal (cardinality X) (cardinality Y) + + sim-cardinality : UU (l1 ⊔ l2) + sim-cardinality = type-Prop sim-prop-cardinality + + is-prop-sim-sim-cardinality : is-prop sim-cardinality + is-prop-sim-sim-cardinality = is-prop-type-Prop sim-prop-cardinality + + eq-compute-sim-prop-cardinality : + sim-prop-cardinality ＝ mere-equiv-Prop (type-Set X) (type-Set Y) + eq-compute-sim-prop-cardinality = + ( htpy-eq + ( triangle-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( sim-prop-Cardinal') + ( X)) + ( cardinality Y)) ∙ + ( eq-compute-sim-prop-Cardinal' X Y) + + eq-compute-sim-cardinality : + sim-cardinality ＝ mere-equiv (type-Set X) (type-Set Y) + eq-compute-sim-cardinality = + ap type-Prop eq-compute-sim-prop-cardinality + + compute-sim-cardinality : + sim-cardinality ≃ mere-equiv (type-Set X) (type-Set Y) + compute-sim-cardinality = equiv-eq eq-compute-sim-cardinality + + unit-sim-cardinality : + mere-equiv (type-Set X) (type-Set Y) → sim-cardinality + unit-sim-cardinality = map-inv-equiv compute-sim-cardinality + + inv-unit-sim-cardinality : + sim-cardinality → mere-equiv (type-Set X) (type-Set Y) + inv-unit-sim-cardinality = pr1 compute-sim-cardinality +``` + +## Properties + +### Equality of cardinalities is equivalent to mere equivalence + +```agda +is-effective-cardinality : + {l : Level} (X Y : Set l) → + (cardinality X ＝ cardinality Y) ≃ mere-equiv (type-Set X) (type-Set Y) +is-effective-cardinality X Y = + ( equiv-trunc-Prop (extensionality-Set X Y)) ∘e + ( is-effective-unit-trunc-Set (Set _) X Y) + +eq-mere-equiv-cardinality : + {l : Level} (X Y : Set l) → + mere-equiv (type-Set X) (type-Set Y) → cardinality X ＝ cardinality Y +eq-mere-equiv-cardinality X Y = map-inv-equiv (is-effective-cardinality X Y) +``` + +### Similarity of cardinals is reflexive + +```agda +refl-sim-cardinality : {l : Level} (X : Set l) → sim-cardinality X X +refl-sim-cardinality X = unit-sim-cardinality X X (refl-mere-equiv (type-Set X)) + +refl-sim-Cardinal : {l : Level} (X : Cardinal l) → sim-Cardinal X X +refl-sim-Cardinal = + apply-dependent-universal-property-trunc-Set' + ( λ X → sim-set-Cardinal X X) + ( refl-sim-cardinality) +``` + +### Similarity of cardinals is symmetric + +```agda +symmetric-sim-cardinality : + {l1 l2 : Level} + (X : Set l1) (Y : Set l2) → + sim-cardinality X Y → sim-cardinality Y X +symmetric-sim-cardinality X Y p = + unit-sim-cardinality Y X + ( symmetric-sim-cardinality' X Y (inv-unit-sim-cardinality X Y p)) + +abstract + symmetric-sim-Cardinal : + {l1 l2 : Level} + (X : Cardinal l1) (Y : Cardinal l2) → + sim-Cardinal X Y → sim-Cardinal Y X + symmetric-sim-Cardinal = + apply-twice-dependent-universal-property-trunc-Set' + ( λ X Y → hom-set-Set (sim-set-Cardinal X Y) (sim-set-Cardinal Y X)) + ( symmetric-sim-cardinality) +``` + +### Similarity of cardinals is transitive + +```agda +transitive-sim-cardinality : + {l1 l2 l3 : Level} + (X : Set l1) (Y : Set l2) (Z : Set l3) → + sim-cardinality Y Z → sim-cardinality X Y → sim-cardinality X Z +transitive-sim-cardinality X Y Z p q = + unit-sim-cardinality X Z + ( transitive-sim-cardinality' X Y Z + ( inv-unit-sim-cardinality Y Z p) + ( inv-unit-sim-cardinality X Y q)) + +abstract + transitive-sim-Cardinal : + {l1 l2 l3 : Level} + (X : Cardinal l1) (Y : Cardinal l2) (Z : Cardinal l3) → + sim-Cardinal Y Z → sim-Cardinal X Y → sim-Cardinal X Z + transitive-sim-Cardinal = + apply-thrice-dependent-universal-property-trunc-Set' + ( λ X Y Z → + hom-set-Set + ( sim-set-Cardinal Y Z) + ( hom-set-Set (sim-set-Cardinal X Y) (sim-set-Cardinal X Z))) + ( transitive-sim-cardinality) +``` + +### Similarity of cardinals is extensional + +```agda +module _ + {l : Level} + where + + sim-eq-Cardinal : + (X Y : Cardinal l) → X ＝ Y → sim-Cardinal X Y + sim-eq-Cardinal X .X refl = refl-sim-Cardinal X + + sim-eq-cardinality : + (X Y : Set l) → cardinality X ＝ cardinality Y → sim-cardinality X Y + sim-eq-cardinality X Y = sim-eq-Cardinal (cardinality X) (cardinality Y) + + eq-sim-cardinality : + (X Y : Set l) → sim-cardinality X Y → cardinality X ＝ cardinality Y + eq-sim-cardinality X Y p = + eq-mere-equiv-cardinality X Y (inv-unit-sim-cardinality X Y p) + + abstract + eq-sim-Cardinal : + (X Y : Cardinal l) → sim-Cardinal X Y → X ＝ Y + eq-sim-Cardinal = + apply-twice-dependent-universal-property-trunc-Set' + ( λ X Y → + hom-set-Set + ( sim-set-Cardinal X Y) + ( set-Prop (Id-Prop (Cardinal-Set l) X Y))) + ( eq-sim-cardinality) + + abstract + antisymmetric-sim-Cardinal : + (X Y : Cardinal l) → sim-Cardinal X Y → sim-Cardinal Y X → X ＝ Y + antisymmetric-sim-Cardinal X Y p _ = eq-sim-Cardinal X Y p +``` + +### Similarity of cardinals is a large similarity relation + +```agda +large-equivalence-relation-Cardinal : Large-Equivalence-Relation (_⊔_) Cardinal +large-equivalence-relation-Cardinal = + λ where + .sim-prop-Large-Equivalence-Relation → sim-prop-Cardinal + .refl-sim-Large-Equivalence-Relation → refl-sim-Cardinal + .symmetric-sim-Large-Equivalence-Relation → symmetric-sim-Cardinal + .transitive-sim-Large-Equivalence-Relation → transitive-sim-Cardinal + +large-similarity-relation-Cardinal : Large-Similarity-Relation (_⊔_) Cardinal +large-similarity-relation-Cardinal = + λ where + .large-equivalence-relation-Large-Similarity-Relation → + large-equivalence-relation-Cardinal + .eq-sim-Large-Similarity-Relation → + eq-sim-Cardinal +``` + +## External links + +- [Cardinality](https://en.wikipedia.org/wiki/Cardinality) at Wikipedia +- [cardinal number](https://ncatlab.org/nlab/show/cardinal+number) at $n$Lab From eff7780101e6fa89b72f2a5ccf06d008fbb48fdd Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Fri, 17 Oct 2025 12:46:47 +0200 Subject: [PATCH 19/50] restore equality of cardinals --- src/set-theory.lagda.md | 1 + 1 file changed, 1 insertion(+) diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index b10e1a3556..8b7a422531 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -51,6 +51,7 @@ open import set-theory.cardinals public open import set-theory.complemented-inequality-cardinals public open import set-theory.countable-sets public open import set-theory.cumulative-hierarchy public +open import set-theory.equality-cardinals public open import set-theory.inequality-cardinals public open import set-theory.infinite-sets public open import set-theory.russells-paradox public From b2c6677ab15cdaeaa00588321db9cb8dca75b035 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Fri, 17 Oct 2025 12:55:35 +0200 Subject: [PATCH 20/50] fix --- src/set-theory/equality-cardinals.lagda.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/set-theory/equality-cardinals.lagda.md b/src/set-theory/equality-cardinals.lagda.md index 3787e99b51..8eea5937e5 100644 --- a/src/set-theory/equality-cardinals.lagda.md +++ b/src/set-theory/equality-cardinals.lagda.md @@ -23,14 +23,14 @@ open import foundation.sets open import foundation.univalence open import foundation.universe-levels -open import set-theory.cardinalities +open import set-theory.cardinals ```
## Idea -Two [cardinals of sets](set-theory.cardinalities.md) `X` and `Y` are +Two [cardinals of sets](set-theory.cardinals.md) `X` and `Y` are {{#concept "similar" Disambiguation="cardinals" Agda=sim-Cardinal}} if there [merely](foundation.inhabited-types.md) is an [equivalence](foundation-core.equivalences.md) between any two representing two From c11a8d0f2e6c9eb34ca6e34d50e89390864c1a1c Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Fri, 17 Oct 2025 12:57:11 +0200 Subject: [PATCH 21/50] fix links --- src/set-theory/inhabited-cardinals.lagda.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/set-theory/inhabited-cardinals.lagda.md b/src/set-theory/inhabited-cardinals.lagda.md index cad4ba4699..592b9ea916 100644 --- a/src/set-theory/inhabited-cardinals.lagda.md +++ b/src/set-theory/inhabited-cardinals.lagda.md @@ -27,10 +27,10 @@ open import set-theory.cardinals ## Idea -A [cardinal of sets](set-theory.cardinals.lagda.md) `κ` is +A [cardinal of sets](set-theory.cardinals.md) `κ` is {{#concept "inhabited" Disambiguation="cardinal" Agda=is-inhabited-Cardinal}}, if any [set](foundation-core.sets.md) in its isomorphism class is -[inhabited](foundation-core.inhabited-types.md). +[inhabited](foundation.inhabited-types.md). ## Definitions From ee8a5cf1cc0572e2e02fa289a92e7b3d513d4578 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Fri, 17 Oct 2025 12:59:45 +0200 Subject: [PATCH 22/50] inhabited cardinals form a set --- src/set-theory/inhabited-cardinals.lagda.md | 8 ++++++++ 1 file changed, 8 insertions(+) diff --git a/src/set-theory/inhabited-cardinals.lagda.md b/src/set-theory/inhabited-cardinals.lagda.md index 592b9ea916..93b06db761 100644 --- a/src/set-theory/inhabited-cardinals.lagda.md +++ b/src/set-theory/inhabited-cardinals.lagda.md @@ -12,6 +12,7 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-types open import foundation.identity-types +open import foundation.subtypes open import foundation.inhabited-types open import foundation.propositional-extensionality open import foundation.propositions @@ -107,6 +108,13 @@ module _ Inhabited-Cardinal : (l : Level) → UU (lsuc l) Inhabited-Cardinal l = Σ (Cardinal l) is-inhabited-Cardinal +is-set-Inhabited-Cardinal : {l : Level} → is-set (Inhabited-Cardinal l) +is-set-Inhabited-Cardinal = + is-set-type-subtype is-inhabited-prop-Cardinal is-set-Cardinal + +Inhabited-Cardinal-Set : (l : Level) → Set (lsuc l) +Inhabited-Cardinal-Set l = (Inhabited-Cardinal l , is-set-Inhabited-Cardinal) + module _ {l : Level} (κ : Inhabited-Cardinal l) where From 4189fea4447ac76aba3349342a74bff7139ec000 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 16 Nov 2025 01:18:53 +0100 Subject: [PATCH 23/50] cardinality inductive sets and many edits --- src/foundation/0-connected-types.lagda.md | 34 +-- src/foundation/axiom-of-choice.lagda.md | 2 +- src/foundation/connected-maps.lagda.md | 112 ++++++-- .../functoriality-truncation.lagda.md | 30 +- src/foundation/mere-embeddings.lagda.md | 31 +++ .../propositional-truncations.lagda.md | 3 +- src/foundation/set-truncations.lagda.md | 8 +- .../truncation-equivalences.lagda.md | 220 +++++++++++---- ...ropositional-truncation-into-sets.lagda.md | 4 +- src/set-theory.lagda.md | 11 +- .../cardinality-inductive-sets.lagda.md | 154 ++++++++++ .../cardinality-projective-sets.lagda.md | 262 +++++++++++++----- .../dependent-products-cardinals.lagda.md | 26 +- .../dependent-sums-cardinals.lagda.md | 86 ++++-- src/set-theory/inhabited-cardinals.lagda.md | 2 +- ...t-truncation-over-finite-products.lagda.md | 33 ++- 16 files changed, 788 insertions(+), 230 deletions(-) create mode 100644 src/set-theory/cardinality-inductive-sets.lagda.md diff --git a/src/foundation/0-connected-types.lagda.md b/src/foundation/0-connected-types.lagda.md index a9f50ba0fe..a156f9b018 100644 --- a/src/foundation/0-connected-types.lagda.md +++ b/src/foundation/0-connected-types.lagda.md @@ -159,22 +159,24 @@ is-trunc-map-ev-is-connected k {A} {B} a H K = ### The dependent universal property of 0-connected types ```agda -equiv-dependent-universal-property-is-0-connected : - {l1 : Level} {A : UU l1} (a : A) → is-0-connected A → - ( {l : Level} (P : A → Prop l) → - ((x : A) → type-Prop (P x)) ≃ type-Prop (P a)) -equiv-dependent-universal-property-is-0-connected a H P = - ( equiv-universal-property-unit (type-Prop (P a))) ∘e - ( equiv-dependent-universal-property-surjection-is-surjective - ( point a) - ( is-surjective-point-is-0-connected a H) - ( P)) - -apply-dependent-universal-property-is-0-connected : - {l1 : Level} {A : UU l1} (a : A) → is-0-connected A → - {l : Level} (P : A → Prop l) → type-Prop (P a) → (x : A) → type-Prop (P x) -apply-dependent-universal-property-is-0-connected a H P = - map-inv-equiv (equiv-dependent-universal-property-is-0-connected a H P) +module _ + {l1 : Level} {A : UU l1} (a : A) (H : is-0-connected A) + {l : Level} (P : A → Prop l) + where + + equiv-dependent-universal-property-is-0-connected : + ((x : A) → type-Prop (P x)) ≃ type-Prop (P a) + equiv-dependent-universal-property-is-0-connected = + ( equiv-universal-property-unit (type-Prop (P a))) ∘e + ( equiv-dependent-universal-property-surjection-is-surjective + ( point a) + ( is-surjective-point-is-0-connected a H) + ( P)) + + apply-dependent-universal-property-is-0-connected : + type-Prop (P a) → (x : A) → type-Prop (P x) + apply-dependent-universal-property-is-0-connected = + map-inv-equiv equiv-dependent-universal-property-is-0-connected ``` ### A type `A` is 0-connected if and only if every fiber inclusion `B a → Σ A B` is surjective diff --git a/src/foundation/axiom-of-choice.lagda.md b/src/foundation/axiom-of-choice.lagda.md index ac46b18ce6..e9397084ae 100644 --- a/src/foundation/axiom-of-choice.lagda.md +++ b/src/foundation/axiom-of-choice.lagda.md @@ -102,7 +102,7 @@ is-set-projective-AC0 : is-set-projective-AC0 ac X A B f h = map-trunc-Prop ( ( map-Σ - ( λ g → ((map-surjection f) ∘ g) ＝ h) + ( λ g → (map-surjection f ∘ g) ＝ h) ( precomp h A) ( λ s H → htpy-postcomp X H h)) ∘ ( section-is-split-surjective (map-surjection f))) diff --git a/src/foundation/connected-maps.lagda.md b/src/foundation/connected-maps.lagda.md index 7613972118..8e3aacd56e 100644 --- a/src/foundation/connected-maps.lagda.md +++ b/src/foundation/connected-maps.lagda.md @@ -54,19 +54,19 @@ if its [fibers](foundation-core.fibers-of-maps.md) are ### Connected maps ```agda -is-connected-map-Prop : - {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} → (A → B) → Prop (l1 ⊔ l2) -is-connected-map-Prop k {B = B} f = - Π-Prop B (λ b → is-connected-Prop k (fiber f b)) - -is-connected-map : - {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} → (A → B) → UU (l1 ⊔ l2) -is-connected-map k f = type-Prop (is-connected-map-Prop k f) - -is-prop-is-connected-map : - {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : A → B) → - is-prop (is-connected-map k f) -is-prop-is-connected-map k f = is-prop-type-Prop (is-connected-map-Prop k f) +module _ + {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : A → B) + where + + is-connected-map-Prop : Prop (l1 ⊔ l2) + is-connected-map-Prop = + Π-Prop B (λ b → is-connected-Prop k (fiber f b)) + + is-connected-map : UU (l1 ⊔ l2) + is-connected-map = type-Prop is-connected-map-Prop + + is-prop-is-connected-map : is-prop is-connected-map + is-prop-is-connected-map = is-prop-type-Prop is-connected-map-Prop ``` ### The type of connected maps between two types @@ -325,6 +325,34 @@ is-connected-map-left-factor k {g = g} {h} H GH z = ( is-connected-equiv' (compute-fiber-comp g h z) (GH z)) ``` +### The class of `k`-equivalences has the 3-for-2 property + +```agda +module _ + {l1 l2 l3 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} {X : UU l3} + (f : A → X) (g : B → X) (h : A → B) (K : f ~ g ∘ h) + where + + abstract + is-connected-map-left-map-triangle : + is-connected-map k h → + is-connected-map k g → + is-connected-map k f + is-connected-map-left-map-triangle H G = + is-connected-map-htpy k K + ( is-connected-map-comp k G H) + + abstract + is-connected-map-right-map-triangle : + is-connected-map k f → + is-connected-map k h → + is-connected-map k g + is-connected-map-right-map-triangle F H = + is-connected-map-left-factor k + ( H) + ( is-connected-map-htpy' k K F) +``` + ### The total map induced by a family of maps is `k`-connected if and only if all maps in the family are `k`-connected ```agda @@ -377,14 +405,56 @@ module _ is-equiv (precomp-Π f (λ b → type-Truncated-Type (P b))) module _ - {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} {f : A → B} + {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} {f : A → B} + (H : is-connected-map k f) where - dependent-universal-property-is-connected-map : - is-connected-map k f → dependent-universal-property-connected-map k f - dependent-universal-property-is-connected-map H P = - is-equiv-precomp-Π-fiber-condition - ( λ b → is-equiv-diagonal-exponential-is-connected (P b) (H b)) + abstract + dependent-universal-property-is-connected-map : + dependent-universal-property-connected-map k f + dependent-universal-property-is-connected-map P = + is-equiv-precomp-Π-fiber-condition + ( λ b → is-equiv-diagonal-exponential-is-connected (P b) (H b)) + + ind-is-connected-map : + {l3 : Level} (P : B → Truncated-Type l3 k) → + ((a : A) → type-Truncated-Type (P (f a))) → + (b : B) → type-Truncated-Type (P b) + ind-is-connected-map P = + map-inv-is-equiv (dependent-universal-property-is-connected-map P) + + compute-ind-is-connected-map : + {l3 : Level} (P : B → Truncated-Type l3 k) → + (g : (a : A) → type-Truncated-Type (P (f a))) → + (x : A) → ind-is-connected-map P g (f x) ＝ g x + compute-ind-is-connected-map P f = + htpy-eq + ( is-section-map-inv-is-equiv + ( dependent-universal-property-is-connected-map P) + ( f)) + +module _ + {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} (f : connected-map k A B) + where + + dup-connected-map : + dependent-universal-property-connected-map k (map-connected-map f) + dup-connected-map = + dependent-universal-property-is-connected-map + ( is-connected-map-connected-map f) + + ind-connected-map : + {l3 : Level} (P : B → Truncated-Type l3 k) → + ((a : A) → type-Truncated-Type (P (map-connected-map f a))) → + (b : B) → type-Truncated-Type (P b) + ind-connected-map = ind-is-connected-map (is-connected-map-connected-map f) + + compute-ind-connected-map : + {l3 : Level} (P : B → Truncated-Type l3 k) → + (g : (a : A) → type-Truncated-Type (P (map-connected-map f a))) → + (x : A) → ind-connected-map P g (map-connected-map f x) ＝ g x + compute-ind-connected-map = + compute-ind-is-connected-map (is-connected-map-connected-map f) module _ {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : connected-map k A B) @@ -397,7 +467,7 @@ module _ pr1 (equiv-dependent-universal-property-is-connected-map P) = precomp-Π (map-connected-map f) (λ b → type-Truncated-Type (P b)) pr2 (equiv-dependent-universal-property-is-connected-map P) = - dependent-universal-property-is-connected-map k + dependent-universal-property-is-connected-map ( is-connected-map-connected-map f) ( P) ``` @@ -510,7 +580,7 @@ abstract ( precomp-Π f (λ b → type-Truncated-Type (P b))) is-trunc-map-precomp-Π-is-connected-map k neg-two-𝕋 H P = is-contr-map-is-equiv - ( dependent-universal-property-is-connected-map k H + ( dependent-universal-property-is-connected-map H ( λ b → pair ( type-Truncated-Type (P b)) diff --git a/src/foundation/functoriality-truncation.lagda.md b/src/foundation/functoriality-truncation.lagda.md index a4b988d7c2..f3df0ea202 100644 --- a/src/foundation/functoriality-truncation.lagda.md +++ b/src/foundation/functoriality-truncation.lagda.md @@ -20,6 +20,7 @@ open import foundation-core.contractible-types open import foundation-core.equivalences open import foundation-core.function-types open import foundation-core.homotopies +open import foundation-core.identity-types open import foundation-core.retractions open import foundation-core.retracts-of-types open import foundation-core.sections @@ -194,13 +195,40 @@ module _ (x : X) → type-trunc k ((x : X) → Y x) → type-trunc k (Y x) map-trunc-ev x = map-trunc k (ev x) + compute-map-trunc-ev : + {X : UU l1} {Y : X → UU l2} → + (x : X) (f : (x : X) → Y x) → + map-trunc-ev x (unit-trunc f) ＝ unit-trunc (f x) + compute-map-trunc-ev x f = coherence-square-map-trunc k (ev x) f +``` + +### Distributivity of truncations over dependent products + +```agda +module _ + {l1 l2 : Level} (k : 𝕋) + where + map-distributive-trunc-Π : {X : UU l1} (Y : X → UU l2) → type-trunc k ((x : X) → Y x) → (x : X) → type-trunc k (Y x) - map-distributive-trunc-Π Y f x = map-trunc-ev x f + map-distributive-trunc-Π Y f x = map-trunc-ev k x f + + compute-distributive-trunc-Π : + {X : UU l1} {Y : X → UU l2} → + (f : (x : X) → Y x) (x : X) → + map-distributive-trunc-Π Y (unit-trunc f) x ＝ unit-trunc (f x) + compute-distributive-trunc-Π f x = compute-map-trunc-ev k x f map-distributive-trunc-function-type : (X : UU l1) (Y : UU l2) → type-trunc k (X → Y) → X → type-trunc k Y map-distributive-trunc-function-type X Y = map-distributive-trunc-Π (λ _ → Y) + + compute-distributive-trunc-function-type : + {X : UU l1} {Y : UU l2} → + (f : X → Y) (x : X) → + map-distributive-trunc-function-type X Y (unit-trunc f) x ＝ + unit-trunc (f x) + compute-distributive-trunc-function-type = compute-distributive-trunc-Π ``` diff --git a/src/foundation/mere-embeddings.lagda.md b/src/foundation/mere-embeddings.lagda.md index 931180da64..68bc5beb44 100644 --- a/src/foundation/mere-embeddings.lagda.md +++ b/src/foundation/mere-embeddings.lagda.md @@ -8,10 +8,13 @@ module foundation.mere-embeddings where ```agda open import foundation.cantor-schroder-bernstein-escardo +open import foundation.dependent-pair-types open import foundation.embeddings +open import foundation.functoriality-dependent-function-types open import foundation.functoriality-propositional-truncation open import foundation.law-of-excluded-middle open import foundation.mere-equivalences +open import foundation.projective-types open import foundation.propositional-truncations open import foundation.universe-levels @@ -72,3 +75,31 @@ antisymmetric-mere-emb : antisymmetric-mere-emb lem = map-binary-trunc-Prop (Cantor-Schröder-Bernstein-Escardó lem) ``` + +### Dependent sums over projective types preserve mere embeddings + +```agda +module _ + {l1 l2 l3 : Level} {X : UU l1} {Y : X → UU l2} {Z : X → UU l3} + where + + mere-emb-tot : + is-projective-Level' (l2 ⊔ l3) X → + ((x : X) → mere-emb (Y x) (Z x)) → + mere-emb (Σ X Y) (Σ X Z) + mere-emb-tot H e = map-trunc-Prop emb-tot (H _ e) +``` + +### Dependent products over projective types preserve mere embeddings + +```agda +module _ + {l1 l2 l3 : Level} {X : UU l1} {Y : X → UU l2} {Z : X → UU l3} + where + + mere-emb-Π : + is-projective-Level' (l2 ⊔ l3) X → + ((x : X) → mere-emb (Y x) (Z x)) → + mere-emb ((x : X) → Y x) ((x : X) → Z x) + mere-emb-Π H e = map-trunc-Prop emb-Π (H _ e) +``` diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md index f38f2062b3..185b30b298 100644 --- a/src/foundation/propositional-truncations.lagda.md +++ b/src/foundation/propositional-truncations.lagda.md @@ -68,8 +68,7 @@ trunc-Prop = trunc neg-one-𝕋 **Notation.** The [box drawings double vertical](https://codepoints.net/U+2551) symbol `║` in the propositional truncation notation `║_║₋₁` can be inserted with -`agda-input` using the escape sequence `\--=` and selecting the second item in -the list. +`agda-input` using the escape sequence `\--=2`. ## Properties diff --git a/src/foundation/set-truncations.lagda.md b/src/foundation/set-truncations.lagda.md index 3059816359..ce22a33fb3 100644 --- a/src/foundation/set-truncations.lagda.md +++ b/src/foundation/set-truncations.lagda.md @@ -68,6 +68,9 @@ is-set-type-trunc-Set = is-trunc-type-trunc unit-trunc-Set : {l : Level} {A : UU l} → A → type-trunc-Set A unit-trunc-Set = unit-trunc +unit-trunc-Set' : {l : Level} (A : UU l) → A → type-trunc-Set A +unit-trunc-Set' A = unit-trunc-Set + is-set-truncation-trunc-Set : {l1 : Level} (A : UU l1) → is-set-truncation (trunc-Set A) unit-trunc-Set is-set-truncation-trunc-Set A = is-truncation-trunc @@ -78,8 +81,7 @@ is-set-truncation-trunc-Set A = is-truncation-trunc **Notation.** The [box drawings double vertical](https://codepoints.net/U+2551) symbol `║` in the set truncation notation `║_║₀` can be inserted with -`agda-input` using the escape sequence `\--=` and selecting the second item in -the list. +`agda-input` using the escape sequence `\--=2`. ## Properties @@ -465,7 +467,7 @@ module _ equiv-unit-trunc-set = equiv-unit-trunc A ``` -### Distributive of set truncation over coproduct +### Distributivity of set truncation over coproduct ```agda module _ diff --git a/src/foundation/truncation-equivalences.lagda.md b/src/foundation/truncation-equivalences.lagda.md index da39546ed6..78b0c0162d 100644 --- a/src/foundation/truncation-equivalences.lagda.md +++ b/src/foundation/truncation-equivalences.lagda.md @@ -12,6 +12,8 @@ open import foundation.connected-maps open import foundation.connected-types open import foundation.contractible-types open import foundation.dependent-pair-types +open import foundation.equivalences-arrows +open import foundation.function-extensionality open import foundation.functoriality-truncation open import foundation.identity-types open import foundation.precomposition-functions-into-subuniverses @@ -60,7 +62,7 @@ truncation-equivalence : truncation-equivalence k A B = Σ (A → B) (is-truncation-equivalence k) module _ - {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} + {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} (f : truncation-equivalence k A B) where @@ -85,39 +87,17 @@ module _ ### A map `f : A → B` is a `k`-equivalence if and only if `- ∘ f : (B → X) → (A → X)` is an equivalence for every `k`-truncated type `X` ```agda -is-equiv-precomp-is-truncation-equivalence : - {l1 l2 l3 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : A → B) → - is-truncation-equivalence k f → - (X : Truncated-Type l3 k) → is-equiv (precomp f (type-Truncated-Type X)) -is-equiv-precomp-is-truncation-equivalence k f H X = - is-equiv-bottom-is-equiv-top-square - ( precomp unit-trunc (type-Truncated-Type X)) - ( precomp unit-trunc (type-Truncated-Type X)) - ( precomp (map-trunc k f) (type-Truncated-Type X)) - ( precomp f (type-Truncated-Type X)) - ( precomp-coherence-square-maps - ( unit-trunc) - ( f) - ( map-trunc k f) - ( unit-trunc) - ( inv-htpy (coherence-square-map-trunc k f)) - ( type-Truncated-Type X)) - ( is-truncation-trunc X) - ( is-truncation-trunc X) - ( is-equiv-precomp-is-equiv (map-trunc k f) H (type-Truncated-Type X)) - -is-truncation-equivalence-is-equiv-precomp : - {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : A → B) → - ( (l : Level) (X : Truncated-Type l k) → - is-equiv (precomp f (type-Truncated-Type X))) → - is-truncation-equivalence k f -is-truncation-equivalence-is-equiv-precomp k {A} {B} f H = - is-equiv-is-equiv-precomp-Truncated-Type k - ( trunc k A) - ( trunc k B) - ( map-trunc k f) - ( λ X → - is-equiv-top-is-equiv-bottom-square +module _ + {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} {f : A → B} + where + + abstract + is-equiv-precomp-is-truncation-equivalence : + is-truncation-equivalence k f → + {l : Level} (X : Truncated-Type l k) → + is-equiv (precomp f (type-Truncated-Type X)) + is-equiv-precomp-is-truncation-equivalence H X = + is-equiv-bottom-is-equiv-top-square ( precomp unit-trunc (type-Truncated-Type X)) ( precomp unit-trunc (type-Truncated-Type X)) ( precomp (map-trunc k f) (type-Truncated-Type X)) @@ -131,7 +111,79 @@ is-truncation-equivalence-is-equiv-precomp k {A} {B} f H = ( type-Truncated-Type X)) ( is-truncation-trunc X) ( is-truncation-trunc X) - ( H _ X)) + ( is-equiv-precomp-is-equiv (map-trunc k f) H (type-Truncated-Type X)) + + abstract + is-truncation-equivalence-is-equiv-precomp : + ( (l : Level) (X : Truncated-Type l k) → + is-equiv (precomp f (type-Truncated-Type X))) → + is-truncation-equivalence k f + is-truncation-equivalence-is-equiv-precomp H = + is-equiv-is-equiv-precomp-Truncated-Type k + ( trunc k A) + ( trunc k B) + ( map-trunc k f) + ( λ X → + is-equiv-top-is-equiv-bottom-square + ( precomp unit-trunc (type-Truncated-Type X)) + ( precomp unit-trunc (type-Truncated-Type X)) + ( precomp (map-trunc k f) (type-Truncated-Type X)) + ( precomp f (type-Truncated-Type X)) + ( precomp-coherence-square-maps + ( unit-trunc) + ( f) + ( map-trunc k f) + ( unit-trunc) + ( inv-htpy (coherence-square-map-trunc k f)) + ( type-Truncated-Type X)) + ( is-truncation-trunc X) + ( is-truncation-trunc X) + ( H _ X)) +``` + +### The recursion principle of truncation equivalences + +```agda +module _ + {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} {f : A → B} + (H : is-truncation-equivalence k f) + where + + rec-is-truncation-equivalence : + {l : Level} (X : Truncated-Type l k) → + (A → type-Truncated-Type X) → B → type-Truncated-Type X + rec-is-truncation-equivalence X = + map-inv-is-equiv (is-equiv-precomp-is-truncation-equivalence H X) + + compute-rec-is-truncation-equivalence : + {l : Level} (X : Truncated-Type l k) → + (g : A → type-Truncated-Type X) (x : A) → + rec-is-truncation-equivalence X g (f x) ＝ g x + compute-rec-is-truncation-equivalence X g = + htpy-eq + ( is-section-map-inv-is-equiv + ( is-equiv-precomp-is-truncation-equivalence H X) + ( g)) + +module _ + {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} + (f : truncation-equivalence k A B) + where + + rec-truncation-equivalence : + {l : Level} (X : Truncated-Type l k) → + (A → type-Truncated-Type X) → B → type-Truncated-Type X + rec-truncation-equivalence = + rec-is-truncation-equivalence + ( is-truncation-equivalence-truncation-equivalence f) + + compute-rec-truncation-equivalence : + {l : Level} (X : Truncated-Type l k) → + (g : A → type-Truncated-Type X) (x : A) → + rec-truncation-equivalence X g (map-truncation-equivalence f x) ＝ g x + compute-rec-truncation-equivalence = + compute-rec-is-truncation-equivalence + ( is-truncation-equivalence-truncation-equivalence f) ``` ### Equivalences are `k`-equivalences for all `k` @@ -182,8 +234,21 @@ module _ is-truncation-equivalence-is-connected-map : is-connected-map k f → is-truncation-equivalence k f is-truncation-equivalence-is-connected-map c = - is-truncation-equivalence-is-equiv-precomp k f - ( λ l X → dependent-universal-property-is-connected-map k c (λ _ → X)) + is-truncation-equivalence-is-equiv-precomp + ( λ l X → dependent-universal-property-is-connected-map c (λ _ → X)) +``` + +### The unit maps of `k`-truncation are `k`-equivalences + +```agda +module _ + {l : Level} {k : 𝕋} {A : UU l} + where + + is-truncation-equivalence-unit-trunc : + is-truncation-equivalence k (unit-trunc {k = k} {A}) + is-truncation-equivalence-unit-trunc = + is-truncation-equivalence-is-equiv-precomp (λ l → is-truncation-trunc) ``` ### The `k`-equivalences are closed under composition @@ -209,16 +274,16 @@ module _ truncation-equivalence k A B → truncation-equivalence k A C pr1 (truncation-equivalence-comp g f) = - map-truncation-equivalence k g ∘ map-truncation-equivalence k f + map-truncation-equivalence g ∘ map-truncation-equivalence f pr2 (truncation-equivalence-comp g f) = is-truncation-equivalence-comp - ( map-truncation-equivalence k g) - ( map-truncation-equivalence k f) - ( is-truncation-equivalence-truncation-equivalence k f) - ( is-truncation-equivalence-truncation-equivalence k g) + ( map-truncation-equivalence g) + ( map-truncation-equivalence f) + ( is-truncation-equivalence-truncation-equivalence f) + ( is-truncation-equivalence-truncation-equivalence g) ``` -### The class of `k`-equivalences has the 3-for-2 property +### Cancellation property of truncation equivalences ```agda module _ @@ -250,6 +315,43 @@ module _ ( e)) ``` +### The class of `k`-equivalences has the 3-for-2 property + +```agda +module _ + {l1 l2 l3 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} {X : UU l3} + (f : A → X) (g : B → X) (h : A → B) (K : f ~ g ∘ h) + where + + abstract + is-truncation-equivalence-top-map-triangle : + is-truncation-equivalence k g → + is-truncation-equivalence k f → + is-truncation-equivalence k h + is-truncation-equivalence-top-map-triangle G F = + is-truncation-equivalence-right-factor g h + ( is-truncation-equivalence-htpy' k K F) + ( G) + + abstract + is-truncation-equivalence-left-map-triangle : + is-truncation-equivalence k h → + is-truncation-equivalence k g → + is-truncation-equivalence k f + is-truncation-equivalence-left-map-triangle H G = + is-truncation-equivalence-htpy k K + ( is-truncation-equivalence-comp g h H G) + + abstract + is-truncation-equivalence-right-map-triangle : + is-truncation-equivalence k f → + is-truncation-equivalence k h → + is-truncation-equivalence k g + is-truncation-equivalence-right-map-triangle F = + is-truncation-equivalence-left-factor g h + ( is-truncation-equivalence-htpy' k K F) +``` + ### Sections of `k`-equivalences are `k`-equivalences ```agda @@ -341,8 +443,7 @@ module _ is-truncation-equivalence-map-Σ-map-base-unit-trunc' : is-truncation-equivalence k map-Σ-map-base-unit-trunc' is-truncation-equivalence-map-Σ-map-base-unit-trunc' = - is-truncation-equivalence-is-equiv-precomp k - ( map-Σ-map-base-unit-trunc') + is-truncation-equivalence-is-equiv-precomp ( λ l (Y , TY) → is-equiv-equiv ( equiv-ev-pair) @@ -365,8 +466,7 @@ module _ is-truncation-equivalence-map-Σ-map-base-unit-trunc : is-truncation-equivalence k map-Σ-map-base-unit-trunc is-truncation-equivalence-map-Σ-map-base-unit-trunc = - is-truncation-equivalence-is-equiv-precomp k - ( map-Σ-map-base-unit-trunc) + is-truncation-equivalence-is-equiv-precomp ( λ l (Y , TY) → is-equiv-equiv {f = precomp (λ x → unit-trunc (pr1 x) , pr2 x) Y} @@ -451,7 +551,7 @@ module _ type-trunc k (fiber f b) ≃ type-trunc k (fiber (map-trunc (succ-𝕋 k) f) (unit-trunc b)) equiv-trunc-fiber-map-trunc-fiber = - equiv-trunc-truncation-equivalence k + equiv-trunc-truncation-equivalence ( truncation-equivalence-fiber-map-trunc-fiber) ``` @@ -478,15 +578,15 @@ module _ truncation-equivalence k A B → is-connected k B → is-connected k A is-connected-truncation-equivalence-is-connected f = is-connected-is-truncation-equivalence-is-connected - ( map-truncation-equivalence k f) - ( is-truncation-equivalence-truncation-equivalence k f) + ( map-truncation-equivalence f) + ( is-truncation-equivalence-truncation-equivalence f) is-connected-truncation-equivalence-is-connected' : truncation-equivalence k A B → is-connected k A → is-connected k B is-connected-truncation-equivalence-is-connected' f = is-connected-is-truncation-equivalence-is-connected' - ( map-truncation-equivalence k f) - ( is-truncation-equivalence-truncation-equivalence k f) + ( map-truncation-equivalence f) + ( is-truncation-equivalence-truncation-equivalence f) ``` ### Every `(k+1)`-equivalence is `k`-connected @@ -684,6 +784,22 @@ module _ ( is-connected-map-section-is-truncation-equivalence-succ k (s , h) e) ``` +### An `n`-truncation equivalence between `n`-truncated types is an equivalence + +```agda +module _ + {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} {f : A → B} + where + + is-equiv-is-truncation-equivalence : + is-trunc k A → is-trunc k B → is-truncation-equivalence k f → is-equiv f + is-equiv-is-truncation-equivalence a b = + is-equiv-source-is-equiv-target-equiv-arrow f (map-trunc k f) + ( equiv-unit-trunc (A , a) , + equiv-unit-trunc (B , b) , + inv-htpy (coherence-square-map-trunc k f)) +``` + ## References - The notion of `k`-equivalence is a special case of the notion of diff --git a/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md b/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md index 27f1fb5de1..137eeeed95 100644 --- a/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md +++ b/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md @@ -42,9 +42,7 @@ is-weakly-constant-map-precomp-unit-trunc-Prop : {l1 l2 : Level} {A : UU l1} {B : UU l2} (g : type-trunc-Prop A → B) → is-weakly-constant-map (g ∘ unit-trunc-Prop) is-weakly-constant-map-precomp-unit-trunc-Prop g x y = - ap - ( g) - ( eq-is-prop (is-prop-type-trunc-Prop)) + ap g (eq-is-prop (is-prop-type-trunc-Prop)) precomp-universal-property-set-quotient-trunc-Prop : {l1 l2 : Level} {A : UU l1} (B : Set l2) → diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index bf75562e57..1f0154f2b9 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -53,6 +53,7 @@ open import set-theory.baire-space public open import set-theory.bounded-increasing-binary-sequences public open import set-theory.cantor-space public open import set-theory.cantors-diagonal-argument public +open import set-theory.cardinality-inductive-sets public open import set-theory.cardinality-projective-sets public open import set-theory.cardinals public open import set-theory.complemented-inequality-cardinals public @@ -61,19 +62,17 @@ open import set-theory.cumulative-hierarchy public open import set-theory.dependent-products-cardinals public open import set-theory.dependent-sums-cardinals public open import set-theory.equality-cardinals public -open import set-theory.inequality-cardinals public -open import set-theory.infinite-sets public -open import set-theory.inhabited-cardinals public -open import set-theory.russells-paradox public -open import set-theory.strict-complemented-boundedness-cardinalities public -open import set-theory.strict-complemented-inequality-cardinalities public open import set-theory.finite-elements-increasing-binary-sequences public open import set-theory.inclusion-natural-numbers-increasing-binary-sequences public open import set-theory.increasing-binary-sequences public +open import set-theory.inequality-cardinals public open import set-theory.inequality-increasing-binary-sequences public open import set-theory.infinite-sets public +open import set-theory.inhabited-cardinals public open import set-theory.positive-elements-increasing-binary-sequences public open import set-theory.russells-paradox public +open import set-theory.strict-complemented-boundedness-cardinalities public +open import set-theory.strict-complemented-inequality-cardinalities public open import set-theory.strict-lower-bounds-increasing-binary-sequences public open import set-theory.uncountable-sets public open import set-theory.zero-cardinal public diff --git a/src/set-theory/cardinality-inductive-sets.lagda.md b/src/set-theory/cardinality-inductive-sets.lagda.md new file mode 100644 index 0000000000..981e877d97 --- /dev/null +++ b/src/set-theory/cardinality-inductive-sets.lagda.md @@ -0,0 +1,154 @@ +# Cardinality-inductive sets + +```agda +module set-theory.cardinality-inductive-sets where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.dependent-pair-types +open import foundation.embeddings +open import foundation.equivalences +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.functoriality-truncation +open import foundation.identity-types +open import foundation.injective-maps +open import foundation.mere-equivalences +open import foundation.retractions +open import foundation.retracts-of-types +open import foundation.set-truncations +open import foundation.sets +open import foundation.truncation-levels +open import foundation.truncations +open import foundation.universe-levels + +open import set-theory.cardinals +``` + +
+ +## Idea + +For every type $X$ there is a map $║X → Set║₀ → (X → \mathrm{Cardinal})$. We +call [sets](foundation-core.sets.md) $X$ for which this map is a retract +{{#concept "cardinality-inductive" Disamibguation="sets" Agda=Cardinality-Inductive-Set}}. +Over such sets we may form +[dependent sum](set-theory.dependent-sums-cardinals.md) and +[dependent product](set-theory.dependent-products-cardinals.md) +[cardinals](set-theory.cardinals.md). + +Note that classically, the universe of sets is itself a set, and so trivially +$║X → \mathrm{Set}║₀ ≃ (X → ║\mathrm{Set}║₀)$. However, with univalence, the +universe of sets $\mathrm{Set}$ brandishes higher structure, and its set +truncation $║Set║₀$ presents cardinals. + +```text + (X → Set) + / \ + surj ∨ \ + ∨ ∨ + ║X → Set║₀ ╰-----> (X → Cardinality) + <<--- +``` + +**Terminology.** This is nonstandard terminology and may be subject to change. + +## Definition + +```agda +module _ + {l1 : Level} (l2 : Level) (X : Set l1) + where + + is-cardinality-inductive-set-Level : UU (l1 ⊔ lsuc l2) + is-cardinality-inductive-set-Level = + retraction + ( map-distributive-trunc-function-type zero-𝕋 (type-Set X) (Set l2)) +``` + +### The universe of cardinality-inductive sets at a universe level + +```agda +Cardinality-Inductive-Set : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) +Cardinality-Inductive-Set l1 l2 = + Σ (Set l1) (is-cardinality-inductive-set-Level l2) + +module _ + {l1 l2 : Level} (X : Cardinality-Inductive-Set l1 l2) + where + + set-Cardinality-Inductive-Set : Set l1 + set-Cardinality-Inductive-Set = pr1 X + + type-Cardinality-Inductive-Set : UU l1 + type-Cardinality-Inductive-Set = type-Set set-Cardinality-Inductive-Set + + is-set-type-Cardinality-Inductive-Set : + is-set type-Cardinality-Inductive-Set + is-set-type-Cardinality-Inductive-Set = + is-set-type-Set set-Cardinality-Inductive-Set + + is-cardinality-inductive-Cardinality-Inductive-Set : + is-cardinality-inductive-set-Level l2 set-Cardinality-Inductive-Set + is-cardinality-inductive-Cardinality-Inductive-Set = pr2 X + + unit-Cardinality-Inductive-Set : + ( type-Cardinality-Inductive-Set → Cardinal l2) → + ║ (type-Cardinality-Inductive-Set → Set l2) ║₀ + unit-Cardinality-Inductive-Set = + map-retraction + ( map-distributive-trunc-function-type zero-𝕋 + ( type-Cardinality-Inductive-Set) + ( Set l2)) + ( is-cardinality-inductive-Cardinality-Inductive-Set) + + is-retraction-unit-Cardinality-Inductive-Set : + is-retraction + ( map-distributive-trunc-function-type zero-𝕋 + ( type-Cardinality-Inductive-Set) + ( Set l2)) + ( unit-Cardinality-Inductive-Set) + is-retraction-unit-Cardinality-Inductive-Set = + is-retraction-map-retraction + ( map-distributive-trunc-function-type zero-𝕋 + ( type-Cardinality-Inductive-Set) + ( Set l2)) + ( is-cardinality-inductive-Cardinality-Inductive-Set) + + retract-Cardinality-Inductive-Set : + ║ (type-Cardinality-Inductive-Set → Set l2) ║₀ retract-of + ( type-Cardinality-Inductive-Set → Cardinal l2) + retract-Cardinality-Inductive-Set = + ( ( map-distributive-trunc-function-type + ( zero-𝕋) + ( type-Cardinality-Inductive-Set) + ( Set l2)) , + ( is-cardinality-inductive-Cardinality-Inductive-Set)) + + compute-unit-Cardinality-Inductive-Set : + (K : type-Cardinality-Inductive-Set → Set l2) → + unit-Cardinality-Inductive-Set (cardinality ∘ K) ＝ unit-trunc-Set K + compute-unit-Cardinality-Inductive-Set K = + equational-reasoning + unit-Cardinality-Inductive-Set (cardinality ∘ K) + ＝ unit-Cardinality-Inductive-Set + ( map-distributive-trunc-function-type zero-𝕋 + ( type-Cardinality-Inductive-Set) + ( Set l2) + ( unit-trunc K)) + by + ap + ( unit-Cardinality-Inductive-Set) + ( inv (eq-htpy (compute-distributive-trunc-function-type zero-𝕋 K))) + ＝ unit-trunc K + by is-retraction-unit-Cardinality-Inductive-Set (unit-trunc K) +``` + +## See also + +- In + [Distributivity of set truncation over finite products](univalent-combinatorics.distributivity-of-set-truncation-over-finite-products.md) + it is demonstrated that finite types are cardinality-inductive. diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index af75900a24..78680fdf5c 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -7,16 +7,29 @@ module set-theory.cardinality-projective-sets where
Imports ```agda +open import foundation.0-connected-maps +open import foundation.action-on-identifications-functions +open import foundation.connected-maps open import foundation.dependent-pair-types +open import foundation.embeddings open import foundation.equivalences +open import foundation.function-extensionality open import foundation.function-types open import foundation.functoriality-truncation +open import foundation.homotopies open import foundation.identity-types +open import foundation.injective-maps open import foundation.mere-equivalences -open import foundation.sections +open import foundation.postcomposition-functions +open import foundation.retractions +open import foundation.retracts-of-types open import foundation.set-truncations open import foundation.sets +open import foundation.surjective-maps +open import foundation.truncated-types +open import foundation.truncation-equivalences open import foundation.truncation-levels +open import foundation.truncations open import foundation.universe-levels open import set-theory.cardinals @@ -26,26 +39,24 @@ open import set-theory.cardinals ## Idea -For every type $X$ there is a map $║ X → Set ║₀ → (X → Cardinal)$. We call sets -$X$ for which this map has a section -{{#concept "cardinality-projective" Disamibguation="sets"}}. Over such types we -may form [dependent sum](set-theory.dependent-sums-cardinals.md) and -[dependent product](set-theory.dependent-products-cardinals.md) cardinals. +A set $I$ is +{{#concept "cardinality-projective" Disamibguation="sets" Agda=Cardinality-Projective-Set}} +if it is projective and the postcomposition map +$$\mathrm{cardinality} ∘ {-} : (I → Set) → (I → \mathrm{Cardinal})$$ is +0-connected. -Note that classically, the universe of sets is itself a set, and so trivially -$║ X → Set ║₀ → (X → ║ Set ║₀)$ is an equivalence. However, with univalence, -meaning that $║ Set ║₀$ is itself the type of cardinalities, this is no longer -the case. +## Definitions -**Terminology.** This is nonstandard terminology and is subject to change. - -## Definition +### The predicate of being cardinality-projective at a universe level ```agda -is-cardinality-projective-set-Level : - {l1 : Level} (l2 : Level) (X : Set l1) → UU (l1 ⊔ lsuc l2) -is-cardinality-projective-set-Level l2 X = - section (map-distributive-trunc-function-type zero-𝕋 (type-Set X) (Set l2)) +module _ + {l1 : Level} (l2 : Level) (I : Set l1) + where + + is-cardinality-projective-set-Level : UU (l1 ⊔ lsuc l2) + is-cardinality-projective-set-Level = + is-0-connected-map (postcomp (type-Set I) (cardinality {l2})) ``` ### The universe of cardinality-projective sets at a universe level @@ -56,11 +67,11 @@ Cardinality-Projective-Set l1 l2 = Σ (Set l1) (is-cardinality-projective-set-Level l2) module _ - {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + {l1 l2 : Level} (I : Cardinality-Projective-Set l1 l2) where set-Cardinality-Projective-Set : Set l1 - set-Cardinality-Projective-Set = pr1 X + set-Cardinality-Projective-Set = pr1 I type-Cardinality-Projective-Set : UU l1 type-Cardinality-Projective-Set = type-Set set-Cardinality-Projective-Set @@ -72,66 +83,183 @@ module _ is-cardinality-projective-Cardinality-Projective-Set : is-cardinality-projective-set-Level l2 set-Cardinality-Projective-Set - is-cardinality-projective-Cardinality-Projective-Set = pr2 X - - map-section-Cardinality-Projective-Set : - ( type-Cardinality-Projective-Set → Cardinal l2) → - ║ (type-Cardinality-Projective-Set → Set l2) ║₀ - map-section-Cardinality-Projective-Set = - map-section - ( map-distributive-trunc-function-type zero-𝕋 - ( type-Cardinality-Projective-Set) - ( Set l2)) - ( is-cardinality-projective-Cardinality-Projective-Set) + is-cardinality-projective-Cardinality-Projective-Set = pr2 I - is-section-map-section-Cardinality-Projective-Set : - is-section - ( map-distributive-trunc-function-type zero-𝕋 - ( type-Cardinality-Projective-Set) - ( Set l2)) - ( map-section-Cardinality-Projective-Set) - is-section-map-section-Cardinality-Projective-Set = - is-section-map-section - ( map-distributive-trunc-function-type zero-𝕋 - ( type-Cardinality-Projective-Set) - ( Set l2)) + is-set-equivalence-postcomp-cardinality-type-Cardinality-Projective-Set : + is-truncation-equivalence zero-𝕋 + ( postcomp type-Cardinality-Projective-Set (cardinality {l2})) + is-set-equivalence-postcomp-cardinality-type-Cardinality-Projective-Set = + is-truncation-equivalence-is-connected-map + ( postcomp type-Cardinality-Projective-Set cardinality) ( is-cardinality-projective-Cardinality-Projective-Set) + + constr-Cardinality-Projective-Set : + {l : Level} → + ((type-Cardinality-Projective-Set → Set l2) → Cardinal l) → + (type-Cardinality-Projective-Set → Cardinal l2) → + Cardinal l + constr-Cardinality-Projective-Set {l} = + rec-is-truncation-equivalence + ( is-set-equivalence-postcomp-cardinality-type-Cardinality-Projective-Set) + ( Cardinal-Set l) + + compute-constr-Cardinality-Projective-Set : + {l : Level} → + (T : (type-Cardinality-Projective-Set → Set l2) → Cardinal l) + (Y : pr1 set-Cardinality-Projective-Set → Set l2) → + constr-Cardinality-Projective-Set T (cardinality ∘ Y) ＝ T Y + compute-constr-Cardinality-Projective-Set {l} = + compute-rec-is-truncation-equivalence + ( is-set-equivalence-postcomp-cardinality-type-Cardinality-Projective-Set) + ( Cardinal-Set l) ``` +## Properties + +### Distributive property of cardinality-projective sets + +A set `I` is cardinality-projective if and only if the distributive map + ```text - (X → Set) ↖ - | \ \ - | \ \ - ∨ ∨ \ - (X → Card) -> ║ X → Set ║₀ - - - ∘ η - (X → Set) - | \ - η | \ - ∘ η - ∨ ↘ - ║X → Set║₀ ---> (X → Card) - <--- - 𝓈 + ║I → Set║₀ → (I → Cardinal) ``` -Naturality says +is an equivalence. + +**Proof.** We have the commuting triangle ```text - ev x - (X → Set) -----> Set - | | - η | | η - ∨ ║ev x║₀ ∨ - ║X → Set║₀ ----> Card + (I → Set) + / \ + / \ + ∨ ∨ + ║I → Set║₀ -----> (I → Cardinal) ``` +where the left vertical map is 0-connected hence by the cancellation/composition +property of 0-connected maps, `I` is cardinality-projective if and only if the +bottom horizontal map is. But the horizontal map is a map between sets, and so +is 0-connected if and only if it is an equivalence. ∎ + ```agda - compute-map-section-at-cardinality-Cardinality-Projective-Set : - (K : type-Cardinality-Projective-Set → Set l2) → - map-section-Cardinality-Projective-Set (cardinality ∘ K) ＝ unit-trunc-Set K - compute-map-section-at-cardinality-Cardinality-Projective-Set = TODO - where postulate TODO : _ +module _ + {l1 l2 : Level} (I : Set l1) + where + + is-set-equivalence-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set : + is-truncation-equivalence zero-𝕋 + ( postcomp (type-Set I) (cardinality {l2})) → + is-truncation-equivalence zero-𝕋 + ( map-distributive-trunc-function-type zero-𝕋 (type-Set I) (Set l2)) + is-set-equivalence-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set + H = + is-truncation-equivalence-right-map-triangle + ( postcomp (type-Set I) cardinality) + ( map-distributive-trunc-function-type zero-𝕋 (type-Set I) (Set l2)) + ( unit-trunc-Set) + ( λ f → inv (eq-htpy (compute-distributive-trunc-function-type zero-𝕋 f))) + ( H) + ( is-truncation-equivalence-unit-trunc) + + is-equiv-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set : + is-truncation-equivalence zero-𝕋 + ( postcomp (type-Set I) (cardinality {l2})) → + is-equiv (map-distributive-trunc-function-type zero-𝕋 (type-Set I) (Set l2)) + is-equiv-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set + H = + is-equiv-is-truncation-equivalence + ( is-set-type-trunc-Set) + ( is-set-function-type is-set-Cardinal) + ( is-set-equivalence-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set + ( H)) + + is-equiv-map-distributive-trunc-set-is-cardinality-projective-set : + is-cardinality-projective-set-Level l2 I → + is-equiv (map-distributive-trunc-function-type zero-𝕋 (type-Set I) (Set l2)) + is-equiv-map-distributive-trunc-set-is-cardinality-projective-set H = + is-equiv-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set + ( is-truncation-equivalence-is-connected-map _ H) + + is-cardinality-projective-set-is-is-equiv-map-distributive-trunc-set : + is-equiv + ( map-distributive-trunc-function-type zero-𝕋 (type-Set I) (Set l2)) → + is-cardinality-projective-set-Level l2 I + is-cardinality-projective-set-is-is-equiv-map-distributive-trunc-set H = + is-connected-map-left-map-triangle + ( postcomp (type-Set I) cardinality) + ( map-distributive-trunc-function-type zero-𝕋 (type-Set I) (Set l2)) + ( unit-trunc-Set) + ( λ f → inv (eq-htpy (compute-distributive-trunc-function-type zero-𝕋 f))) + ( is-connected-map-unit-trunc zero-𝕋) + ( is-connected-map-is-equiv H) + +is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set : + {l1 l2 : Level} (I : Cardinality-Projective-Set l1 l2) → + is-equiv + ( map-distributive-trunc-function-type zero-𝕋 + ( type-Cardinality-Projective-Set I) + ( Set l2)) +is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set (I , H) = + is-equiv-map-distributive-trunc-set-is-cardinality-projective-set I H +``` + +We call the inverse map to the distributive law the "unit map" of the +cardinality-projective set. + +```agda +module _ + {l1 l2 : Level} (I : Cardinality-Projective-Set l1 l2) + (let I' = type-Cardinality-Projective-Set I) + where + + unit-Cardinality-Projective-Set : + (I' → Cardinal l2) → ║ (I' → Set l2) ║₀ + unit-Cardinality-Projective-Set = + map-inv-is-equiv + ( is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set I) + + is-retraction-unit-Cardinality-Projective-Set : + is-retraction + ( map-distributive-trunc-function-type zero-𝕋 I' (Set l2)) + ( unit-Cardinality-Projective-Set) + is-retraction-unit-Cardinality-Projective-Set = + is-retraction-map-inv-is-equiv + ( is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set I) + + compute-unit-Cardinality-Projective-Set : + (K : I' → Set l2) → + unit-Cardinality-Projective-Set (cardinality ∘ K) ＝ unit-trunc-Set K + compute-unit-Cardinality-Projective-Set K = + equational-reasoning + unit-Cardinality-Projective-Set (cardinality ∘ K) + ＝ unit-Cardinality-Projective-Set + ( map-distributive-trunc-function-type zero-𝕋 I' + ( Set l2) + ( unit-trunc K)) + by + ap + ( unit-Cardinality-Projective-Set) + ( inv (eq-htpy (compute-distributive-trunc-function-type zero-𝕋 K))) + ＝ unit-trunc K + by is-retraction-unit-Cardinality-Projective-Set (unit-trunc K) +``` + +### A set is cardinality-projective if the postcomposition map is a set-truncation equivalence + +```agda +module _ + {l1 l2 : Level} (I : Set l1) + where + + is-cardinality-projective-set-is-set-equivalence-postcomp-cardinality-Set : + is-truncation-equivalence zero-𝕋 + ( postcomp (type-Set I) (cardinality {l2})) → + is-cardinality-projective-set-Level l2 I + is-cardinality-projective-set-is-set-equivalence-postcomp-cardinality-Set H = + is-cardinality-projective-set-is-is-equiv-map-distributive-trunc-set I + ( is-equiv-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set + ( I) + ( H)) ``` ## See also diff --git a/src/set-theory/dependent-products-cardinals.lagda.md b/src/set-theory/dependent-products-cardinals.lagda.md index 8ad271a71e..226b4038c7 100644 --- a/src/set-theory/dependent-products-cardinals.lagda.md +++ b/src/set-theory/dependent-products-cardinals.lagda.md @@ -39,7 +39,7 @@ open import foundation-core.propositions open import logic.propositionally-decidable-types -open import set-theory.cardinality-projective-sets +open import set-theory.cardinality-inductive-sets open import set-theory.cardinals ``` @@ -58,32 +58,32 @@ representing sets $Kᵢ$. ```agda module _ - {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + {l1 l2 : Level} (X : Cardinality-Inductive-Set l1 l2) where Π-Cardinal : - (type-Cardinality-Projective-Set X → Cardinal l2) → Cardinal (l1 ⊔ l2) + (type-Cardinality-Inductive-Set X → Cardinal l2) → Cardinal (l1 ⊔ l2) Π-Cardinal Y = map-trunc-Set - ( Π-Set (set-Cardinality-Projective-Set X)) - ( map-section-Cardinality-Projective-Set X Y) + ( Π-Set (set-Cardinality-Inductive-Set X)) + ( unit-Cardinality-Inductive-Set X Y) compute-Π-Cardinal : - (K : type-Cardinality-Projective-Set X → Set l2) → + (K : type-Cardinality-Inductive-Set X → Set l2) → Π-Cardinal (cardinality ∘ K) ＝ - cardinality (Π-Set (set-Cardinality-Projective-Set X) K) + cardinality (Π-Set (set-Cardinality-Inductive-Set X) K) compute-Π-Cardinal K = equational-reasoning map-trunc-Set - ( Π-Set (set-Cardinality-Projective-Set X)) - ( map-section-Cardinality-Projective-Set X (cardinality ∘ K)) + ( Π-Set (set-Cardinality-Inductive-Set X)) + ( unit-Cardinality-Inductive-Set X (cardinality ∘ K)) ＝ map-trunc-Set (Π-Set (pr1 X)) (unit-trunc-Set K) by ap - ( map-trunc-Set (Π-Set (set-Cardinality-Projective-Set X))) - ( compute-map-section-at-cardinality-Cardinality-Projective-Set X K) + ( map-trunc-Set (Π-Set (set-Cardinality-Inductive-Set X))) + ( compute-unit-Cardinality-Inductive-Set X K) ＝ cardinality - ( Π-Set (set-Cardinality-Projective-Set X) K) + ( Π-Set (set-Cardinality-Inductive-Set X) K) by - naturality-unit-trunc-Set (Π-Set (set-Cardinality-Projective-Set X)) K + naturality-unit-trunc-Set (Π-Set (set-Cardinality-Inductive-Set X)) K ``` diff --git a/src/set-theory/dependent-sums-cardinals.lagda.md b/src/set-theory/dependent-sums-cardinals.lagda.md index e9cf56a520..f1778191f0 100644 --- a/src/set-theory/dependent-sums-cardinals.lagda.md +++ b/src/set-theory/dependent-sums-cardinals.lagda.md @@ -10,6 +10,7 @@ module set-theory.dependent-sums-cardinals where open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions +open import foundation.connected-maps open import foundation.coproduct-types open import foundation.decidable-propositions open import foundation.decidable-subtypes @@ -19,13 +20,16 @@ open import foundation.discrete-types open import foundation.double-negation open import foundation.function-extensionality open import foundation.function-types +open import foundation.functoriality-propositional-truncation open import foundation.functoriality-set-truncation open import foundation.identity-types open import foundation.isolated-elements open import foundation.logical-equivalences +open import foundation.mere-embeddings open import foundation.negated-equality open import foundation.negation open import foundation.powersets +open import foundation.projective-types open import foundation.propositional-truncations open import foundation.sections open import foundation.set-truncations @@ -39,7 +43,7 @@ open import foundation-core.propositions open import logic.propositionally-decidable-types -open import set-theory.cardinality-projective-sets +open import set-theory.cardinality-inductive-sets open import set-theory.cardinals open import set-theory.inequality-cardinals ``` @@ -49,58 +53,80 @@ open import set-theory.inequality-cardinals ## Idea Given a family of cardinals $κ : I → \mathrm{Cardinal}$ over a -[cardinality-projective set](set-theory.cardinality-projective-sets.md) $I$, -then we may define the {{#concept "dependent sum cardinal"}} $Σ_{i∈I}κᵢ$, as the -cardinality of the [dependent sum](foundation.dependent-pair-types.md) of any -family of representing sets $Kᵢ$. +[cardinality-inductive set](set-theory.cardinality-inductive-sets.md) $I$, then +we may define the {{#concept "dependent sum cardinal" Agda=Σ-Cardinal}} +$Σ_{i∈I}κᵢ$, as the cardinality of the +[dependent sum](foundation.dependent-pair-types.md) of any family of +representing sets $Kᵢ$. ## Definitions ```agda module _ - {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + {l1 l2 : Level} (X : Set l1) + where + + cardinality-Σ : (type-Set X → Set l2) → Cardinal (l1 ⊔ l2) + cardinality-Σ Y = cardinality (Σ-Set X Y) + +module _ + {l1 l2 : Level} (X : Cardinality-Inductive-Set l1 l2) + (let set-X = set-Cardinality-Inductive-Set X) + (let type-X = type-Cardinality-Inductive-Set X) where Σ-Cardinal : - (type-Cardinality-Projective-Set X → Cardinal l2) → Cardinal (l1 ⊔ l2) - Σ-Cardinal Y = - map-trunc-Set - ( Σ-Set (set-Cardinality-Projective-Set X)) - ( map-section-Cardinality-Projective-Set X Y) + (type-X → Cardinal l2) → Cardinal (l1 ⊔ l2) + Σ-Cardinal K = + map-trunc-Set (Σ-Set set-X) (unit-Cardinality-Inductive-Set X K) compute-Σ-Cardinal : - (K : type-Cardinality-Projective-Set X → Set l2) → - Σ-Cardinal (cardinality ∘ K) ＝ - cardinality (Σ-Set (set-Cardinality-Projective-Set X) K) - compute-Σ-Cardinal K = + (Y : type-X → Set l2) → + Σ-Cardinal (cardinality ∘ Y) ＝ cardinality (Σ-Set set-X Y) + compute-Σ-Cardinal Y = equational-reasoning - map-trunc-Set - ( Σ-Set (set-Cardinality-Projective-Set X)) - ( map-section-Cardinality-Projective-Set X (cardinality ∘ K)) - ＝ map-trunc-Set (Σ-Set (pr1 X)) (unit-trunc-Set K) + Σ-Cardinal (cardinality ∘ Y) + ＝ map-trunc-Set (Σ-Set set-X) (unit-trunc-Set Y) by ap - ( map-trunc-Set (Σ-Set (set-Cardinality-Projective-Set X))) - ( compute-map-section-at-cardinality-Cardinality-Projective-Set X K) - ＝ cardinality - ( Σ-Set (set-Cardinality-Projective-Set X) K) - by - naturality-unit-trunc-Set (Σ-Set (set-Cardinality-Projective-Set X)) K + ( map-trunc-Set (Σ-Set set-X)) + ( compute-unit-Cardinality-Inductive-Set X Y) + ＝ cardinality (Σ-Set set-X Y) + by naturality-unit-trunc-Set (Σ-Set set-X) Y ``` ## Properties -### Inequality is preserved under dependent sums +### Inequality is preserved under dependent sums over projective types ```agda module _ - {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + {l1 l2 : Level} (X : Set l1) + (is-projective-X : is-projective-Level' l2 (type-Set X)) + where + + leq-cardinality-Σ : + (K P : type-Set X → Set l2) → + ((i : type-Set X) → leq-cardinality (K i) (P i)) → + leq-cardinality (Σ-Set X K) (Σ-Set X P) + leq-cardinality-Σ K P f = + unit-leq-cardinality + ( Σ-Set X K) + ( Σ-Set X P) + ( mere-emb-tot + ( is-projective-X) + ( λ x → inv-unit-leq-cardinality (K x) (P x) (f x))) + +module _ + {l1 l2 : Level} (X : Cardinality-Inductive-Set l1 l2) + (let type-X = type-Cardinality-Inductive-Set X) + (is-projective-X : is-projective-Level' l2 (type-Cardinality-Inductive-Set X)) where leq-Σ-Cardinal : - (K P : type-Cardinality-Projective-Set X → Cardinal l2) → - ((i : type-Cardinality-Projective-Set X) → leq-Cardinal (K i) (P i)) → + (K P : type-X → Cardinal l2) → + ((i : type-X) → leq-Cardinal (K i) (P i)) → leq-Cardinal (Σ-Cardinal X K) (Σ-Cardinal X P) leq-Σ-Cardinal K P f = {! !} - -- proof somehow proceeds by using that since `X` is cardinality-projective, it suffices to show this for families of sets, and then it's just an easy fact of dependent sums. + -- proof somehow proceeds by using that since `X` is cardinality-inductive, it suffices to show this for families of sets, and then it's just an easy fact of dependent sums. ``` diff --git a/src/set-theory/inhabited-cardinals.lagda.md b/src/set-theory/inhabited-cardinals.lagda.md index 93b06db761..0d0d56ae63 100644 --- a/src/set-theory/inhabited-cardinals.lagda.md +++ b/src/set-theory/inhabited-cardinals.lagda.md @@ -12,12 +12,12 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-types open import foundation.identity-types -open import foundation.subtypes open import foundation.inhabited-types open import foundation.propositional-extensionality open import foundation.propositions open import foundation.set-truncations open import foundation.sets +open import foundation.subtypes open import foundation.univalence open import foundation.universe-levels diff --git a/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md b/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md index 8ef38ae7ee..69782fd17e 100644 --- a/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md +++ b/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md @@ -146,15 +146,15 @@ module _ distributive-trunc-Π-count-Set : count A → is-contr - ( Σ ( ( type-trunc-Set ((x : A) → B x)) ≃ - ( (x : A) → type-trunc-Set (B x))) + ( Σ ( ║ ((x : A) → B x) ║₀ ≃ + ( (x : A) → ║ B x ║₀)) ( λ e → ( map-equiv e ∘ unit-trunc-Set) ~ ( map-Π (λ x → unit-trunc-Set)))) distributive-trunc-Π-count-Set (pair k e) = is-contr-equiv - ( Σ ( ( type-trunc-Set ((x : A) → B x)) ≃ - ( (x : Fin k) → type-trunc-Set (B (map-equiv e x)))) + ( Σ ( ( ║ ((x : A) → B x) ║₀) ≃ + ( (x : Fin k) → ║ B (map-equiv e x) ║₀)) ( λ f → ( map-equiv f ∘ unit-trunc-Set) ~ ( map-Π (λ x → unit-trunc-Set) ∘ precomp-Π (map-equiv e) B))) @@ -164,7 +164,7 @@ module _ ( map-Π (λ x → unit-trunc-Set) ∘ precomp-Π (map-equiv e) B)) ( equiv-postcomp-equiv ( equiv-precomp-Π e (type-trunc-Set ∘ B)) - ( type-trunc-Set ((x : A) → B x))) + ( ║ ((x : A) → B x) ║₀)) ( λ f → equiv-Π-equiv-family ( λ h → @@ -175,8 +175,8 @@ module _ map-Π (λ y → unit-trunc-Set) h x))) ∘e ( equiv-funext)))) ( is-contr-equiv' - ( Σ ( ( type-trunc-Set ((x : Fin k) → B (map-equiv e x))) ≃ - ( (x : Fin k) → type-trunc-Set (B (map-equiv e x)))) + ( Σ ( ( ║ ((x : Fin k) → B (map-equiv e x)) ║₀) ≃ + ( (x : Fin k) → ║ B (map-equiv e x) ║₀)) ( λ f → ( map-equiv f ∘ unit-trunc-Set) ~ ( map-Π (λ x → unit-trunc-Set)))) @@ -186,7 +186,7 @@ module _ ( map-Π (λ x → unit-trunc-Set) ∘ precomp-Π (map-equiv e) B)) ( equiv-precomp-equiv ( equiv-trunc-Set (equiv-precomp-Π e B)) - ( (x : Fin k) → type-trunc-Set (B (map-equiv e x)))) + ( (x : Fin k) → ║ B (map-equiv e x) ║₀)) ( λ f → equiv-Π ( λ h → @@ -248,12 +248,12 @@ module _ where equiv-distributive-trunc-Π-count-Set : - ( type-trunc-Set ((x : A) → B x)) ≃ ((x : A) → type-trunc-Set (B x)) + ║ ((x : A) → B x) ║₀ ≃ ((x : A) → ║ B x ║₀) equiv-distributive-trunc-Π-count-Set = pr1 (center (distributive-trunc-Π-count-Set B c)) map-equiv-distributive-trunc-Π-count-Set : - ( type-trunc-Set ((x : A) → B x)) → ((x : A) → type-trunc-Set (B x)) + ║ ((x : A) → B x) ║₀ → ((x : A) → ║ B x ║₀) map-equiv-distributive-trunc-Π-count-Set = map-equiv equiv-distributive-trunc-Π-count-Set @@ -270,8 +270,8 @@ module _ abstract distributive-trunc-Π-is-finite-Set : is-contr - ( Σ ( ( type-trunc-Set ((x : A) → B x)) ≃ - ( (x : A) → type-trunc-Set (B x))) + ( Σ ( ( ║ ((x : A) → B x) ║₀) ≃ + ( (x : A) → ║ B x ║₀)) ( λ e → ( map-equiv e ∘ unit-trunc-Set) ~ ( map-Π (λ x → unit-trunc-Set)))) @@ -281,12 +281,12 @@ module _ ( distributive-trunc-Π-count-Set B) equiv-distributive-trunc-Π-is-finite-Set : - ( type-trunc-Set ((x : A) → B x)) ≃ ((x : A) → type-trunc-Set (B x)) + ║ ((x : A) → B x) ║₀ ≃ ((x : A) → ║ B x ║₀) equiv-distributive-trunc-Π-is-finite-Set = pr1 (center distributive-trunc-Π-is-finite-Set) map-equiv-distributive-trunc-Π-is-finite-Set : - ( type-trunc-Set ((x : A) → B x)) → ((x : A) → type-trunc-Set (B x)) + ║ ((x : A) → B x) ║₀ → ((x : A) → ║ B x ║₀) map-equiv-distributive-trunc-Π-is-finite-Set = map-equiv equiv-distributive-trunc-Π-is-finite-Set @@ -296,3 +296,8 @@ module _ triangle-distributive-trunc-Π-is-finite-Set = pr2 (center distributive-trunc-Π-is-finite-Set) ``` + +## See also + +- This in particular means that finite sets are + [cardinality-projective](set-theory.cardinality-projective-sets.md). From 1818d4e596f063c3764c85812fb368cfbd7b0889 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 16 Nov 2025 01:20:14 +0100 Subject: [PATCH 24/50] remove an import --- src/set-theory/cardinality-projective-sets.lagda.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index 78680fdf5c..fa73472d13 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -7,7 +7,6 @@ module set-theory.cardinality-projective-sets where
Imports ```agda -open import foundation.0-connected-maps open import foundation.action-on-identifications-functions open import foundation.connected-maps open import foundation.dependent-pair-types @@ -56,7 +55,7 @@ module _ is-cardinality-projective-set-Level : UU (l1 ⊔ lsuc l2) is-cardinality-projective-set-Level = - is-0-connected-map (postcomp (type-Set I) (cardinality {l2})) + is-connected-map zero-𝕋 (postcomp (type-Set I) (cardinality {l2})) ``` ### The universe of cardinality-projective sets at a universe level From d46c3708f2d0c2f52885ccff95faee0adf268af0 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 16 Nov 2025 01:28:27 +0100 Subject: [PATCH 25/50] cardinality-preprojectivity --- .../cardinality-projective-sets.lagda.md | 61 +++++++++++++------ 1 file changed, 44 insertions(+), 17 deletions(-) diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index fa73472d13..42f81714c5 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -13,6 +13,8 @@ open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences open import foundation.function-extensionality +open import foundation.cartesian-product-types +open import foundation.projective-types open import foundation.function-types open import foundation.functoriality-truncation open import foundation.homotopies @@ -46,6 +48,18 @@ $$\mathrm{cardinality} ∘ {-} : (I → Set) → (I → \mathrm{Cardinal})$$ is ## Definitions +### The predicate of being cardinality-preprojective at a universe level + +```agda +module _ + {l1 : Level} (l2 : Level) (I : Set l1) + where + + is-cardinality-preprojective-set-Level : UU (l1 ⊔ lsuc l2) + is-cardinality-preprojective-set-Level = + is-connected-map zero-𝕋 (postcomp (type-Set I) (cardinality {l2})) +``` + ### The predicate of being cardinality-projective at a universe level ```agda @@ -55,7 +69,8 @@ module _ is-cardinality-projective-set-Level : UU (l1 ⊔ lsuc l2) is-cardinality-projective-set-Level = - is-connected-map zero-𝕋 (postcomp (type-Set I) (cardinality {l2})) + is-connected-map zero-𝕋 (postcomp (type-Set I) (cardinality {l2})) × + is-projective-Level' l2 (type-Set I) ``` ### The universe of cardinality-projective sets at a universe level @@ -84,13 +99,25 @@ module _ is-cardinality-projective-set-Level l2 set-Cardinality-Projective-Set is-cardinality-projective-Cardinality-Projective-Set = pr2 I + is-cardinality-preprojective-Cardinality-Projective-Set : + is-cardinality-preprojective-set-Level l2 set-Cardinality-Projective-Set + is-cardinality-preprojective-Cardinality-Projective-Set = + pr1 is-cardinality-projective-Cardinality-Projective-Set + + + is-projective-Cardinality-Projective-Set : + is-projective-Level' l2 type-Cardinality-Projective-Set + is-projective-Cardinality-Projective-Set = + pr2 is-cardinality-projective-Cardinality-Projective-Set + + is-set-equivalence-postcomp-cardinality-type-Cardinality-Projective-Set : is-truncation-equivalence zero-𝕋 ( postcomp type-Cardinality-Projective-Set (cardinality {l2})) is-set-equivalence-postcomp-cardinality-type-Cardinality-Projective-Set = is-truncation-equivalence-is-connected-map ( postcomp type-Cardinality-Projective-Set cardinality) - ( is-cardinality-projective-Cardinality-Projective-Set) + ( is-cardinality-preprojective-Cardinality-Projective-Set) constr-Cardinality-Projective-Set : {l : Level} → @@ -115,9 +142,9 @@ module _ ## Properties -### Distributive property of cardinality-projective sets +### Distributive property of cardinality-preprojective sets -A set `I` is cardinality-projective if and only if the distributive map +A set `I` is cardinality-preprojective if and only if the distributive map ```text ║I → Set║₀ → (I → Cardinal) @@ -172,18 +199,18 @@ module _ ( is-set-equivalence-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set ( H)) - is-equiv-map-distributive-trunc-set-is-cardinality-projective-set : - is-cardinality-projective-set-Level l2 I → + is-equiv-map-distributive-trunc-set-is-cardinality-preprojective-set : + is-cardinality-preprojective-set-Level l2 I → is-equiv (map-distributive-trunc-function-type zero-𝕋 (type-Set I) (Set l2)) - is-equiv-map-distributive-trunc-set-is-cardinality-projective-set H = + is-equiv-map-distributive-trunc-set-is-cardinality-preprojective-set H = is-equiv-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set ( is-truncation-equivalence-is-connected-map _ H) - is-cardinality-projective-set-is-is-equiv-map-distributive-trunc-set : + is-cardinality-preprojective-set-is-is-equiv-map-distributive-trunc-set : is-equiv ( map-distributive-trunc-function-type zero-𝕋 (type-Set I) (Set l2)) → - is-cardinality-projective-set-Level l2 I - is-cardinality-projective-set-is-is-equiv-map-distributive-trunc-set H = + is-cardinality-preprojective-set-Level l2 I + is-cardinality-preprojective-set-is-is-equiv-map-distributive-trunc-set H = is-connected-map-left-map-triangle ( postcomp (type-Set I) cardinality) ( map-distributive-trunc-function-type zero-𝕋 (type-Set I) (Set l2)) @@ -198,8 +225,8 @@ is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set : ( map-distributive-trunc-function-type zero-𝕋 ( type-Cardinality-Projective-Set I) ( Set l2)) -is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set (I , H) = - is-equiv-map-distributive-trunc-set-is-cardinality-projective-set I H +is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set (I , (H , _)) = + is-equiv-map-distributive-trunc-set-is-cardinality-preprojective-set I H ``` We call the inverse map to the distributive law the "unit map" of the @@ -243,19 +270,19 @@ module _ by is-retraction-unit-Cardinality-Projective-Set (unit-trunc K) ``` -### A set is cardinality-projective if the postcomposition map is a set-truncation equivalence +### A set is cardinality-preprojective if the postcomposition map is a set-truncation equivalence ```agda module _ {l1 l2 : Level} (I : Set l1) where - is-cardinality-projective-set-is-set-equivalence-postcomp-cardinality-Set : + is-cardinality-preprojective-set-is-set-equivalence-postcomp-cardinality-Set : is-truncation-equivalence zero-𝕋 ( postcomp (type-Set I) (cardinality {l2})) → - is-cardinality-projective-set-Level l2 I - is-cardinality-projective-set-is-set-equivalence-postcomp-cardinality-Set H = - is-cardinality-projective-set-is-is-equiv-map-distributive-trunc-set I + is-cardinality-preprojective-set-Level l2 I + is-cardinality-preprojective-set-is-set-equivalence-postcomp-cardinality-Set H = + is-cardinality-preprojective-set-is-is-equiv-map-distributive-trunc-set I ( is-equiv-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set ( I) ( H)) From 851990066a82a436b3a0eb6d31e50bbb1c5456b7 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 16 Nov 2025 01:33:51 +0100 Subject: [PATCH 26/50] edits --- .../cardinality-projective-sets.lagda.md | 9 +++---- .../dependent-products-cardinals.lagda.md | 26 ++++++++----------- 2 files changed, 15 insertions(+), 20 deletions(-) diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index 42f81714c5..83bd31d3cd 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -8,13 +8,12 @@ module set-theory.cardinality-projective-sets where ```agda open import foundation.action-on-identifications-functions +open import foundation.cartesian-product-types open import foundation.connected-maps open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences open import foundation.function-extensionality -open import foundation.cartesian-product-types -open import foundation.projective-types open import foundation.function-types open import foundation.functoriality-truncation open import foundation.homotopies @@ -22,6 +21,7 @@ open import foundation.identity-types open import foundation.injective-maps open import foundation.mere-equivalences open import foundation.postcomposition-functions +open import foundation.projective-types open import foundation.retractions open import foundation.retracts-of-types open import foundation.set-truncations @@ -104,13 +104,11 @@ module _ is-cardinality-preprojective-Cardinality-Projective-Set = pr1 is-cardinality-projective-Cardinality-Projective-Set - is-projective-Cardinality-Projective-Set : is-projective-Level' l2 type-Cardinality-Projective-Set is-projective-Cardinality-Projective-Set = pr2 is-cardinality-projective-Cardinality-Projective-Set - is-set-equivalence-postcomp-cardinality-type-Cardinality-Projective-Set : is-truncation-equivalence zero-𝕋 ( postcomp type-Cardinality-Projective-Set (cardinality {l2})) @@ -281,7 +279,8 @@ module _ is-truncation-equivalence zero-𝕋 ( postcomp (type-Set I) (cardinality {l2})) → is-cardinality-preprojective-set-Level l2 I - is-cardinality-preprojective-set-is-set-equivalence-postcomp-cardinality-Set H = + is-cardinality-preprojective-set-is-set-equivalence-postcomp-cardinality-Set + H = is-cardinality-preprojective-set-is-is-equiv-map-distributive-trunc-set I ( is-equiv-map-distributive-trunc-set-is-set-equivalence-postcomp-cardinality-Set ( I) diff --git a/src/set-theory/dependent-products-cardinals.lagda.md b/src/set-theory/dependent-products-cardinals.lagda.md index 226b4038c7..0c391f5c52 100644 --- a/src/set-theory/dependent-products-cardinals.lagda.md +++ b/src/set-theory/dependent-products-cardinals.lagda.md @@ -48,8 +48,8 @@ open import set-theory.cardinals ## Idea Given a family of cardinals $κ : I → \mathrm{Cardinal}$ over a -[cardinality-projective set](set-theory.cardinality-projective-sets.md) $I$, -then we may define the {{#concept "dependent product cardinal" Agda=Π-Cardinal}} +[cardinality-inductive set](set-theory.cardinality-inductive-sets.md) $I$, then +we may define the {{#concept "dependent product cardinal" Agda=Π-Cardinal}} $Π_{i∈I}κᵢ$, as the cardinality of the [dependent product](foundation.dependent-function-types.md) of any family of representing sets $Kᵢ$. @@ -59,31 +59,27 @@ representing sets $Kᵢ$. ```agda module _ {l1 l2 : Level} (X : Cardinality-Inductive-Set l1 l2) + (let set-X = set-Cardinality-Inductive-Set X) where Π-Cardinal : - (type-Cardinality-Inductive-Set X → Cardinal l2) → Cardinal (l1 ⊔ l2) + (type-Set set-X → Cardinal l2) → Cardinal (l1 ⊔ l2) Π-Cardinal Y = - map-trunc-Set - ( Π-Set (set-Cardinality-Inductive-Set X)) - ( unit-Cardinality-Inductive-Set X Y) + map-trunc-Set (Π-Set set-X) (unit-Cardinality-Inductive-Set X Y) compute-Π-Cardinal : (K : type-Cardinality-Inductive-Set X → Set l2) → - Π-Cardinal (cardinality ∘ K) ＝ - cardinality (Π-Set (set-Cardinality-Inductive-Set X) K) + Π-Cardinal (cardinality ∘ K) ＝ cardinality (Π-Set set-X K) compute-Π-Cardinal K = equational-reasoning map-trunc-Set - ( Π-Set (set-Cardinality-Inductive-Set X)) + ( Π-Set set-X) ( unit-Cardinality-Inductive-Set X (cardinality ∘ K)) - ＝ map-trunc-Set (Π-Set (pr1 X)) (unit-trunc-Set K) + ＝ map-trunc-Set (Π-Set set-X) (unit-trunc-Set K) by ap - ( map-trunc-Set (Π-Set (set-Cardinality-Inductive-Set X))) + ( map-trunc-Set (Π-Set set-X)) ( compute-unit-Cardinality-Inductive-Set X K) - ＝ cardinality - ( Π-Set (set-Cardinality-Inductive-Set X) K) - by - naturality-unit-trunc-Set (Π-Set (set-Cardinality-Inductive-Set X)) K + ＝ cardinality (Π-Set set-X K) + by naturality-unit-trunc-Set (Π-Set set-X) K ``` From 022bb874e122915956ed0e8bccb90bc25228f2ec Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 16 Nov 2025 02:35:07 +0100 Subject: [PATCH 27/50] rename cardinality-recursive sets --- src/set-theory.lagda.md | 2 +- .../cardinality-projective-sets.lagda.md | 65 ++++++++---- ...md => cardinality-recursive-sets.lagda.md} | 98 +++++++++---------- 3 files changed, 96 insertions(+), 69 deletions(-) rename src/set-theory/{cardinality-inductive-sets.lagda.md => cardinality-recursive-sets.lagda.md} (52%) diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index 1f0154f2b9..f57fc0043e 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -53,7 +53,7 @@ open import set-theory.baire-space public open import set-theory.bounded-increasing-binary-sequences public open import set-theory.cantor-space public open import set-theory.cantors-diagonal-argument public -open import set-theory.cardinality-inductive-sets public +open import set-theory.cardinality-recursive-sets public open import set-theory.cardinality-projective-sets public open import set-theory.cardinals public open import set-theory.complemented-inequality-cardinals public diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index 83bd31d3cd..728209cf2a 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -34,6 +34,7 @@ open import foundation.truncations open import foundation.universe-levels open import set-theory.cardinals +open import set-theory.cardinality-recursive-sets ```
@@ -117,6 +118,27 @@ module _ ( postcomp type-Cardinality-Projective-Set cardinality) ( is-cardinality-preprojective-Cardinality-Projective-Set) + ind-Cardinality-Projective-Set : + {l3 : Level} + (P : (type-Cardinality-Projective-Set → Cardinal l2) → Set l3) → + ( (Y : type-Cardinality-Projective-Set → Set l2) → + type-Set (P (cardinality ∘ Y))) → + (X : type-Cardinality-Projective-Set → Cardinal l2) → type-Set (P X) + ind-Cardinality-Projective-Set = + ind-is-connected-map is-cardinality-preprojective-Cardinality-Projective-Set + + compute-ind-Cardinality-Projective-Set : + {l3 : Level} + (P : (type-Cardinality-Projective-Set → Cardinal l2) → Set l3) + (T : + (Y : type-Cardinality-Projective-Set → Set l2) → + type-Set (P (cardinality ∘ Y))) + (Y : type-Cardinality-Projective-Set → Set l2) → + ind-Cardinality-Projective-Set P T (cardinality ∘ Y) ＝ T Y + compute-ind-Cardinality-Projective-Set = + compute-ind-is-connected-map + ( is-cardinality-preprojective-Cardinality-Projective-Set) + constr-Cardinality-Projective-Set : {l : Level} → ((type-Cardinality-Projective-Set → Set l2) → Cardinal l) → @@ -130,7 +152,7 @@ module _ compute-constr-Cardinality-Projective-Set : {l : Level} → (T : (type-Cardinality-Projective-Set → Set l2) → Cardinal l) - (Y : pr1 set-Cardinality-Projective-Set → Set l2) → + (Y : type-Cardinality-Projective-Set → Set l2) → constr-Cardinality-Projective-Set T (cardinality ∘ Y) ＝ T Y compute-constr-Cardinality-Projective-Set {l} = compute-rec-is-truncation-equivalence @@ -227,8 +249,11 @@ is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set (I , (H , _)) = is-equiv-map-distributive-trunc-set-is-cardinality-preprojective-set I H ``` +### Cardinality-projective sets are cardinality-recursive + We call the inverse map to the distributive law the "unit map" of the -cardinality-projective set. +cardinality-projective set, and this map gives an induction principle for +constructing cardinals over $I$. ```agda module _ @@ -236,36 +261,38 @@ module _ (let I' = type-Cardinality-Projective-Set I) where + is-cardinality-recursive-Cardinality-Projective-Set : + is-cardinality-recursive-set-Level l2 (set-Cardinality-Projective-Set I) + is-cardinality-recursive-Cardinality-Projective-Set = + retraction-is-equiv + ( is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set I) + + cardinality-recursive-set-Cardinality-Projective-Set : + Cardinality-Recursive-Set l1 l2 + cardinality-recursive-set-Cardinality-Projective-Set = + ( set-Cardinality-Projective-Set I , + is-cardinality-recursive-Cardinality-Projective-Set) + unit-Cardinality-Projective-Set : (I' → Cardinal l2) → ║ (I' → Set l2) ║₀ unit-Cardinality-Projective-Set = - map-inv-is-equiv - ( is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set I) + unit-Cardinality-Recursive-Set + ( cardinality-recursive-set-Cardinality-Projective-Set) is-retraction-unit-Cardinality-Projective-Set : is-retraction ( map-distributive-trunc-function-type zero-𝕋 I' (Set l2)) ( unit-Cardinality-Projective-Set) is-retraction-unit-Cardinality-Projective-Set = - is-retraction-map-inv-is-equiv - ( is-equiv-map-distributive-trunc-set-Cardinality-Projective-Set I) + is-retraction-unit-Cardinality-Recursive-Set + ( cardinality-recursive-set-Cardinality-Projective-Set) compute-unit-Cardinality-Projective-Set : (K : I' → Set l2) → unit-Cardinality-Projective-Set (cardinality ∘ K) ＝ unit-trunc-Set K - compute-unit-Cardinality-Projective-Set K = - equational-reasoning - unit-Cardinality-Projective-Set (cardinality ∘ K) - ＝ unit-Cardinality-Projective-Set - ( map-distributive-trunc-function-type zero-𝕋 I' - ( Set l2) - ( unit-trunc K)) - by - ap - ( unit-Cardinality-Projective-Set) - ( inv (eq-htpy (compute-distributive-trunc-function-type zero-𝕋 K))) - ＝ unit-trunc K - by is-retraction-unit-Cardinality-Projective-Set (unit-trunc K) + compute-unit-Cardinality-Projective-Set = + compute-unit-Cardinality-Recursive-Set + ( cardinality-recursive-set-Cardinality-Projective-Set) ``` ### A set is cardinality-preprojective if the postcomposition map is a set-truncation equivalence diff --git a/src/set-theory/cardinality-inductive-sets.lagda.md b/src/set-theory/cardinality-recursive-sets.lagda.md similarity index 52% rename from src/set-theory/cardinality-inductive-sets.lagda.md rename to src/set-theory/cardinality-recursive-sets.lagda.md index 981e877d97..5670b4f196 100644 --- a/src/set-theory/cardinality-inductive-sets.lagda.md +++ b/src/set-theory/cardinality-recursive-sets.lagda.md @@ -1,7 +1,7 @@ -# Cardinality-inductive sets +# Cardinality-recursive sets ```agda -module set-theory.cardinality-inductive-sets where +module set-theory.cardinality-recursive-sets where ```
Imports @@ -34,7 +34,7 @@ open import set-theory.cardinals For every type $X$ there is a map $║X → Set║₀ → (X → \mathrm{Cardinal})$. We call [sets](foundation-core.sets.md) $X$ for which this map is a retract -{{#concept "cardinality-inductive" Disamibguation="sets" Agda=Cardinality-Inductive-Set}}. +{{#concept "cardinality-recursive" Disamibguation="sets" Agda=Cardinality-Recursive-Set}}. Over such sets we may form [dependent sum](set-theory.dependent-sums-cardinals.md) and [dependent product](set-theory.dependent-products-cardinals.md) @@ -63,92 +63,92 @@ module _ {l1 : Level} (l2 : Level) (X : Set l1) where - is-cardinality-inductive-set-Level : UU (l1 ⊔ lsuc l2) - is-cardinality-inductive-set-Level = + is-cardinality-recursive-set-Level : UU (l1 ⊔ lsuc l2) + is-cardinality-recursive-set-Level = retraction ( map-distributive-trunc-function-type zero-𝕋 (type-Set X) (Set l2)) ``` -### The universe of cardinality-inductive sets at a universe level +### The universe of cardinality-recursive sets at a universe level ```agda -Cardinality-Inductive-Set : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) -Cardinality-Inductive-Set l1 l2 = - Σ (Set l1) (is-cardinality-inductive-set-Level l2) +Cardinality-Recursive-Set : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) +Cardinality-Recursive-Set l1 l2 = + Σ (Set l1) (is-cardinality-recursive-set-Level l2) module _ - {l1 l2 : Level} (X : Cardinality-Inductive-Set l1 l2) + {l1 l2 : Level} (X : Cardinality-Recursive-Set l1 l2) where - set-Cardinality-Inductive-Set : Set l1 - set-Cardinality-Inductive-Set = pr1 X + set-Cardinality-Recursive-Set : Set l1 + set-Cardinality-Recursive-Set = pr1 X - type-Cardinality-Inductive-Set : UU l1 - type-Cardinality-Inductive-Set = type-Set set-Cardinality-Inductive-Set + type-Cardinality-Recursive-Set : UU l1 + type-Cardinality-Recursive-Set = type-Set set-Cardinality-Recursive-Set - is-set-type-Cardinality-Inductive-Set : - is-set type-Cardinality-Inductive-Set - is-set-type-Cardinality-Inductive-Set = - is-set-type-Set set-Cardinality-Inductive-Set + is-set-type-Cardinality-Recursive-Set : + is-set type-Cardinality-Recursive-Set + is-set-type-Cardinality-Recursive-Set = + is-set-type-Set set-Cardinality-Recursive-Set - is-cardinality-inductive-Cardinality-Inductive-Set : - is-cardinality-inductive-set-Level l2 set-Cardinality-Inductive-Set - is-cardinality-inductive-Cardinality-Inductive-Set = pr2 X + is-cardinality-recursive-Cardinality-Recursive-Set : + is-cardinality-recursive-set-Level l2 set-Cardinality-Recursive-Set + is-cardinality-recursive-Cardinality-Recursive-Set = pr2 X - unit-Cardinality-Inductive-Set : - ( type-Cardinality-Inductive-Set → Cardinal l2) → - ║ (type-Cardinality-Inductive-Set → Set l2) ║₀ - unit-Cardinality-Inductive-Set = + unit-Cardinality-Recursive-Set : + ( type-Cardinality-Recursive-Set → Cardinal l2) → + ║ (type-Cardinality-Recursive-Set → Set l2) ║₀ + unit-Cardinality-Recursive-Set = map-retraction ( map-distributive-trunc-function-type zero-𝕋 - ( type-Cardinality-Inductive-Set) + ( type-Cardinality-Recursive-Set) ( Set l2)) - ( is-cardinality-inductive-Cardinality-Inductive-Set) + ( is-cardinality-recursive-Cardinality-Recursive-Set) - is-retraction-unit-Cardinality-Inductive-Set : + is-retraction-unit-Cardinality-Recursive-Set : is-retraction ( map-distributive-trunc-function-type zero-𝕋 - ( type-Cardinality-Inductive-Set) + ( type-Cardinality-Recursive-Set) ( Set l2)) - ( unit-Cardinality-Inductive-Set) - is-retraction-unit-Cardinality-Inductive-Set = + ( unit-Cardinality-Recursive-Set) + is-retraction-unit-Cardinality-Recursive-Set = is-retraction-map-retraction ( map-distributive-trunc-function-type zero-𝕋 - ( type-Cardinality-Inductive-Set) + ( type-Cardinality-Recursive-Set) ( Set l2)) - ( is-cardinality-inductive-Cardinality-Inductive-Set) + ( is-cardinality-recursive-Cardinality-Recursive-Set) - retract-Cardinality-Inductive-Set : - ║ (type-Cardinality-Inductive-Set → Set l2) ║₀ retract-of - ( type-Cardinality-Inductive-Set → Cardinal l2) - retract-Cardinality-Inductive-Set = + retract-Cardinality-Recursive-Set : + ║ (type-Cardinality-Recursive-Set → Set l2) ║₀ retract-of + ( type-Cardinality-Recursive-Set → Cardinal l2) + retract-Cardinality-Recursive-Set = ( ( map-distributive-trunc-function-type ( zero-𝕋) - ( type-Cardinality-Inductive-Set) + ( type-Cardinality-Recursive-Set) ( Set l2)) , - ( is-cardinality-inductive-Cardinality-Inductive-Set)) + ( is-cardinality-recursive-Cardinality-Recursive-Set)) - compute-unit-Cardinality-Inductive-Set : - (K : type-Cardinality-Inductive-Set → Set l2) → - unit-Cardinality-Inductive-Set (cardinality ∘ K) ＝ unit-trunc-Set K - compute-unit-Cardinality-Inductive-Set K = + compute-unit-Cardinality-Recursive-Set : + (K : type-Cardinality-Recursive-Set → Set l2) → + unit-Cardinality-Recursive-Set (cardinality ∘ K) ＝ unit-trunc-Set K + compute-unit-Cardinality-Recursive-Set K = equational-reasoning - unit-Cardinality-Inductive-Set (cardinality ∘ K) - ＝ unit-Cardinality-Inductive-Set + unit-Cardinality-Recursive-Set (cardinality ∘ K) + ＝ unit-Cardinality-Recursive-Set ( map-distributive-trunc-function-type zero-𝕋 - ( type-Cardinality-Inductive-Set) + ( type-Cardinality-Recursive-Set) ( Set l2) ( unit-trunc K)) by ap - ( unit-Cardinality-Inductive-Set) + ( unit-Cardinality-Recursive-Set) ( inv (eq-htpy (compute-distributive-trunc-function-type zero-𝕋 K))) ＝ unit-trunc K - by is-retraction-unit-Cardinality-Inductive-Set (unit-trunc K) + by is-retraction-unit-Cardinality-Recursive-Set (unit-trunc K) ``` ## See also - In [Distributivity of set truncation over finite products](univalent-combinatorics.distributivity-of-set-truncation-over-finite-products.md) - it is demonstrated that finite types are cardinality-inductive. + it is demonstrated that finite types are cardinality-recursive. From e3e5a102f3c4b3d683300b241c0398d566953fc1 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 16 Nov 2025 02:37:53 +0100 Subject: [PATCH 28/50] fixes --- ...phisms-with-respect-to-truncated-types.lagda.md | 7 +++---- src/set-theory.lagda.md | 2 +- .../cardinality-projective-sets.lagda.md | 14 +++++++------- .../truncated-acyclic-maps.lagda.md | 4 +--- 4 files changed, 12 insertions(+), 15 deletions(-) diff --git a/src/foundation/epimorphisms-with-respect-to-truncated-types.lagda.md b/src/foundation/epimorphisms-with-respect-to-truncated-types.lagda.md index bec1d7a0cf..1e4127dda2 100644 --- a/src/foundation/epimorphisms-with-respect-to-truncated-types.lagda.md +++ b/src/foundation/epimorphisms-with-respect-to-truncated-types.lagda.md @@ -95,7 +95,7 @@ is-epimorphism-is-truncation-equivalence-Truncated-Type : is-truncation-equivalence k f → is-epimorphism-Truncated-Type k f is-epimorphism-is-truncation-equivalence-Truncated-Type k f H X = - is-emb-is-equiv (is-equiv-precomp-is-truncation-equivalence k f H X) + is-emb-is-equiv (is-equiv-precomp-is-truncation-equivalence H X) ``` ### A map is a `k`-epimorphism if and only if its `k`-truncation is a `k`-epimorphism @@ -283,8 +283,7 @@ module _ is-epimorphism-Truncated-Type k f → is-truncation-equivalence k (codiagonal-map f) is-truncation-equivalence-codiagonal-map-is-epimorphism-Truncated-Type e = - is-truncation-equivalence-is-equiv-precomp k - ( codiagonal-map f) + is-truncation-equivalence-is-equiv-precomp ( λ l X → is-equiv-right-factor ( ( horizontal-map-cocone f f) ∘ @@ -330,7 +329,7 @@ module _ ( is-equiv-comp ( map-equiv (equiv-up-pushout f f (type-Truncated-Type X))) ( precomp (codiagonal-map f) (type-Truncated-Type X)) - ( is-equiv-precomp-is-truncation-equivalence k (codiagonal-map f) e X) + (is-equiv-precomp-is-truncation-equivalence e X) ( is-equiv-map-equiv (equiv-up-pushout f f (type-Truncated-Type X)))) is-epimorphism-is-truncation-equivalence-codiagonal-map-Truncated-Type : diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index f57fc0043e..5f90d24015 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -53,8 +53,8 @@ open import set-theory.baire-space public open import set-theory.bounded-increasing-binary-sequences public open import set-theory.cantor-space public open import set-theory.cantors-diagonal-argument public -open import set-theory.cardinality-recursive-sets public open import set-theory.cardinality-projective-sets public +open import set-theory.cardinality-recursive-sets public open import set-theory.cardinals public open import set-theory.complemented-inequality-cardinals public open import set-theory.countable-sets public diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index 728209cf2a..2434ce4eab 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -33,8 +33,8 @@ open import foundation.truncation-levels open import foundation.truncations open import foundation.universe-levels -open import set-theory.cardinals open import set-theory.cardinality-recursive-sets +open import set-theory.cardinals ```
@@ -128,12 +128,12 @@ module _ ind-is-connected-map is-cardinality-preprojective-Cardinality-Projective-Set compute-ind-Cardinality-Projective-Set : - {l3 : Level} - (P : (type-Cardinality-Projective-Set → Cardinal l2) → Set l3) - (T : - (Y : type-Cardinality-Projective-Set → Set l2) → - type-Set (P (cardinality ∘ Y))) - (Y : type-Cardinality-Projective-Set → Set l2) → + {l3 : Level} + (P : (type-Cardinality-Projective-Set → Cardinal l2) → Set l3) + (T : + (Y : type-Cardinality-Projective-Set → Set l2) → + type-Set (P (cardinality ∘ Y))) + (Y : type-Cardinality-Projective-Set → Set l2) → ind-Cardinality-Projective-Set P T (cardinality ∘ Y) ＝ T Y compute-ind-Cardinality-Projective-Set = compute-ind-is-connected-map diff --git a/src/synthetic-homotopy-theory/truncated-acyclic-maps.lagda.md b/src/synthetic-homotopy-theory/truncated-acyclic-maps.lagda.md index de8f09d97b..297572055d 100644 --- a/src/synthetic-homotopy-theory/truncated-acyclic-maps.lagda.md +++ b/src/synthetic-homotopy-theory/truncated-acyclic-maps.lagda.md @@ -407,9 +407,7 @@ module _ is-truncation-equivalence k f → is-truncated-acyclic-map k f is-truncated-acyclic-map-is-truncation-equivalence e = is-truncated-acyclic-map-is-epimorphism-Truncated-Type f - ( λ C → - is-emb-is-equiv - ( is-equiv-precomp-is-truncation-equivalence k f e C)) + ( λ C → is-emb-is-equiv (is-equiv-precomp-is-truncation-equivalence e C)) ``` ### `k`-acyclic maps are closed under pullbacks From d9fb6fdbd6a8bcc63c9cf497e5cc3b3ca81dd753 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 16 Nov 2025 02:37:58 +0100 Subject: [PATCH 29/50] dependent sums and products of inhabited cardinals --- .../dependent-products-cardinals.lagda.md | 91 +++++++++++--- .../dependent-sums-cardinals.lagda.md | 112 ++++++++++++++---- 2 files changed, 168 insertions(+), 35 deletions(-) diff --git a/src/set-theory/dependent-products-cardinals.lagda.md b/src/set-theory/dependent-products-cardinals.lagda.md index 0c391f5c52..e8386f8516 100644 --- a/src/set-theory/dependent-products-cardinals.lagda.md +++ b/src/set-theory/dependent-products-cardinals.lagda.md @@ -21,6 +21,7 @@ open import foundation.function-extensionality open import foundation.function-types open import foundation.functoriality-set-truncation open import foundation.identity-types +open import foundation.inhabited-types open import foundation.isolated-elements open import foundation.logical-equivalences open import foundation.negated-equality @@ -31,6 +32,7 @@ open import foundation.sections open import foundation.set-truncations open import foundation.sets open import foundation.surjective-maps +open import foundation.transport-along-identifications open import foundation.universe-levels open import foundation-core.empty-types @@ -39,8 +41,10 @@ open import foundation-core.propositions open import logic.propositionally-decidable-types -open import set-theory.cardinality-inductive-sets +open import set-theory.cardinality-projective-sets +open import set-theory.cardinality-recursive-sets open import set-theory.cardinals +open import set-theory.inhabited-cardinals ```
@@ -48,38 +52,97 @@ open import set-theory.cardinals ## Idea Given a family of cardinals $κ : I → \mathrm{Cardinal}$ over a -[cardinality-inductive set](set-theory.cardinality-inductive-sets.md) $I$, then -we may define the {{#concept "dependent product cardinal" Agda=Π-Cardinal}} +[cardinality-recursive set](set-theory.cardinality-recursive-sets.md) $I$, then +we may define the {{#concept "dependent product cardinal" Agda=Π-Cardinal'}} $Π_{i∈I}κᵢ$, as the cardinality of the [dependent product](foundation.dependent-function-types.md) of any family of representing sets $Kᵢ$. ## Definitions +### Dependent products of cardinals over cardinality-recursive sets + ```agda module _ - {l1 l2 : Level} (X : Cardinality-Inductive-Set l1 l2) - (let set-X = set-Cardinality-Inductive-Set X) + {l1 l2 : Level} (X : Cardinality-Recursive-Set l1 l2) + (let set-X = set-Cardinality-Recursive-Set X) where - Π-Cardinal : + Π-Cardinal' : (type-Set set-X → Cardinal l2) → Cardinal (l1 ⊔ l2) - Π-Cardinal Y = - map-trunc-Set (Π-Set set-X) (unit-Cardinality-Inductive-Set X Y) + Π-Cardinal' Y = + map-trunc-Set (Π-Set set-X) (unit-Cardinality-Recursive-Set X Y) - compute-Π-Cardinal : - (K : type-Cardinality-Inductive-Set X → Set l2) → - Π-Cardinal (cardinality ∘ K) ＝ cardinality (Π-Set set-X K) - compute-Π-Cardinal K = + compute-Π-Cardinal' : + (K : type-Cardinality-Recursive-Set X → Set l2) → + Π-Cardinal' (cardinality ∘ K) ＝ cardinality (Π-Set set-X K) + compute-Π-Cardinal' K = equational-reasoning map-trunc-Set ( Π-Set set-X) - ( unit-Cardinality-Inductive-Set X (cardinality ∘ K)) + ( unit-Cardinality-Recursive-Set X (cardinality ∘ K)) ＝ map-trunc-Set (Π-Set set-X) (unit-trunc-Set K) by ap ( map-trunc-Set (Π-Set set-X)) - ( compute-unit-Cardinality-Inductive-Set X K) + ( compute-unit-Cardinality-Recursive-Set X K) ＝ cardinality (Π-Set set-X K) by naturality-unit-trunc-Set (Π-Set set-X) K ``` + +### Dependent products of cardinals over cardinality-projective sets + +```agda +module _ + {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + where + + Π-Cardinal : + (type-Cardinality-Projective-Set X → Cardinal l2) → Cardinal (l1 ⊔ l2) + Π-Cardinal = + Π-Cardinal' (cardinality-recursive-set-Cardinality-Projective-Set X) + + compute-Π-Cardinal : + (Y : type-Cardinality-Projective-Set X → Set l2) → + Π-Cardinal (cardinality ∘ Y) ＝ + cardinality (Π-Set (set-Cardinality-Projective-Set X) Y) + compute-Π-Cardinal = + compute-Π-Cardinal' (cardinality-recursive-set-Cardinality-Projective-Set X) +``` + +## Properties + +### Dependent products of inhabited cardinals are inhabited + +```agda +module _ + {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + (let type-X = type-Cardinality-Projective-Set X) + where + + is-inhabited-Π-Cardinal : + (K : type-X → Cardinal l2) → + (is-inhabited-K : (x : type-X) → is-inhabited-Cardinal (K x)) → + is-inhabited-Cardinal (Π-Cardinal X K) + is-inhabited-Π-Cardinal = + ind-Cardinality-Projective-Set X + ( λ K → + set-Prop + ( function-Prop + ( (x : type-X) → is-inhabited-Cardinal (K x)) + ( is-inhabited-prop-Cardinal (Π-Cardinal X K)))) + ( λ Y y → + inv-tr + ( is-inhabited-Cardinal) + ( compute-Π-Cardinal X Y) + ( unit-is-inhabited-cardinality + ( Π-Set (set-Cardinality-Projective-Set X) Y) + ( is-projective-Cardinality-Projective-Set X + ( type-Set ∘ Y) + ( λ x → inv-unit-is-inhabited-cardinality (Y x) (y x))))) + + Π-Inhabited-Cardinal : + (type-X → Inhabited-Cardinal l2) → Inhabited-Cardinal (l1 ⊔ l2) + Π-Inhabited-Cardinal K = + ( Π-Cardinal X (pr1 ∘ K) , is-inhabited-Π-Cardinal (pr1 ∘ K) (pr2 ∘ K)) +``` diff --git a/src/set-theory/dependent-sums-cardinals.lagda.md b/src/set-theory/dependent-sums-cardinals.lagda.md index f1778191f0..f8d08b4919 100644 --- a/src/set-theory/dependent-sums-cardinals.lagda.md +++ b/src/set-theory/dependent-sums-cardinals.lagda.md @@ -23,6 +23,7 @@ open import foundation.function-types open import foundation.functoriality-propositional-truncation open import foundation.functoriality-set-truncation open import foundation.identity-types +open import foundation.inhabited-types open import foundation.isolated-elements open import foundation.logical-equivalences open import foundation.mere-embeddings @@ -35,6 +36,7 @@ open import foundation.sections open import foundation.set-truncations open import foundation.sets open import foundation.surjective-maps +open import foundation.transport-along-identifications open import foundation.universe-levels open import foundation-core.empty-types @@ -43,9 +45,11 @@ open import foundation-core.propositions open import logic.propositionally-decidable-types -open import set-theory.cardinality-inductive-sets +open import set-theory.cardinality-projective-sets +open import set-theory.cardinality-recursive-sets open import set-theory.cardinals open import set-theory.inequality-cardinals +open import set-theory.inhabited-cardinals ``` @@ -53,14 +57,16 @@ open import set-theory.inequality-cardinals ## Idea Given a family of cardinals $κ : I → \mathrm{Cardinal}$ over a -[cardinality-inductive set](set-theory.cardinality-inductive-sets.md) $I$, then -we may define the {{#concept "dependent sum cardinal" Agda=Σ-Cardinal}} +[cardinality-recursive set](set-theory.cardinality-recursive-sets.md) $I$, then +we may define the {{#concept "dependent sum cardinal" Agda=Σ-Cardinal'}} $Σ_{i∈I}κᵢ$, as the cardinality of the [dependent sum](foundation.dependent-pair-types.md) of any family of representing sets $Kᵢ$. ## Definitions +### The cardinality of a dependent sum of sets + ```agda module _ {l1 l2 : Level} (X : Set l1) @@ -68,35 +74,95 @@ module _ cardinality-Σ : (type-Set X → Set l2) → Cardinal (l1 ⊔ l2) cardinality-Σ Y = cardinality (Σ-Set X Y) +``` +### Dependent sums of cardinals over cardinality-recursive sets + +```agda module _ - {l1 l2 : Level} (X : Cardinality-Inductive-Set l1 l2) - (let set-X = set-Cardinality-Inductive-Set X) - (let type-X = type-Cardinality-Inductive-Set X) + {l1 l2 : Level} (X : Cardinality-Recursive-Set l1 l2) + (let set-X = set-Cardinality-Recursive-Set X) + (let type-X = type-Cardinality-Recursive-Set X) where - Σ-Cardinal : + Σ-Cardinal' : (type-X → Cardinal l2) → Cardinal (l1 ⊔ l2) - Σ-Cardinal K = - map-trunc-Set (Σ-Set set-X) (unit-Cardinality-Inductive-Set X K) + Σ-Cardinal' K = + map-trunc-Set (Σ-Set set-X) (unit-Cardinality-Recursive-Set X K) - compute-Σ-Cardinal : + compute-Σ-Cardinal' : (Y : type-X → Set l2) → - Σ-Cardinal (cardinality ∘ Y) ＝ cardinality (Σ-Set set-X Y) - compute-Σ-Cardinal Y = + Σ-Cardinal' (cardinality ∘ Y) ＝ cardinality (Σ-Set set-X Y) + compute-Σ-Cardinal' Y = equational-reasoning - Σ-Cardinal (cardinality ∘ Y) + Σ-Cardinal' (cardinality ∘ Y) ＝ map-trunc-Set (Σ-Set set-X) (unit-trunc-Set Y) by ap ( map-trunc-Set (Σ-Set set-X)) - ( compute-unit-Cardinality-Inductive-Set X Y) + ( compute-unit-Cardinality-Recursive-Set X Y) ＝ cardinality (Σ-Set set-X Y) by naturality-unit-trunc-Set (Σ-Set set-X) Y ``` +### Dependent sums of cardinals over cardinality-projective sets + +```agda +module _ + {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + where + + Σ-Cardinal : + (type-Cardinality-Projective-Set X → Cardinal l2) → Cardinal (l1 ⊔ l2) + Σ-Cardinal = + Σ-Cardinal' (cardinality-recursive-set-Cardinality-Projective-Set X) + + compute-Σ-Cardinal : + (Y : type-Cardinality-Projective-Set X → Set l2) → + Σ-Cardinal (cardinality ∘ Y) ＝ + cardinality (Σ-Set (set-Cardinality-Projective-Set X) Y) + compute-Σ-Cardinal = + compute-Σ-Cardinal' (cardinality-recursive-set-Cardinality-Projective-Set X) +``` + ## Properties +### Dependent sums of inhabited cardinals are inhabited + +```agda +module _ + {l1 l2 : Level} (X : Cardinality-Projective-Set l1 l2) + (let type-X = type-Cardinality-Projective-Set X) + (|x| : is-inhabited type-X) + where + + is-inhabited-Σ-Cardinal : + (K : type-X → Cardinal l2) → + (is-inhabited-K : (x : type-X) → is-inhabited-Cardinal (K x)) → + is-inhabited-Cardinal (Σ-Cardinal X K) + is-inhabited-Σ-Cardinal = + ind-Cardinality-Projective-Set X + ( λ K → + set-Prop + ( function-Prop + ( (x : type-X) → is-inhabited-Cardinal (K x)) + ( is-inhabited-prop-Cardinal (Σ-Cardinal X K)))) + ( λ Y y → + inv-tr + ( is-inhabited-Cardinal) + ( compute-Σ-Cardinal X Y) + ( unit-is-inhabited-cardinality + ( Σ-Set (set-Cardinality-Projective-Set X) Y) + ( is-inhabited-Σ + ( |x|) + ( λ x → inv-unit-is-inhabited-cardinality (Y x) (y x))))) + + Σ-Inhabited-Cardinal : + (type-X → Inhabited-Cardinal l2) → Inhabited-Cardinal (l1 ⊔ l2) + Σ-Inhabited-Cardinal K = + ( Σ-Cardinal X (pr1 ∘ K) , is-inhabited-Σ-Cardinal (pr1 ∘ K) (pr2 ∘ K)) +``` + ### Inequality is preserved under dependent sums over projective types ```agda @@ -116,17 +182,21 @@ module _ ( mere-emb-tot ( is-projective-X) ( λ x → inv-unit-leq-cardinality (K x) (P x) (f x))) +``` + +TODO +```text module _ - {l1 l2 : Level} (X : Cardinality-Inductive-Set l1 l2) - (let type-X = type-Cardinality-Inductive-Set X) - (is-projective-X : is-projective-Level' l2 (type-Cardinality-Inductive-Set X)) + {l1 l2 : Level} (X : Cardinality-Recursive-Set l1 l2) + (let type-X = type-Cardinality-Recursive-Set X) + (is-projective-X : is-projective-Level' l2 (type-Cardinality-Recursive-Set X)) where - leq-Σ-Cardinal : + leq-Σ-Cardinal' : (K P : type-X → Cardinal l2) → ((i : type-X) → leq-Cardinal (K i) (P i)) → - leq-Cardinal (Σ-Cardinal X K) (Σ-Cardinal X P) - leq-Σ-Cardinal K P f = {! !} - -- proof somehow proceeds by using that since `X` is cardinality-inductive, it suffices to show this for families of sets, and then it's just an easy fact of dependent sums. + leq-Cardinal (Σ-Cardinal' X K) (Σ-Cardinal' X P) + leq-Σ-Cardinal' K P f = {! !} + -- proof somehow proceeds by using that since `X` is cardinality-recursive, it suffices to show this for families of sets, and then it's just an easy fact of dependent sums. ``` From 211a48b06b1b209e7d7c532496fa68f015557f17 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 16 Nov 2025 02:50:37 +0100 Subject: [PATCH 30/50] names --- src/set-theory.lagda.md | 4 ++-- src/set-theory/dependent-products-cardinals.lagda.md | 2 +- src/set-theory/dependent-sums-cardinals.lagda.md | 2 +- ....md => strict-complemented-boundedness-cardinals.lagda.md} | 2 +- ...a.md => strict-complemented-inequality-cardinals.lagda.md} | 2 +- 5 files changed, 6 insertions(+), 6 deletions(-) rename src/set-theory/{strict-complemented-boundedness-cardinalities.lagda.md => strict-complemented-boundedness-cardinals.lagda.md} (98%) rename src/set-theory/{strict-complemented-inequality-cardinalities.lagda.md => strict-complemented-inequality-cardinals.lagda.md} (99%) diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index 5f90d24015..8243f73f42 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -71,8 +71,8 @@ open import set-theory.infinite-sets public open import set-theory.inhabited-cardinals public open import set-theory.positive-elements-increasing-binary-sequences public open import set-theory.russells-paradox public -open import set-theory.strict-complemented-boundedness-cardinalities public -open import set-theory.strict-complemented-inequality-cardinalities public +open import set-theory.strict-complemented-boundedness-cardinals public +open import set-theory.strict-complemented-inequality-cardinals public open import set-theory.strict-lower-bounds-increasing-binary-sequences public open import set-theory.uncountable-sets public open import set-theory.zero-cardinal public diff --git a/src/set-theory/dependent-products-cardinals.lagda.md b/src/set-theory/dependent-products-cardinals.lagda.md index e8386f8516..8d187aff03 100644 --- a/src/set-theory/dependent-products-cardinals.lagda.md +++ b/src/set-theory/dependent-products-cardinals.lagda.md @@ -122,7 +122,7 @@ module _ is-inhabited-Π-Cardinal : (K : type-X → Cardinal l2) → - (is-inhabited-K : (x : type-X) → is-inhabited-Cardinal (K x)) → + ((x : type-X) → is-inhabited-Cardinal (K x)) → is-inhabited-Cardinal (Π-Cardinal X K) is-inhabited-Π-Cardinal = ind-Cardinality-Projective-Set X diff --git a/src/set-theory/dependent-sums-cardinals.lagda.md b/src/set-theory/dependent-sums-cardinals.lagda.md index f8d08b4919..0d5e7486f0 100644 --- a/src/set-theory/dependent-sums-cardinals.lagda.md +++ b/src/set-theory/dependent-sums-cardinals.lagda.md @@ -138,7 +138,7 @@ module _ is-inhabited-Σ-Cardinal : (K : type-X → Cardinal l2) → - (is-inhabited-K : (x : type-X) → is-inhabited-Cardinal (K x)) → + ((x : type-X) → is-inhabited-Cardinal (K x)) → is-inhabited-Cardinal (Σ-Cardinal X K) is-inhabited-Σ-Cardinal = ind-Cardinality-Projective-Set X diff --git a/src/set-theory/strict-complemented-boundedness-cardinalities.lagda.md b/src/set-theory/strict-complemented-boundedness-cardinals.lagda.md similarity index 98% rename from src/set-theory/strict-complemented-boundedness-cardinalities.lagda.md rename to src/set-theory/strict-complemented-boundedness-cardinals.lagda.md index 5faef5d29a..0e2dbbef0e 100644 --- a/src/set-theory/strict-complemented-boundedness-cardinalities.lagda.md +++ b/src/set-theory/strict-complemented-boundedness-cardinals.lagda.md @@ -1,7 +1,7 @@ # Strict complemented boundedness on cardinals ```agda -module set-theory.strict-complemented-boundedness-cardinalities where +module set-theory.strict-complemented-boundedness-cardinals where ```
Imports diff --git a/src/set-theory/strict-complemented-inequality-cardinalities.lagda.md b/src/set-theory/strict-complemented-inequality-cardinals.lagda.md similarity index 99% rename from src/set-theory/strict-complemented-inequality-cardinalities.lagda.md rename to src/set-theory/strict-complemented-inequality-cardinals.lagda.md index 1220acac0f..36429748aa 100644 --- a/src/set-theory/strict-complemented-inequality-cardinalities.lagda.md +++ b/src/set-theory/strict-complemented-inequality-cardinals.lagda.md @@ -27,7 +27,7 @@ open import foundation.universe-levels open import set-theory.cardinals open import set-theory.complemented-inequality-cardinals open import set-theory.inequality-cardinals -open import set-theory.strict-complemented-boundedness-cardinalities +open import set-theory.strict-complemented-boundedness-cardinals ```
From 67e70bc76fd6edd872c8113151693af2aec21ef7 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 16 Nov 2025 12:08:32 +0100 Subject: [PATCH 31/50] finite sets are cardinality-projective --- .../transpositions.lagda.md | 4 +- src/foundation/projective-types.lagda.md | 28 +++- .../cardinality-projective-sets.lagda.md | 120 +++++++++++++++++- .../counting-decidable-subtypes.lagda.md | 2 +- src/univalent-combinatorics/counting.lagda.md | 7 +- ...t-truncation-over-finite-products.lagda.md | 71 ++++++++++- .../finite-choice.lagda.md | 4 + .../finite-types.lagda.md | 2 +- ...initely-many-connected-components.lagda.md | 2 +- ...tations-complete-undirected-graph.lagda.md | 56 ++++---- .../pigeonhole-principle.lagda.md | 4 +- .../retracts-of-finite-types.lagda.md | 2 +- 12 files changed, 255 insertions(+), 47 deletions(-) diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md index a38bb0720c..2b61f3fe91 100644 --- a/src/finite-group-theory/transpositions.lagda.md +++ b/src/finite-group-theory/transpositions.lagda.md @@ -446,11 +446,11 @@ module _ apply-universal-property-trunc-Prop (pr2 Y) ( coproduct-Prop ( Id-Prop - ( pair X (is-set-count eX)) + ( set-type-count eX) ( pr1 two-elements-transposition) ( x)) ( Id-Prop - ( pair X (is-set-count eX)) + ( set-type-count eX) ( pr1 (pr2 two-elements-transposition)) ( x)) ( λ q r → diff --git a/src/foundation/projective-types.lagda.md b/src/foundation/projective-types.lagda.md index 7e0578ef97..5a2fd4906d 100644 --- a/src/foundation/projective-types.lagda.md +++ b/src/foundation/projective-types.lagda.md @@ -10,12 +10,14 @@ module foundation.projective-types where open import elementary-number-theory.natural-numbers open import foundation.connected-maps +open import foundation.dependent-pair-types open import foundation.inhabited-types open import foundation.postcomposition-functions open import foundation.surjective-maps open import foundation.truncation-levels open import foundation.universe-levels +open import foundation-core.propositions open import foundation-core.sets open import foundation-core.truncated-types ``` @@ -90,11 +92,27 @@ is-trunc-projective k X = {l2 l3 : Level} → is-trunc-projective-Level l2 l3 k ### Alternative statement of set-projectivity ```agda -is-projective-Level' : {l1 : Level} (l2 : Level) → UU l1 → UU (l1 ⊔ lsuc l2) -is-projective-Level' l2 X = - (P : X → UU l2) → - ((x : X) → is-inhabited (P x)) → - is-inhabited ((x : X) → (P x)) +module _ + {l1 : Level} (l2 : Level) (X : UU l1) + where + + is-projective-Level' : UU (l1 ⊔ lsuc l2) + is-projective-Level' = + (P : X → UU l2) → + ((x : X) → is-inhabited (P x)) → + is-inhabited ((x : X) → (P x)) + + abstract + is-prop-is-projective-Level' : is-prop is-projective-Level' + is-prop-is-projective-Level' = + is-prop-Π + ( λ P → + is-prop-function-type + ( is-property-is-inhabited ((x : X) → P x))) + + is-projective-prop-Level' : Prop (l1 ⊔ lsuc l2) + is-projective-prop-Level' = + ( is-projective-Level' , is-prop-is-projective-Level') is-projective' : {l1 : Level} → UU l1 → UUω is-projective' X = {l2 : Level} → is-projective-Level' l2 X diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index 2434ce4eab..b2eb5ae54c 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -7,6 +7,8 @@ module set-theory.cardinality-projective-sets where
Imports ```agda +open import elementary-number-theory.natural-numbers + open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.connected-maps @@ -22,6 +24,8 @@ open import foundation.injective-maps open import foundation.mere-equivalences open import foundation.postcomposition-functions open import foundation.projective-types +open import foundation.propositional-truncations +open import foundation.propositions open import foundation.retractions open import foundation.retracts-of-types open import foundation.set-truncations @@ -35,6 +39,12 @@ open import foundation.universe-levels open import set-theory.cardinality-recursive-sets open import set-theory.cardinals + +open import univalent-combinatorics.counting +open import univalent-combinatorics.distributivity-of-set-truncation-over-finite-products +open import univalent-combinatorics.finite-choice +open import univalent-combinatorics.finite-types +open import univalent-combinatorics.standard-finite-types ```
@@ -59,6 +69,16 @@ module _ is-cardinality-preprojective-set-Level : UU (l1 ⊔ lsuc l2) is-cardinality-preprojective-set-Level = is-connected-map zero-𝕋 (postcomp (type-Set I) (cardinality {l2})) + + is-prop-is-cardinality-preprojective-set-Level : + is-prop is-cardinality-preprojective-set-Level + is-prop-is-cardinality-preprojective-set-Level = + is-prop-is-connected-map zero-𝕋 (postcomp (type-Set I) cardinality) + + is-cardinality-preprojective-set-prop-Level : Prop (l1 ⊔ lsuc l2) + is-cardinality-preprojective-set-prop-Level = + ( is-cardinality-preprojective-set-Level , + is-prop-is-cardinality-preprojective-set-Level) ``` ### The predicate of being cardinality-projective at a universe level @@ -72,6 +92,18 @@ module _ is-cardinality-projective-set-Level = is-connected-map zero-𝕋 (postcomp (type-Set I) (cardinality {l2})) × is-projective-Level' l2 (type-Set I) + + is-prop-is-cardinality-projective-set-Level : + is-prop is-cardinality-projective-set-Level + is-prop-is-cardinality-projective-set-Level = + is-prop-product + ( is-prop-is-cardinality-preprojective-set-Level l2 I) + ( is-prop-is-projective-Level' l2 (type-Set I)) + + is-cardinality-projective-set-prop-Level : Prop (l1 ⊔ lsuc l2) + is-cardinality-projective-set-prop-Level = + ( is-cardinality-projective-set-Level , + is-prop-is-cardinality-projective-set-Level) ``` ### The universe of cardinality-projective sets at a universe level @@ -295,7 +327,7 @@ module _ ( cardinality-recursive-set-Cardinality-Projective-Set) ``` -### A set is cardinality-preprojective if the postcomposition map is a set-truncation equivalence +### A set is cardinality-preprojective if the postcomposition map is a set-equivalence ```agda module _ @@ -314,6 +346,92 @@ module _ ( H)) ``` +### The standard finite sets are cardinality-projective + +```agda +module _ + {l : Level} (n : ℕ) + where + + abstract + is-cardinality-preprojective-Fin : + is-cardinality-preprojective-set-Level l (Fin-Set n) + is-cardinality-preprojective-Fin = + is-connected-map-left-map-triangle + ( postcomp (Fin n) cardinality) + ( map-equiv-distributive-trunc-Π-Fin-Set n (λ _ → Set l)) + ( unit-trunc-Set) + ( inv-htpy (triangle-distributive-trunc-Π-Fin-Set n (λ _ → Set l))) + ( is-connected-map-unit-trunc zero-𝕋) + ( is-connected-map-is-equiv + ( is-equiv-map-equiv-distributive-trunc-Π-Fin-Set n (λ _ → Set l))) + + is-cardinality-projective-Fin : + is-cardinality-projective-set-Level l (Fin-Set n) + is-cardinality-projective-Fin = + ( is-cardinality-preprojective-Fin , (λ P → finite-choice-Fin n)) + + cardinality-projective-set-Fin : Cardinality-Projective-Set lzero l + cardinality-projective-set-Fin = (Fin-Set n , is-cardinality-projective-Fin) +``` + +### Sets equipped with counting are cardinality-projective + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (c : count A) + where + + abstract + is-cardinality-preprojective-set-count : + is-cardinality-preprojective-set-Level l2 (set-type-count c) + is-cardinality-preprojective-set-count = + is-connected-map-left-map-triangle + ( postcomp A cardinality) + ( map-equiv-distributive-trunc-Π-count-Set c (λ _ → Set l2)) + ( unit-trunc-Set) + ( inv-htpy (triangle-distributive-trunc-Π-count-Set c (λ _ → Set l2))) + ( is-connected-map-unit-trunc zero-𝕋) + ( is-connected-map-is-equiv + ( is-equiv-map-equiv-distributive-trunc-Π-count-Set c (λ _ → Set l2))) + + is-cardinality-projective-set-count : + is-cardinality-projective-set-Level l2 (set-type-count c) + is-cardinality-projective-set-count = + ( is-cardinality-preprojective-set-count , (λ P → finite-choice-count c)) + + cardinality-projective-set-count : Cardinality-Projective-Set l1 l2 + cardinality-projective-set-count = + ( set-type-count c , is-cardinality-projective-set-count) +``` + +### Finite sets are cardinality-projective + +```agda +module _ + {l1 l2 : Level} (A : Finite-Type l1) + where + + abstract + is-cardinality-preprojective-set-Finite-Type : + is-cardinality-preprojective-set-Level l2 (set-Finite-Type A) + is-cardinality-preprojective-set-Finite-Type = + rec-trunc-Prop + ( is-cardinality-preprojective-set-prop-Level l2 (set-Finite-Type A)) + ( is-cardinality-preprojective-set-count) + ( is-finite-type-Finite-Type A) + + is-cardinality-projective-set-Finite-Type : + is-cardinality-projective-set-Level l2 (set-Finite-Type A) + is-cardinality-projective-set-Finite-Type = + ( is-cardinality-preprojective-set-Finite-Type , + ( λ P → finite-choice (is-finite-type-Finite-Type A))) + + cardinality-projective-set-Finite-Type : Cardinality-Projective-Set l1 l2 + cardinality-projective-set-Finite-Type = + ( set-Finite-Type A , is-cardinality-projective-set-Finite-Type) +``` + ## See also - In diff --git a/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md b/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md index 9a0db96729..95ed6cfe55 100644 --- a/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md +++ b/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md @@ -118,7 +118,7 @@ is-decidable-count-subtype P e f x = pair ( pr1 y ＝ x) ( pair - ( is-set-count e (pr1 y) x) + ( is-set-type-count e (pr1 y) x) ( has-decidable-equality-count e (pr1 y) x))) ( f))) ``` diff --git a/src/univalent-combinatorics/counting.lagda.md b/src/univalent-combinatorics/counting.lagda.md index 71aae67e0f..a0f7e1317b 100644 --- a/src/univalent-combinatorics/counting.lagda.md +++ b/src/univalent-combinatorics/counting.lagda.md @@ -71,12 +71,15 @@ module _ inv-equiv-count : X ≃ Fin number-of-elements-count inv-equiv-count = inv-equiv equiv-count - is-set-count : is-set X - is-set-count = + is-set-type-count : is-set X + is-set-type-count = is-set-equiv' ( Fin number-of-elements-count) ( equiv-count) ( is-set-Fin number-of-elements-count) + + set-type-count : Set l + set-type-count = (X , is-set-type-count) ``` ## Properties diff --git a/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md b/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md index 69782fd17e..ee730f0bd7 100644 --- a/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md +++ b/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md @@ -132,6 +132,11 @@ module _ map-equiv-distributive-trunc-Π-Fin-Set = map-equiv equiv-distributive-trunc-Π-Fin-Set + is-equiv-map-equiv-distributive-trunc-Π-Fin-Set : + is-equiv map-equiv-distributive-trunc-Π-Fin-Set + is-equiv-map-equiv-distributive-trunc-Π-Fin-Set = + is-equiv-map-equiv equiv-distributive-trunc-Π-Fin-Set + triangle-distributive-trunc-Π-Fin-Set : ( map-equiv-distributive-trunc-Π-Fin-Set ∘ unit-trunc-Set) ~ ( map-Π (λ x → unit-trunc-Set)) @@ -242,9 +247,15 @@ module _ ( x)))))))) ∘e ( equiv-funext)))) ( distributive-trunc-Π-Fin-Set k (B ∘ map-equiv e))) +``` + +## Corollaries +### Set-truncation distributes over sets equipped with counting + +```agda module _ - {l1 l2 : Level} {A : UU l1} (B : A → UU l2) (c : count A) + {l1 l2 : Level} {A : UU l1} (c : count A) (B : A → UU l2) where equiv-distributive-trunc-Π-count-Set : @@ -257,14 +268,23 @@ module _ map-equiv-distributive-trunc-Π-count-Set = map-equiv equiv-distributive-trunc-Π-count-Set + is-equiv-map-equiv-distributive-trunc-Π-count-Set : + is-equiv map-equiv-distributive-trunc-Π-count-Set + is-equiv-map-equiv-distributive-trunc-Π-count-Set = + is-equiv-map-equiv equiv-distributive-trunc-Π-count-Set + triangle-distributive-trunc-Π-count-Set : ( map-equiv-distributive-trunc-Π-count-Set ∘ unit-trunc-Set) ~ ( map-Π (λ x → unit-trunc-Set)) triangle-distributive-trunc-Π-count-Set = pr2 (center (distributive-trunc-Π-count-Set B c)) +``` + +### Set-truncation distributes over finite sets +```agda module _ - {l1 l2 : Level} {A : UU l1} (B : A → UU l2) (H : is-finite A) + {l1 l2 : Level} {A : UU l1} (H : is-finite A) (B : A → UU l2) where abstract @@ -290,14 +310,59 @@ module _ map-equiv-distributive-trunc-Π-is-finite-Set = map-equiv equiv-distributive-trunc-Π-is-finite-Set + is-equiv-map-equiv-distributive-trunc-Π-is-finite-Set : + is-equiv map-equiv-distributive-trunc-Π-is-finite-Set + is-equiv-map-equiv-distributive-trunc-Π-is-finite-Set = + is-equiv-map-equiv equiv-distributive-trunc-Π-is-finite-Set + triangle-distributive-trunc-Π-is-finite-Set : ( map-equiv-distributive-trunc-Π-is-finite-Set ∘ unit-trunc-Set) ~ ( map-Π (λ x → unit-trunc-Set)) triangle-distributive-trunc-Π-is-finite-Set = pr2 (center distributive-trunc-Π-is-finite-Set) + +module _ + {l1 l2 : Level} (A : Finite-Type l1) (B : type-Finite-Type A → UU l2) + (let H = is-finite-type-Finite-Type A) + where + + distributive-trunc-Π-Finite-Type : + is-contr + ( Σ ( ( ║ ((x : type-Finite-Type A) → B x) ║₀) ≃ + ( (x : type-Finite-Type A) → ║ B x ║₀)) + ( λ e → + ( map-equiv e ∘ unit-trunc-Set) ~ + ( map-Π (λ x → unit-trunc-Set)))) + distributive-trunc-Π-Finite-Type = + distributive-trunc-Π-is-finite-Set H B + + equiv-distributive-trunc-Π-Finite-Type : + ║ ((x : type-Finite-Type A) → B x) ║₀ ≃ + ((x : type-Finite-Type A) → ║ B x ║₀) + equiv-distributive-trunc-Π-Finite-Type = + equiv-distributive-trunc-Π-is-finite-Set H B + + map-equiv-distributive-trunc-Π-Finite-Type : + ║ ((x : type-Finite-Type A) → B x) ║₀ → + ((x : type-Finite-Type A) → ║ B x ║₀) + map-equiv-distributive-trunc-Π-Finite-Type = + map-equiv-distributive-trunc-Π-is-finite-Set H B + + is-equiv-map-equiv-distributive-trunc-Π-Finite-Type : + is-equiv map-equiv-distributive-trunc-Π-Finite-Type + is-equiv-map-equiv-distributive-trunc-Π-Finite-Type = + is-equiv-map-equiv-distributive-trunc-Π-is-finite-Set H B + + triangle-distributive-trunc-Π-Finite-Type : + ( map-equiv-distributive-trunc-Π-Finite-Type ∘ unit-trunc-Set) ~ + ( map-Π (λ x → unit-trunc-Set)) + triangle-distributive-trunc-Π-Finite-Type = + triangle-distributive-trunc-Π-is-finite-Set H B ``` ## See also -- This in particular means that finite sets are +- [Finite choice](univalent-combinatorics.finite-choice.md) for distributivity + of propositional truncation over finite products. +- Distributivity of set-truncation in particular means that finite sets are [cardinality-projective](set-theory.cardinality-projective-sets.md). diff --git a/src/univalent-combinatorics/finite-choice.lagda.md b/src/univalent-combinatorics/finite-choice.lagda.md index 43f0971a20..e9d809bc9b 100644 --- a/src/univalent-combinatorics/finite-choice.lagda.md +++ b/src/univalent-combinatorics/finite-choice.lagda.md @@ -193,3 +193,7 @@ module _ ( double-negation-type-Prop ((x : X) → Y x)) ( λ e → finite-double-negation-choice-count e H) ``` + +## See also + +- [Distributivity of set truncation over finite products](univalent-combinatorics.distributivity-of-set-truncation-over-finite-products.md) diff --git a/src/univalent-combinatorics/finite-types.lagda.md b/src/univalent-combinatorics/finite-types.lagda.md index 4dae49548c..b3428795af 100644 --- a/src/univalent-combinatorics/finite-types.lagda.md +++ b/src/univalent-combinatorics/finite-types.lagda.md @@ -450,7 +450,7 @@ abstract is-set-is-finite {l} {X} H = apply-universal-property-trunc-Prop H ( is-set-Prop X) - ( λ e → is-set-count e) + ( λ e → is-set-type-count e) is-set-type-Finite-Type : {l : Level} (X : Finite-Type l) → is-set (type-Finite-Type X) diff --git a/src/univalent-combinatorics/finitely-many-connected-components.lagda.md b/src/univalent-combinatorics/finitely-many-connected-components.lagda.md index b2d49dc3a5..b93098a62c 100644 --- a/src/univalent-combinatorics/finitely-many-connected-components.lagda.md +++ b/src/univalent-combinatorics/finitely-many-connected-components.lagda.md @@ -214,7 +214,7 @@ has-finitely-many-connected-components-finite-Π : has-finitely-many-connected-components ((a : A) → B a) has-finitely-many-connected-components-finite-Π {B = B} H K = is-finite-equiv' - ( equiv-distributive-trunc-Π-is-finite-Set B H) + ( equiv-distributive-trunc-Π-is-finite-Set H B) ( is-finite-Π H K) ``` diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md index adf834e7da..6d1ada5aae 100644 --- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md +++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md @@ -899,10 +899,10 @@ module _ ( eq-pair-Σ ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX (pr1 (two-elements-transposition eX Y)) i))) + ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) i))) ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX (pr1 (two-elements-transposition eX Y)) j))))) ∙ + ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) j))))) ∙ ( r1) cases-inward-edge-left-two-elements-orientation-count i j np Y x nq nr (inr (pair r1 r2)) = @@ -928,7 +928,7 @@ module _ { y = inl r1} ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX (pr1 (two-elements-transposition eX Y)) i)))) ∙ + ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) i)))) ∙ ( r2) inward-edge-left-two-elements-orientation-count : @@ -1013,7 +1013,7 @@ module _ ( λ r → nr (inv r))))} ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX + ( is-set-type-count eX ( map-equiv ( transposition ( standard-2-Element-Decidable-Subtype @@ -1096,7 +1096,7 @@ module _ ( eq-pair-Σ ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX + ( is-set-type-count eX ( map-equiv ( transposition ( standard-2-Element-Decidable-Subtype @@ -1106,7 +1106,7 @@ module _ ( pr1 (two-elements-transposition eX Y))))) ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX + ( is-set-type-count eX ( map-equiv ( transposition ( standard-2-Element-Decidable-Subtype @@ -1173,10 +1173,10 @@ module _ ( eq-pair-Σ ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX (pr1 (two-elements-transposition eX Y)) i))) + ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) i))) ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX (pr1 (two-elements-transposition eX Y)) j))))) ∙ + ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) j))))) ∙ ( r1) cases-inward-edge-right-two-elements-orientation-count i j np Y x nq nr (inr (pair r1 r2)) = @@ -1205,10 +1205,10 @@ module _ ( eq-pair-Σ ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX (pr1 (two-elements-transposition eX Y)) i))) + ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) i))) ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX (pr1 (two-elements-transposition eX Y)) j))))) ∙ + ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) j))))) ∙ ( ( ap ( λ d → pr1 @@ -1225,7 +1225,7 @@ module _ { y = inr (λ q → nq (inv r2 ∙ q))} ( eq-is-prop ( is-prop-is-decidable - ( is-set-count + ( is-set-type-count ( eX) ( pr1 (pr2 (two-elements-transposition eX Y))) ( i))))) ∙ @@ -1312,7 +1312,7 @@ module _ ( λ r → nr (inv r))))} ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX + ( is-set-type-count eX ( map-equiv ( transposition ( standard-2-Element-Decidable-Subtype @@ -1395,7 +1395,7 @@ module _ ( eq-pair-Σ ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX + ( is-set-type-count eX ( map-equiv ( transposition ( standard-2-Element-Decidable-Subtype @@ -1405,7 +1405,7 @@ module _ ( pr1 (two-elements-transposition eX Y))))) ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX + ( is-set-type-count eX ( map-equiv ( transposition ( standard-2-Element-Decidable-Subtype @@ -1765,7 +1765,7 @@ module _ ( R) ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX + ( is-set-type-count eX ( map-equiv ( transposition ( standard-2-Element-Decidable-Subtype @@ -1787,8 +1787,8 @@ module _ ( has-decidable-equality-count eX x i) ( has-decidable-equality-count eX x j)} ( eq-pair-Σ - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x i))) - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x j)))) ∙ + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x i))) + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x j)))) ∙ ap ( λ k → pr1 @@ -2019,7 +2019,7 @@ module _ ( R) ( eq-is-prop ( is-prop-is-decidable - ( is-set-count eX + ( is-set-type-count eX ( map-equiv ( transposition ( standard-2-Element-Decidable-Subtype @@ -2041,8 +2041,8 @@ module _ ( has-decidable-equality-count eX x j) ( has-decidable-equality-count eX x i)} ( eq-pair-Σ - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x j))) - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x i)))) ∙ + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x j))) + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x i)))) ∙ ( ap ( λ k → pr1 @@ -2707,8 +2707,8 @@ module _ ( has-decidable-equality-count eX x j)} { y = pair (inr nq) (inr nr)} ( eq-pair-Σ - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x i))) - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x j)))))))) ∙ + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x i))) + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x j)))))))) ∙ ( ( is-fixed-point-standard-transposition ( has-decidable-equality-count eX) ( np) @@ -2729,8 +2729,8 @@ module _ ( has-decidable-equality-count eX x j) ( has-decidable-equality-count eX x i)} ( eq-pair-Σ - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x j))) - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x i))))) ∙ + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x j))) + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x i))))) ∙ ( ap ( λ k → pr1 @@ -3050,8 +3050,8 @@ module _ ( has-decidable-equality-count eX y i)) { y = pair (inr nq) (inr nr)} ( eq-pair-Σ - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x i))) - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x j))))) ∙ + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x i))) + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x j))))) ∙ ( ap ( λ D → cases-orientation-two-elements-count j i @@ -3066,8 +3066,8 @@ module _ ( has-decidable-equality-count eX x j) ( has-decidable-equality-count eX x i)} ( eq-pair-Σ - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x j))) - ( eq-is-prop (is-prop-is-decidable (is-set-count eX x i)))) ∙ + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x j))) + ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x i)))) ∙ ( ap ( λ w → cases-orientation-two-elements-count j i diff --git a/src/univalent-combinatorics/pigeonhole-principle.lagda.md b/src/univalent-combinatorics/pigeonhole-principle.lagda.md index dc8a1e1e99..c007d7dc83 100644 --- a/src/univalent-combinatorics/pigeonhole-principle.lagda.md +++ b/src/univalent-combinatorics/pigeonhole-principle.lagda.md @@ -205,7 +205,7 @@ module _ {f : A → B} → is-injective f → number-of-elements-count eA ≤-ℕ number-of-elements-count eB leq-is-injective-count H = - leq-is-emb-count (is-emb-is-injective (is-set-count eB) H) + leq-is-emb-count (is-emb-is-injective (is-set-type-count eB) H) ``` #### There is no embedding `A ↪ B` between types equipped with a counting if the number of elements of `B` is strictly less than the number of elements of `A` @@ -237,7 +237,7 @@ module _ number-of-elements-count eB <-ℕ number-of-elements-count eA → is-not-injective f is-not-injective-le-count f p H = - is-not-emb-le-count f p (is-emb-is-injective (is-set-count eB) H) + is-not-emb-le-count f p (is-emb-is-injective (is-set-type-count eB) H) ``` #### There is no embedding `ℕ ↪ A` into a type equipped with a counting diff --git a/src/univalent-combinatorics/retracts-of-finite-types.lagda.md b/src/univalent-combinatorics/retracts-of-finite-types.lagda.md index d7b3d888e4..497a61f0f3 100644 --- a/src/univalent-combinatorics/retracts-of-finite-types.lagda.md +++ b/src/univalent-combinatorics/retracts-of-finite-types.lagda.md @@ -64,7 +64,7 @@ abstract {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : count B) (i : A → B) → retraction i → is-emb i is-emb-retract-count e i R = - is-emb-is-injective (is-set-count e) (is-injective-retraction i R) + is-emb-is-injective (is-set-type-count e) (is-injective-retraction i R) emb-retract-count : {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : count B) (i : A → B) → From 5c16bb9818d36d39fde52eae91bec83a51bc07f0 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 16 Nov 2025 12:09:12 +0100 Subject: [PATCH 32/50] indent --- src/set-theory/cardinality-projective-sets.lagda.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index b2eb5ae54c..813e72daea 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -166,7 +166,7 @@ module _ (Y : type-Cardinality-Projective-Set → Set l2) → type-Set (P (cardinality ∘ Y))) (Y : type-Cardinality-Projective-Set → Set l2) → - ind-Cardinality-Projective-Set P T (cardinality ∘ Y) ＝ T Y + ind-Cardinality-Projective-Set P T (cardinality ∘ Y) ＝ T Y compute-ind-Cardinality-Projective-Set = compute-ind-is-connected-map ( is-cardinality-preprojective-Cardinality-Projective-Set) From 11e82af3e5f58853650f1856063fae032e525225 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 15:10:45 +0000 Subject: [PATCH 33/50] =?UTF-8?q?prove=20K=C5=91nig's=20theorem?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../wikipedia-list-of-theorems.lagda.md | 9 + .../cardinality-projective-sets.lagda.md | 15 ++ .../dependent-products-cardinals.lagda.md | 9 + src/set-theory/konigs-theorem.lagda.md | 134 ++++++++++++ ...complemented-inequality-cardinals.lagda.md | 19 +- ...rict-indexed-inequality-cardinals.lagda.md | 202 ++++++++++++++++++ 6 files changed, 378 insertions(+), 10 deletions(-) create mode 100644 src/set-theory/konigs-theorem.lagda.md create mode 100644 src/set-theory/strict-indexed-inequality-cardinals.lagda.md diff --git a/src/literature/wikipedia-list-of-theorems.lagda.md b/src/literature/wikipedia-list-of-theorems.lagda.md index 0c7846e7b9..b9cf8ec25c 100644 --- a/src/literature/wikipedia-list-of-theorems.lagda.md +++ b/src/literature/wikipedia-list-of-theorems.lagda.md @@ -140,6 +140,15 @@ open import order-theory.knaster-tarski-fixed-point-theorem using greatest-fixed-point-knaster-tarski-Suplattice) ``` +### König's theorem {#Q1077462} + +**Author:** [Fredrik Bakke](https://www.ntnu.edu/employees/fredrik.bakke) + +```agda +open import set-theory.konigs-theorem using + ( Königs-Theorem) +``` + ### Lawvere's fixed point theorem {#Q15809744} **Author:** [Egbert Rijke](https://egbertrijke.github.io) diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index 813e72daea..16fe26fadc 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -171,6 +171,21 @@ module _ compute-ind-is-connected-map ( is-cardinality-preprojective-Cardinality-Projective-Set) + apply-twice-ind-Cardinality-Projective-Set : + {l3 : Level} + (P : (X Y : type-Cardinality-Projective-Set → Cardinal l2) → Set l3) → + ( (X Y : type-Cardinality-Projective-Set → Set l2) → + type-Set (P (cardinality ∘ X) (cardinality ∘ Y))) → + (X Y : type-Cardinality-Projective-Set → Cardinal l2) → type-Set (P X Y) + apply-twice-ind-Cardinality-Projective-Set P h X = + ind-Cardinality-Projective-Set + ( P X) + ( λ Y → + ind-Cardinality-Projective-Set + ( λ X → P X (cardinality ∘ Y)) + ( λ X → h X Y) + ( X)) + constr-Cardinality-Projective-Set : {l : Level} → ((type-Cardinality-Projective-Set → Set l2) → Cardinal l) → diff --git a/src/set-theory/dependent-products-cardinals.lagda.md b/src/set-theory/dependent-products-cardinals.lagda.md index 8d187aff03..3dc5cbc232 100644 --- a/src/set-theory/dependent-products-cardinals.lagda.md +++ b/src/set-theory/dependent-products-cardinals.lagda.md @@ -60,6 +60,15 @@ representing sets $Kᵢ$. ## Definitions +```agda +module _ + {l1 l2 : Level} (X : Set l1) + where + + cardinality-Π : (type-Set X → Set l2) → Cardinal (l1 ⊔ l2) + cardinality-Π Y = cardinality (Π-Set X Y) +``` + ### Dependent products of cardinals over cardinality-recursive sets ```agda diff --git a/src/set-theory/konigs-theorem.lagda.md b/src/set-theory/konigs-theorem.lagda.md new file mode 100644 index 0000000000..66cbe731b4 --- /dev/null +++ b/src/set-theory/konigs-theorem.lagda.md @@ -0,0 +1,134 @@ +# Kőnig's theorem + +```agda +module set-theory.konigs-theorem where +``` + +
Imports + +```agda +open import foundation.binary-transport +open import foundation.dependent-pair-types +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.functoriality-propositional-truncation +open import foundation.identity-types +open import foundation.nonsurjective-maps +open import foundation.projective-types +open import foundation.propositions +open import foundation.sets +open import foundation.universe-levels + +open import set-theory.cardinality-projective-sets +open import set-theory.cardinals +open import set-theory.dependent-products-cardinals +open import set-theory.dependent-sums-cardinals +open import set-theory.strict-indexed-inequality-cardinals +``` + +
+ +## Idea + +{{#concept "Kőnig's theorem" WD="König's theorem" WDID=Q1077462 Disambiguation="for cardinals" Agda=Kőnigs-Theorem}} +states that for any pair of families of [cardinals](set-theory.cardinals.md) $A$ +and $B$ over $I$ such that $Aᵢ < Bᵢ$ for all $i$, we have that +$$∑_{i:I}Aᵢ < ∏_{i:I}Bᵢ.$$ + +In constructive mathematics we have to be more mindful of our statements. Here +$I$ is any +[cardinality-projective set](set-theory.cardinality-projective-sets.md) and by +$Aᵢ < Bᵢ$ we mean that $Bᵢ$ is [inhabited](foundation.inhabited-types.md) and +that for every map $f : Aᵢ → Bᵢ$ there +[exists](foundation.existential-quantification.md) an element of $Bᵢ$ that $f$ +does not hit. + +## Lemma + +Given a projective type $I$ and a pair of families of types $A$ and $B$ over $I$ +such that for every $i : I$ every function from $Aᵢ$ to $Bᵢ$ misses an element, +then every function from $ΣᵢAᵢ$ to $ΠᵢBᵢ$ misses an element. + +```agda +module _ + {l1 l2 l3 : Level} + {I : UU l1} (p : is-projective-Level' (l2 ⊔ l3) I) + {A : I → UU l2} {B : I → UU l3} + where + + is-nonsurjective-map-Σ-Π-is-projective-base' : + (H : (i : I) → (f : A i → B i) → is-nonsurjective f) + (g : Σ I A → ((i : I) → B i)) → is-nonsurjective g + is-nonsurjective-map-Σ-Π-is-projective-base' H g = + map-trunc-Prop + ( λ h → (pr1 ∘ h , (λ ((i , a) , r) → pr2 (h i) (a , htpy-eq r i)))) + ( p (λ i → nonim (λ a → g (i , a) i)) (λ i → H i (λ a → g (i , a) i))) +``` + +## Theorem + +```agda +module _ + {l1 l2 : Level} + (I : Set l1) + (is-projective-I : is-projective-Level' l2 (type-Set I)) + where + + cardinality-Kőnigs-Theorem' : + (A B : type-Set I → Set l2) → + ((i : type-Set I) → le-indexed-cardinality' (A i) (B i)) → + le-indexed-cardinality' (Σ-Set I A) (Π-Set I B) + cardinality-Kőnigs-Theorem' A B p = + ( is-projective-I (type-Set ∘ B) (pr1 ∘ p) , + is-nonsurjective-map-Σ-Π-is-projective-base' is-projective-I (pr2 ∘ p)) + + cardinality-Kőnigs-Theorem : + (A B : type-Set I → Set l2) → + ((i : type-Set I) → le-indexed-cardinality (A i) (B i)) → + le-indexed-cardinality (Σ-Set I A) (Π-Set I B) + cardinality-Kőnigs-Theorem A B p = + unit-le-indexed-cardinality + ( Σ-Set I A) + ( Π-Set I B) + ( cardinality-Kőnigs-Theorem' A B + ( λ i → inv-unit-le-indexed-cardinality (A i) (B i) (p i))) + +module _ + {l1 l2 : Level} + (I : Cardinality-Projective-Set l1 l2) + (let type-I = type-Cardinality-Projective-Set I) + (let set-I = set-Cardinality-Projective-Set I) + where + + Kőnigs-Theorem : + (A B : type-I → Cardinal l2) → + ((i : type-I) → le-indexed-Cardinal (A i) (B i)) → + le-indexed-Cardinal (Σ-Cardinal I A) (Π-Cardinal I B) + Kőnigs-Theorem = + apply-twice-ind-Cardinality-Projective-Set I + ( λ A B → + set-Prop + ( function-Prop + ( (i : type-I) → le-indexed-Cardinal (A i) (B i)) + ( le-indexed-prop-Cardinal (Σ-Cardinal I A) (Π-Cardinal I B)))) + ( λ A B p → + binary-tr le-indexed-Cardinal + ( inv (compute-Σ-Cardinal I A)) + ( inv (compute-Π-Cardinal I B)) + ( cardinality-Kőnigs-Theorem + ( set-I) + ( is-projective-Cardinality-Projective-Set I) + ( A) + ( B) + ( p))) +``` + +## Comments + +More generally, for every localization `L` there is an `L`-modal Kőnig's +theorem. + +## External links + +- [Kőnig's theorem (set theory)]() + on Wikipedia diff --git a/src/set-theory/strict-complemented-inequality-cardinals.lagda.md b/src/set-theory/strict-complemented-inequality-cardinals.lagda.md index 36429748aa..5a9391091a 100644 --- a/src/set-theory/strict-complemented-inequality-cardinals.lagda.md +++ b/src/set-theory/strict-complemented-inequality-cardinals.lagda.md @@ -1,7 +1,7 @@ # Strict complemented inequality on cardinals ```agda -module set-theory.strict-complemented-inequality-cardinalities where +module set-theory.strict-complemented-inequality-cardinals where ```
Imports @@ -99,17 +99,16 @@ module _ le-complemented-prop-cardinality : Set l1 → Set l2 → Prop (l1 ⊔ l2) le-complemented-prop-cardinality X Y = - ( leq-complemented-prop-cardinality X Y) ∧ - ( strictly-complemented-bounded-prop-cardinality X Y) + le-complemented-prop-Cardinal (cardinality X) (cardinality Y) - le-complemented-cardinality' : Set l1 → Set l2 → UU (l1 ⊔ l2) - le-complemented-cardinality' X Y = - type-Prop (le-complemented-prop-cardinality' X Y) + le-complemented-cardinality : Set l1 → Set l2 → UU (l1 ⊔ l2) + le-complemented-cardinality X Y = + type-Prop (le-complemented-prop-cardinality X Y) - is-prop-le-complemented-cardinality' : - (X : Set l1) (Y : Set l2) → is-prop (le-complemented-cardinality' X Y) - is-prop-le-complemented-cardinality' X Y = - is-prop-type-Prop (le-complemented-prop-cardinality' X Y) + is-prop-le-complemented-cardinality : + (X : Set l1) (Y : Set l2) → is-prop (le-complemented-cardinality X Y) + is-prop-le-complemented-cardinality X Y = + is-prop-type-Prop (le-complemented-prop-cardinality X Y) ``` ### Inequality on cardinals is reflexive diff --git a/src/set-theory/strict-indexed-inequality-cardinals.lagda.md b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md new file mode 100644 index 0000000000..61baee92a8 --- /dev/null +++ b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md @@ -0,0 +1,202 @@ +# Strict indexed inequality on cardinals + +```agda +module set-theory.strict-indexed-inequality-cardinals where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.conjunction +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.function-extensionality +open import foundation.inhabited-types +open import foundation.propositional-truncations +open import foundation.functoriality-propositional-truncation +open import foundation.identity-types +open import foundation.function-types +open import foundation.large-binary-relations +open import foundation.cartesian-product-types +open import foundation.law-of-excluded-middle +open import foundation.empty-types +open import foundation.mere-embeddings +open import foundation.nonsurjective-maps +open import foundation.negation +open import foundation.propositional-extensionality +open import foundation.propositions +open import foundation.set-truncations +open import foundation.sets +open import foundation.univalence +open import foundation.universe-levels + +open import set-theory.cardinals +open import set-theory.inhabited-cardinals +open import set-theory.complemented-inequality-cardinals +open import set-theory.inequality-cardinals +``` + +
+ +## Idea + +We may say a [cardinal of sets](set-theory.cardinals.md) `X` is +{{#concept "indexed less than" Disambiguation="cardinal of sets" Agda=le-indexed-Cardinal}} +a cardinal `Y` if `Y` is inhabited and any map `f` of +[sets](foundation-core.sets.md) from the isomorphism class represented by `X` +into sets in the isomorphism class represented by `Y` is +[nonsurjective](foundation.nonsurjective-maps.md) in the sense that there exists +an element in `Y` that `f` does not hit. This is a positive way of saying that +`X` is less than `Y`. This defines the +{{#concept "strict indexing ordering" Disambiguation="on cardinalities of sets"}} +on cardinalities of sets. + +## Definition + +### The underlying strict indexed inequality between cardinalities of sets + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + le-indexed-cardinality' : UU (l1 ⊔ l2) + le-indexed-cardinality' = + is-inhabited (type-Set Y) × + ((f : type-Set X → type-Set Y) → is-nonsurjective f) + + le-indexed-prop-cardinality' : Prop (l1 ⊔ l2) + le-indexed-prop-cardinality' = + product-Prop + ( is-inhabited-Prop (type-Set Y)) + ( Π-Prop + ( type-Set X → type-Set Y) + ( is-nonsurjective-Prop)) + + is-prop-le-indexed-cardinality' : + is-prop le-indexed-cardinality' + is-prop-le-indexed-cardinality' = + is-prop-type-Prop le-indexed-prop-cardinality' +``` + +### Strict indexed inequality of a cardinal with respect to a set + +```agda +module _ + {l1 l2 : Level} (X : Set l1) + where + + le-indexed-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) + le-indexed-prop-Cardinal' = + map-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( le-indexed-prop-cardinality' X) + + le-indexed-Cardinal' : Cardinal l2 → UU (l1 ⊔ l2) + le-indexed-Cardinal' Y = + type-Prop (le-indexed-prop-Cardinal' Y) + + compute-le-indexed-prop-Cardinal' : + (Y : Set l2) → + le-indexed-prop-Cardinal' (cardinality Y) ＝ + le-indexed-prop-cardinality' X Y + compute-le-indexed-prop-Cardinal' = + triangle-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( le-indexed-prop-cardinality' X) +``` + +### Strict indexed inequality of cardinals + +```agda +module _ + {l1 l2 : Level} + where + + le-indexed-prop-Cardinal : + Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) + le-indexed-prop-Cardinal = + map-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( le-indexed-prop-Cardinal') + + le-indexed-Cardinal : Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) + le-indexed-Cardinal X Y = type-Prop (le-indexed-prop-Cardinal X Y) +``` + +### Strict indexed inequality of cardinalities of sets + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + le-indexed-prop-cardinality : Prop (l1 ⊔ l2) + le-indexed-prop-cardinality = + le-indexed-prop-Cardinal (cardinality X) (cardinality Y) + + le-indexed-cardinality : UU (l1 ⊔ l2) + le-indexed-cardinality = type-Prop le-indexed-prop-cardinality + + is-prop-le-indexed-cardinality : is-prop le-indexed-cardinality + is-prop-le-indexed-cardinality = is-prop-type-Prop le-indexed-prop-cardinality + + eq-compute-le-indexed-prop-cardinality : + le-indexed-prop-cardinality ＝ le-indexed-prop-cardinality' X Y + eq-compute-le-indexed-prop-cardinality = + ( htpy-eq + ( triangle-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( le-indexed-prop-Cardinal') + ( X)) + ( cardinality Y)) ∙ + ( compute-le-indexed-prop-Cardinal' X Y) + + eq-compute-le-indexed-cardinality : + le-indexed-cardinality ＝ le-indexed-cardinality' X Y + eq-compute-le-indexed-cardinality = + ap type-Prop eq-compute-le-indexed-prop-cardinality + + compute-le-indexed-cardinality : + le-indexed-cardinality ≃ le-indexed-cardinality' X Y + compute-le-indexed-cardinality = + equiv-eq eq-compute-le-indexed-cardinality + + unit-le-indexed-cardinality : + le-indexed-cardinality' X Y → le-indexed-cardinality + unit-le-indexed-cardinality = map-inv-equiv compute-le-indexed-cardinality + + inv-unit-le-indexed-cardinality : + le-indexed-cardinality → le-indexed-cardinality' X Y + inv-unit-le-indexed-cardinality = map-equiv compute-le-indexed-cardinality +``` + +## Properties + +## Strict indexed inequality is irreflexive + +```agda +module _ + {l : Level} + where abstract + + irreflexive-le-indexed-cardinality' : + (A : Set l) → ¬ le-indexed-cardinality' A A + irreflexive-le-indexed-cardinality' A p = + rec-trunc-Prop empty-Prop (λ (x , p) → p (x , refl)) (pr2 p id) + + irreflexive-le-indexed-cardinality : + (A : Set l) → ¬ le-indexed-cardinality A A + irreflexive-le-indexed-cardinality A = + map-neg + ( inv-unit-le-indexed-cardinality A A) + ( irreflexive-le-indexed-cardinality' A) + + irreflexive-le-indexed-Cardinal : + (A : Cardinal l) → ¬ le-indexed-Cardinal A A + irreflexive-le-indexed-Cardinal = + apply-dependent-universal-property-trunc-Set' + ( λ X → set-Prop (neg-Prop (le-indexed-prop-Cardinal X X))) + ( irreflexive-le-indexed-cardinality) +``` From 5203d49398c2ca66d9b9c1f4130f898e55f2bd08 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 15:10:55 +0000 Subject: [PATCH 34/50] fix --- src/set-theory/konigs-theorem.lagda.md | 22 +++++++++++----------- 1 file changed, 11 insertions(+), 11 deletions(-) diff --git a/src/set-theory/konigs-theorem.lagda.md b/src/set-theory/konigs-theorem.lagda.md index 66cbe731b4..e6de008b90 100644 --- a/src/set-theory/konigs-theorem.lagda.md +++ b/src/set-theory/konigs-theorem.lagda.md @@ -30,7 +30,7 @@ open import set-theory.strict-indexed-inequality-cardinals ## Idea -{{#concept "Kőnig's theorem" WD="König's theorem" WDID=Q1077462 Disambiguation="for cardinals" Agda=Kőnigs-Theorem}} +{{#concept "Kőnig's theorem" WD="König's theorem" WDID=Q1077462 Disambiguation="for cardinals" Agda=Königs-Theorem}} states that for any pair of families of [cardinals](set-theory.cardinals.md) $A$ and $B$ over $I$ such that $Aᵢ < Bᵢ$ for all $i$, we have that $$∑_{i:I}Aᵢ < ∏_{i:I}Bᵢ.$$ @@ -74,23 +74,23 @@ module _ (is-projective-I : is-projective-Level' l2 (type-Set I)) where - cardinality-Kőnigs-Theorem' : + cardinality-Königs-Theorem' : (A B : type-Set I → Set l2) → ((i : type-Set I) → le-indexed-cardinality' (A i) (B i)) → le-indexed-cardinality' (Σ-Set I A) (Π-Set I B) - cardinality-Kőnigs-Theorem' A B p = + cardinality-Königs-Theorem' A B p = ( is-projective-I (type-Set ∘ B) (pr1 ∘ p) , is-nonsurjective-map-Σ-Π-is-projective-base' is-projective-I (pr2 ∘ p)) - cardinality-Kőnigs-Theorem : + cardinality-Königs-Theorem : (A B : type-Set I → Set l2) → ((i : type-Set I) → le-indexed-cardinality (A i) (B i)) → le-indexed-cardinality (Σ-Set I A) (Π-Set I B) - cardinality-Kőnigs-Theorem A B p = + cardinality-Königs-Theorem A B p = unit-le-indexed-cardinality ( Σ-Set I A) ( Π-Set I B) - ( cardinality-Kőnigs-Theorem' A B + ( cardinality-Königs-Theorem' A B ( λ i → inv-unit-le-indexed-cardinality (A i) (B i) (p i))) module _ @@ -100,11 +100,11 @@ module _ (let set-I = set-Cardinality-Projective-Set I) where - Kőnigs-Theorem : + Königs-Theorem : (A B : type-I → Cardinal l2) → ((i : type-I) → le-indexed-Cardinal (A i) (B i)) → le-indexed-Cardinal (Σ-Cardinal I A) (Π-Cardinal I B) - Kőnigs-Theorem = + Königs-Theorem = apply-twice-ind-Cardinality-Projective-Set I ( λ A B → set-Prop @@ -115,7 +115,7 @@ module _ binary-tr le-indexed-Cardinal ( inv (compute-Σ-Cardinal I A)) ( inv (compute-Π-Cardinal I B)) - ( cardinality-Kőnigs-Theorem + ( cardinality-Königs-Theorem ( set-I) ( is-projective-Cardinality-Projective-Set I) ( A) @@ -125,8 +125,8 @@ module _ ## Comments -More generally, for every localization `L` there is an `L`-modal Kőnig's -theorem. +More generally, for every localization `L` contained in `Set` there is an +`L`-modal Kőnig's theorem. ## External links From 27878684241b59562565be23515147ba6adc38d2 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 15:22:11 +0000 Subject: [PATCH 35/50] delete silly files --- ...omplemented-boundedness-cardinals.lagda.md | 129 ------------ ...complemented-inequality-cardinals.lagda.md | 197 ------------------ 2 files changed, 326 deletions(-) delete mode 100644 src/set-theory/strict-complemented-boundedness-cardinals.lagda.md delete mode 100644 src/set-theory/strict-complemented-inequality-cardinals.lagda.md diff --git a/src/set-theory/strict-complemented-boundedness-cardinals.lagda.md b/src/set-theory/strict-complemented-boundedness-cardinals.lagda.md deleted file mode 100644 index 0e2dbbef0e..0000000000 --- a/src/set-theory/strict-complemented-boundedness-cardinals.lagda.md +++ /dev/null @@ -1,129 +0,0 @@ -# Strict complemented boundedness on cardinals - -```agda -module set-theory.strict-complemented-boundedness-cardinals where -``` - -
Imports - -```agda -open import foundation.action-on-identifications-functions -open import foundation.decidable-embeddings -open import foundation.dependent-pair-types -open import foundation.equivalences -open import foundation.function-extensionality -open import foundation.function-types -open import foundation.identity-types -open import foundation.large-binary-relations -open import foundation.law-of-excluded-middle -open import foundation.mere-embeddings -open import foundation.nonsurjective-maps -open import foundation.propositional-extensionality -open import foundation.propositions -open import foundation.set-truncations -open import foundation.sets -open import foundation.univalence -open import foundation.universe-levels - -open import set-theory.cardinals -open import set-theory.inequality-cardinals -``` - -
- -## Idea - -We say a [cardinal of sets](set-theory.cardinals.md) `X` is -{{#concept "strictly complemented bounded" Agda=strictly-complemented-bounded-Cardinal}} -by a cardinal `Y` if every decidable embedding of a -[set](foundation-core.sets.md) in the isomorphism class represented by `X` into -a set in the isomorphism class represented by `Y` is -[nonsurjective](foundation.nonsurjective-maps.md) in the sense that there is an -element in `Y` it does not hit. - -## Definition - -### Strict complemented boundedness of cardinality of a set with respect to a set - -```agda -module _ - {l1 l2 : Level} (X : Set l1) (Y : Set l2) - where - - strictly-complemented-bounded-cardinality : UU (l1 ⊔ l2) - strictly-complemented-bounded-cardinality = - (f : type-Set X ↪ᵈ type-Set Y) → is-nonsurjective (map-decidable-emb f) - - strictly-complemented-bounded-prop-cardinality : Prop (l1 ⊔ l2) - strictly-complemented-bounded-prop-cardinality = - Π-Prop - ( type-Set X ↪ᵈ type-Set Y) - ( is-nonsurjective-Prop ∘ map-decidable-emb) - - is-prop-strictly-complemented-bounded-cardinality : - is-prop strictly-complemented-bounded-cardinality - is-prop-strictly-complemented-bounded-cardinality = - is-prop-type-Prop strictly-complemented-bounded-prop-cardinality -``` - -### Strict complemented boundedness of a cardinal with respect to a set - -```agda -module _ - {l1 l2 : Level} (X : Set l1) - where - - strictly-complemented-bounded-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) - strictly-complemented-bounded-prop-Cardinal' = - map-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( strictly-complemented-bounded-prop-cardinality X) - - strictly-complemented-bounded-Cardinal' : Cardinal l2 → UU (l1 ⊔ l2) - strictly-complemented-bounded-Cardinal' Y = - type-Prop (strictly-complemented-bounded-prop-Cardinal' Y) - - compute-strictly-complemented-bounded-prop-Cardinal' : - (Y : Set l2) → - strictly-complemented-bounded-prop-Cardinal' (cardinality Y) ＝ - strictly-complemented-bounded-prop-cardinality X Y - compute-strictly-complemented-bounded-prop-Cardinal' = - triangle-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( strictly-complemented-bounded-prop-cardinality X) -``` - -### Strict complemented boundedness of cardinals - -```agda -module _ - {l1 l2 : Level} - where - - strictly-complemented-bounded-prop-Cardinal : - Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) - strictly-complemented-bounded-prop-Cardinal = - map-universal-property-trunc-Set - ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( strictly-complemented-bounded-prop-Cardinal') - - strictly-complemented-bounded-Cardinal : - Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) - strictly-complemented-bounded-Cardinal X Y = - type-Prop (strictly-complemented-bounded-prop-Cardinal X Y) - - compute-strictly-complemented-bounded-prop-Cardinal : - (X : Set l1) (Y : Set l2) → - strictly-complemented-bounded-prop-Cardinal - ( cardinality X) - ( cardinality Y) ＝ - strictly-complemented-bounded-prop-cardinality X Y - compute-strictly-complemented-bounded-prop-Cardinal X Y = - ( htpy-eq - ( triangle-universal-property-trunc-Set - ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( strictly-complemented-bounded-prop-Cardinal') - ( X)) - ( cardinality Y)) ∙ - ( compute-strictly-complemented-bounded-prop-Cardinal' X Y) -``` diff --git a/src/set-theory/strict-complemented-inequality-cardinals.lagda.md b/src/set-theory/strict-complemented-inequality-cardinals.lagda.md deleted file mode 100644 index 5a9391091a..0000000000 --- a/src/set-theory/strict-complemented-inequality-cardinals.lagda.md +++ /dev/null @@ -1,197 +0,0 @@ -# Strict complemented inequality on cardinals - -```agda -module set-theory.strict-complemented-inequality-cardinals where -``` - -
Imports - -```agda -open import foundation.action-on-identifications-functions -open import foundation.conjunction -open import foundation.dependent-pair-types -open import foundation.equivalences -open import foundation.function-extensionality -open import foundation.identity-types -open import foundation.large-binary-relations -open import foundation.law-of-excluded-middle -open import foundation.mere-embeddings -open import foundation.nonsurjective-maps -open import foundation.propositional-extensionality -open import foundation.propositions -open import foundation.set-truncations -open import foundation.sets -open import foundation.univalence -open import foundation.universe-levels - -open import set-theory.cardinals -open import set-theory.complemented-inequality-cardinals -open import set-theory.inequality-cardinals -open import set-theory.strict-complemented-boundedness-cardinals -``` - -
- -## Idea - -We may say a [cardinal of sets](set-theory.cardinals.md) `X` is -{{#concept "less than" Agda=le-complemented-Cardinal}} a cardinal `Y` if any -[set](foundation-core.sets.md) in the isomorphism class represented by `X` -embeds as a complemented subset into any set in the isomorphism class -represented by `Y`, and every decidable embedding `X ↪ Y` is -[nonsurjective](foundation.nonsurjective-maps.md) in the sense that there is an -element in `Y` they do not hit. This is a positive way of saying that `X` and -`Y` are different. This defines a -{{#concept "strict ordering" Disambiguation="on cardinalities of sets"}} on -cardinalities of sets. - -## Definition - -### The underlying strict complemented inequality of cardinalitiess - -```agda -module _ - {l1 l2 : Level} - where - - le-complemented-prop-cardinality' : Set l1 → Set l2 → Prop (l1 ⊔ l2) - le-complemented-prop-cardinality' X Y = - ( leq-complemented-prop-cardinality X Y) ∧ - ( strictly-complemented-bounded-prop-cardinality X Y) - - le-complemented-cardinality' : Set l1 → Set l2 → UU (l1 ⊔ l2) - le-complemented-cardinality' X Y = - type-Prop (le-complemented-prop-cardinality' X Y) - - is-prop-le-complemented-cardinality' : - (X : Set l1) (Y : Set l2) → is-prop (le-complemented-cardinality' X Y) - is-prop-le-complemented-cardinality' X Y = - is-prop-type-Prop (le-complemented-prop-cardinality' X Y) -``` - -### Strict complemented inequality of cardinals - -```agda -module _ - {l1 l2 : Level} - where - - le-complemented-prop-Cardinal : Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) - le-complemented-prop-Cardinal X Y = - ( leq-complemented-prop-Cardinal X Y) ∧ - ( strictly-complemented-bounded-prop-Cardinal X Y) - - le-complemented-Cardinal : Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) - le-complemented-Cardinal X Y = type-Prop (le-complemented-prop-Cardinal X Y) - - is-prop-le-complemented-Cardinal : - {X : Cardinal l1} {Y : Cardinal l2} → is-prop (le-complemented-Cardinal X Y) - is-prop-le-complemented-Cardinal {X} {Y} = - is-prop-type-Prop (le-complemented-prop-Cardinal X Y) -``` - -### Strict complemented inequality of cardinalitiess - -```agda -module _ - {l1 l2 : Level} - where - - le-complemented-prop-cardinality : Set l1 → Set l2 → Prop (l1 ⊔ l2) - le-complemented-prop-cardinality X Y = - le-complemented-prop-Cardinal (cardinality X) (cardinality Y) - - le-complemented-cardinality : Set l1 → Set l2 → UU (l1 ⊔ l2) - le-complemented-cardinality X Y = - type-Prop (le-complemented-prop-cardinality X Y) - - is-prop-le-complemented-cardinality : - (X : Set l1) (Y : Set l2) → is-prop (le-complemented-cardinality X Y) - is-prop-le-complemented-cardinality X Y = - is-prop-type-Prop (le-complemented-prop-cardinality X Y) -``` - -### Inequality on cardinals is reflexive - -```text -refl-le-complemented-cardinality : is-reflexive-Large-Relation Set le-complemented-cardinality -refl-le-complemented-cardinality A = unit-le-complemented-cardinality A A (refl-mere-emb (type-Set A)) - -refl-le-complemented-Cardinal : is-reflexive-Large-Relation Cardinal le-complemented-Cardinal -refl-le-complemented-Cardinal = - apply-dependent-universal-property-trunc-Set' - ( λ X → set-Prop (le-complemented-prop-Cardinal X X)) - ( refl-le-complemented-cardinality) -``` - -### Inequality on cardinals is transitive - -```text -module _ - {l1 l2 l3 : Level} - where - - transitive-le-complemented-cardinality : - (X : Set l1) (Y : Set l2) (Z : Set l3) → - le-complemented-cardinality Y Z → le-complemented-cardinality X Y → le-complemented-cardinality X Z - transitive-le-complemented-cardinality X Y Z Y≤Z X≤Y = - unit-le-complemented-cardinality X Z - ( transitive-mere-emb - ( inv-unit-le-complemented-cardinality Y Z Y≤Z) - ( inv-unit-le-complemented-cardinality X Y X≤Y)) - - transitive-le-complemented-Cardinal : - (X : Cardinal l1) (Y : Cardinal l2) (Z : Cardinal l3) → - le-complemented-Cardinal Y Z → le-complemented-Cardinal X Y → le-complemented-Cardinal X Z - transitive-le-complemented-Cardinal = - apply-thrice-dependent-universal-property-trunc-Set' - ( λ X Y Z → - ( le-complemented-Cardinal Y Z → le-complemented-Cardinal X Y → le-complemented-Cardinal X Z) , - ( is-set-function-type - ( is-set-function-type - ( is-set-is-prop is-prop-le-complemented-Cardinal)))) - ( transitive-le-complemented-cardinality) -``` - -## Properties - -### Assuming excluded middle, then inequality is antisymmetric - -Using the previous result and assuming excluded middle, we can conclude -`le-complemented-Cardinal` is a partial order by showing that it is -antisymmetric. - -```text -module _ - {l : Level} (lem : LEM l) - where - - antisymmetric-le-complemented-cardinality : - (X Y : Set l) → - le-complemented-cardinality X Y → - le-complemented-cardinality Y X → - cardinality X ＝ cardinality Y - antisymmetric-le-complemented-cardinality X Y X≤Y Y≤X = - eq-mere-equiv-cardinality X Y - ( antisymmetric-mere-emb - ( lem) - ( inv-unit-le-complemented-cardinality X Y X≤Y) - ( inv-unit-le-complemented-cardinality Y X Y≤X)) - - antisymmetric-le-complemented-Cardinal : - (X Y : Cardinal l) → - le-complemented-Cardinal X Y → le-complemented-Cardinal Y X → X ＝ Y - antisymmetric-le-complemented-Cardinal = - apply-twice-dependent-universal-property-trunc-Set' - ( λ X Y → - set-Prop - ( function-Prop - ( le-complemented-Cardinal X Y) - ( function-Prop (le-complemented-Cardinal Y X) (Id-Prop (Cardinal-Set l) X Y)))) - ( antisymmetric-le-complemented-cardinality) -``` - -## External links - -- [Cardinality](https://en.wikipedia.org/wiki/Cardinality) at Wikipedia -- [cardinal number](https://ncatlab.org/nlab/show/cardinal+number) at $n$Lab From 14225f8853f9239d8ae15492e673556bc0edadee Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 15:35:01 +0000 Subject: [PATCH 36/50] fixes --- src/foundation/axiom-of-choice.lagda.md | 2 +- src/foundation/connected-maps.lagda.md | 61 ++++++---- ...ropositional-truncation-into-sets.lagda.md | 2 +- src/set-theory.lagda.md | 5 +- src/set-theory/addition-cardinals.lagda.md | 112 ------------------ ...rict-indexed-inequality-cardinals.lagda.md | 14 +-- 6 files changed, 48 insertions(+), 148 deletions(-) delete mode 100644 src/set-theory/addition-cardinals.lagda.md diff --git a/src/foundation/axiom-of-choice.lagda.md b/src/foundation/axiom-of-choice.lagda.md index e9397084ae..61c0164da4 100644 --- a/src/foundation/axiom-of-choice.lagda.md +++ b/src/foundation/axiom-of-choice.lagda.md @@ -102,7 +102,7 @@ is-set-projective-AC0 : is-set-projective-AC0 ac X A B f h = map-trunc-Prop ( ( map-Σ - ( λ g → (map-surjection f ∘ g) ＝ h) + ( λ g → map-surjection f ∘ g ＝ h) ( precomp h A) ( λ s H → htpy-postcomp X H h)) ∘ ( section-is-split-surjective (map-surjection f))) diff --git a/src/foundation/connected-maps.lagda.md b/src/foundation/connected-maps.lagda.md index 8e3aacd56e..3afb570f37 100644 --- a/src/foundation/connected-maps.lagda.md +++ b/src/foundation/connected-maps.lagda.md @@ -325,7 +325,7 @@ is-connected-map-left-factor k {g = g} {h} H GH z = ( is-connected-equiv' (compute-fiber-comp g h z) (GH z)) ``` -### The class of `k`-equivalences has the 3-for-2 property +### Composition and cancellation in commuting triangles ```agda module _ @@ -416,12 +416,46 @@ module _ is-equiv-precomp-Π-fiber-condition ( λ b → is-equiv-diagonal-exponential-is-connected (P b) (H b)) +module _ + {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} (f : connected-map k A B) + where + + dup-connected-map : + dependent-universal-property-connected-map k (map-connected-map f) + dup-connected-map = + dependent-universal-property-is-connected-map + ( is-connected-map-connected-map f) + +module _ + {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : connected-map k A B) + where + + equiv-dependent-universal-property-is-connected-map : + {l3 : Level} (P : B → Truncated-Type l3 k) → + ((b : B) → type-Truncated-Type (P b)) ≃ + ((a : A) → type-Truncated-Type (P (map-connected-map f a))) + pr1 (equiv-dependent-universal-property-is-connected-map P) = + precomp-Π (map-connected-map f) (λ b → type-Truncated-Type (P b)) + pr2 (equiv-dependent-universal-property-is-connected-map P) = + dependent-universal-property-is-connected-map + ( is-connected-map-connected-map f) + ( P) +``` + +### The induction principle for connected maps + +```agda +module _ + {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} {f : A → B} + (H : is-connected-map k f) + where + ind-is-connected-map : {l3 : Level} (P : B → Truncated-Type l3 k) → ((a : A) → type-Truncated-Type (P (f a))) → (b : B) → type-Truncated-Type (P b) ind-is-connected-map P = - map-inv-is-equiv (dependent-universal-property-is-connected-map P) + map-inv-is-equiv (dependent-universal-property-is-connected-map H P) compute-ind-is-connected-map : {l3 : Level} (P : B → Truncated-Type l3 k) → @@ -430,19 +464,13 @@ module _ compute-ind-is-connected-map P f = htpy-eq ( is-section-map-inv-is-equiv - ( dependent-universal-property-is-connected-map P) + ( dependent-universal-property-is-connected-map H P) ( f)) module _ {l1 l2 : Level} {k : 𝕋} {A : UU l1} {B : UU l2} (f : connected-map k A B) where - dup-connected-map : - dependent-universal-property-connected-map k (map-connected-map f) - dup-connected-map = - dependent-universal-property-is-connected-map - ( is-connected-map-connected-map f) - ind-connected-map : {l3 : Level} (P : B → Truncated-Type l3 k) → ((a : A) → type-Truncated-Type (P (map-connected-map f a))) → @@ -455,21 +483,6 @@ module _ (x : A) → ind-connected-map P g (map-connected-map f x) ＝ g x compute-ind-connected-map = compute-ind-is-connected-map (is-connected-map-connected-map f) - -module _ - {l1 l2 : Level} (k : 𝕋) {A : UU l1} {B : UU l2} (f : connected-map k A B) - where - - equiv-dependent-universal-property-is-connected-map : - {l3 : Level} (P : B → Truncated-Type l3 k) → - ((b : B) → type-Truncated-Type (P b)) ≃ - ((a : A) → type-Truncated-Type (P (map-connected-map f a))) - pr1 (equiv-dependent-universal-property-is-connected-map P) = - precomp-Π (map-connected-map f) (λ b → type-Truncated-Type (P b)) - pr2 (equiv-dependent-universal-property-is-connected-map P) = - dependent-universal-property-is-connected-map - ( is-connected-map-connected-map f) - ( P) ``` ### A map that satisfies the dependent universal property for connected maps is a connected map diff --git a/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md b/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md index 137eeeed95..f9a568c844 100644 --- a/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md +++ b/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md @@ -42,7 +42,7 @@ is-weakly-constant-map-precomp-unit-trunc-Prop : {l1 l2 : Level} {A : UU l1} {B : UU l2} (g : type-trunc-Prop A → B) → is-weakly-constant-map (g ∘ unit-trunc-Prop) is-weakly-constant-map-precomp-unit-trunc-Prop g x y = - ap g (eq-is-prop (is-prop-type-trunc-Prop)) + ap g (eq-is-prop is-prop-type-trunc-Prop) precomp-universal-property-set-quotient-trunc-Prop : {l1 l2 : Level} {A : UU l1} (B : Set l2) → diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index 8243f73f42..089be79c69 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -48,7 +48,6 @@ theory, and [Russell's paradox](set-theory.russells-paradox.md). ```agda module set-theory where -open import set-theory.addition-cardinals public open import set-theory.baire-space public open import set-theory.bounded-increasing-binary-sequences public open import set-theory.cantor-space public @@ -69,10 +68,10 @@ open import set-theory.inequality-cardinals public open import set-theory.inequality-increasing-binary-sequences public open import set-theory.infinite-sets public open import set-theory.inhabited-cardinals public +open import set-theory.konigs-theorem public open import set-theory.positive-elements-increasing-binary-sequences public open import set-theory.russells-paradox public -open import set-theory.strict-complemented-boundedness-cardinals public -open import set-theory.strict-complemented-inequality-cardinals public +open import set-theory.strict-indexed-inequality-cardinals public open import set-theory.strict-lower-bounds-increasing-binary-sequences public open import set-theory.uncountable-sets public open import set-theory.zero-cardinal public diff --git a/src/set-theory/addition-cardinals.lagda.md b/src/set-theory/addition-cardinals.lagda.md deleted file mode 100644 index c936ee99df..0000000000 --- a/src/set-theory/addition-cardinals.lagda.md +++ /dev/null @@ -1,112 +0,0 @@ -# Addition of cardinals - -```agda -module set-theory.addition-cardinals where -``` - -
Imports - -```agda -open import elementary-number-theory.natural-numbers - -open import foundation.action-on-identifications-functions -open import foundation.coproduct-types -open import foundation.decidable-propositions -open import foundation.decidable-subtypes -open import foundation.decidable-types -open import foundation.dependent-pair-types -open import foundation.discrete-types -open import foundation.double-negation -open import foundation.equality-coproduct-types -open import foundation.function-extensionality -open import foundation.function-types -open import foundation.functoriality-set-truncation -open import foundation.identity-types -open import foundation.isolated-elements -open import foundation.logical-equivalences -open import foundation.negated-equality -open import foundation.negation -open import foundation.powersets -open import foundation.propositional-truncations -open import foundation.sections -open import foundation.set-truncations -open import foundation.sets -open import foundation.surjective-maps -open import foundation.universe-levels - -open import foundation-core.empty-types -open import foundation-core.fibers-of-maps -open import foundation-core.propositions - -open import logic.propositionally-decidable-types - -open import set-theory.cardinality-projective-sets -open import set-theory.cardinals -open import set-theory.equality-cardinals -open import set-theory.inequality-cardinals -open import set-theory.zero-cardinal -``` - -
- -## Idea - -Given two [cardinals](set-theory.cardinals.md) `A` and `B`, we may form their -{{#concept "sum" Disambiguation="cardinal" Agda=add-Cardinal}}, or -**coproduct** cardinal, `A + B`. It is defined on isomorphism classes as -[coproducts](foundation-core.coproduct-types.md) of -[sets](foundation-core.sets.md). - -## Definitions - -### The underlying addition of cardinalities - -```agda -module _ - {l1 l2 : Level} - where - - add-cardinality' : Set l1 → Set l2 → Cardinal (l1 ⊔ l2) - add-cardinality' X Y = cardinality (coproduct-Set X Y) -``` - -### Addition of cardinals - -```agda -module _ - {l1 l2 : Level} - where - - add-Cardinal : Cardinal l1 → Cardinal l2 → Cardinal (l1 ⊔ l2) - add-Cardinal = binary-map-trunc-Set coproduct-Set -``` - -### Addition of cardinalities - -```agda -module _ - {l1 l2 : Level} - where - - add-cardinality : Set l1 → Set l2 → Cardinal (l1 ⊔ l2) - add-cardinality X Y = add-Cardinal (cardinality X) (cardinality Y) - - compute-add-cardinality : - (X : Set l1) (Y : Set l2) → - add-Cardinal (cardinality X) (cardinality Y) ＝ add-cardinality' X Y - compute-add-cardinality = {! !} -``` - -## Properties - -### The zero cardinal is the additive unit - -```agda -module _ - {l : Level} - where - - left-unit-law-add-Cardinal : - {l : Level} (X : Cardinal l) → sim-Cardinal (add-Cardinal zero-Cardinal X) X - left-unit-law-add-Cardinal X = {! !} -``` diff --git a/src/set-theory/strict-indexed-inequality-cardinals.lagda.md b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md index 61baee92a8..6ef8a6370a 100644 --- a/src/set-theory/strict-indexed-inequality-cardinals.lagda.md +++ b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md @@ -8,23 +8,23 @@ module set-theory.strict-indexed-inequality-cardinals where ```agda open import foundation.action-on-identifications-functions +open import foundation.cartesian-product-types open import foundation.conjunction open import foundation.dependent-pair-types +open import foundation.empty-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.inhabited-types -open import foundation.propositional-truncations +open import foundation.function-types open import foundation.functoriality-propositional-truncation open import foundation.identity-types -open import foundation.function-types +open import foundation.inhabited-types open import foundation.large-binary-relations -open import foundation.cartesian-product-types open import foundation.law-of-excluded-middle -open import foundation.empty-types open import foundation.mere-embeddings -open import foundation.nonsurjective-maps open import foundation.negation +open import foundation.nonsurjective-maps open import foundation.propositional-extensionality +open import foundation.propositional-truncations open import foundation.propositions open import foundation.set-truncations open import foundation.sets @@ -32,9 +32,9 @@ open import foundation.univalence open import foundation.universe-levels open import set-theory.cardinals -open import set-theory.inhabited-cardinals open import set-theory.complemented-inequality-cardinals open import set-theory.inequality-cardinals +open import set-theory.inhabited-cardinals ```
From e2fe8a3b1a398e46ed02439bddf8649aefb6be21 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 21:33:26 +0000 Subject: [PATCH 37/50] optimize imports nonsurjective maps --- src/foundation/nonsurjective-maps.lagda.md | 30 ---------------------- 1 file changed, 30 deletions(-) diff --git a/src/foundation/nonsurjective-maps.lagda.md b/src/foundation/nonsurjective-maps.lagda.md index 79e022788a..66c93e20a2 100644 --- a/src/foundation/nonsurjective-maps.lagda.md +++ b/src/foundation/nonsurjective-maps.lagda.md @@ -7,63 +7,33 @@ module foundation.nonsurjective-maps where
Imports ```agda -open import foundation.action-on-identifications-functions -open import foundation.connected-maps -open import foundation.contractible-types open import foundation.coproduct-types open import foundation.decidable-maps open import foundation.dependent-pair-types -open import foundation.diagonal-maps-of-types open import foundation.disjunction open import foundation.double-negation -open import foundation.embeddings open import foundation.empty-types -open import foundation.equality-cartesian-product-types open import foundation.existential-quantification -open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-propositional-truncation open import foundation.fundamental-theorem-of-identity-types -open import foundation.homotopy-induction -open import foundation.identity-types -open import foundation.inhabited-types open import foundation.injective-maps open import foundation.negation -open import foundation.postcomposition-dependent-functions open import foundation.propositional-truncations open import foundation.split-surjective-maps open import foundation.structure-identity-principle -open import foundation.subtype-identity-principle open import foundation.surjective-maps -open import foundation.truncated-types -open import foundation.univalence -open import foundation.universal-property-family-of-fibers-of-maps -open import foundation.universal-property-propositional-truncation open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.constant-maps open import foundation-core.contractible-maps -open import foundation-core.equivalences open import foundation-core.fibers-of-maps open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types -open import foundation-core.homotopies -open import foundation-core.precomposition-dependent-functions -open import foundation-core.propositional-maps open import foundation-core.propositions -open import foundation-core.retracts-of-types -open import foundation-core.sections -open import foundation-core.sets -open import foundation-core.subtypes -open import foundation-core.torsorial-type-families -open import foundation-core.truncated-maps -open import foundation-core.truncation-levels open import logic.propositionally-decidable-maps - -open import orthogonal-factorization-systems.extensions-maps ```
From 169af11d565689771cdde462fd3d6e071e69ed16 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Wed, 29 Oct 2025 14:33:53 +0100 Subject: [PATCH 38/50] typecheck! --- src/foundation/mere-decidable-embeddings.lagda.md | 11 ++++++----- .../complemented-inequality-cardinals.lagda.md | 8 ++++---- 2 files changed, 10 insertions(+), 9 deletions(-) diff --git a/src/foundation/mere-decidable-embeddings.lagda.md b/src/foundation/mere-decidable-embeddings.lagda.md index 2412f4d2cf..f2a3e80d3f 100644 --- a/src/foundation/mere-decidable-embeddings.lagda.md +++ b/src/foundation/mere-decidable-embeddings.lagda.md @@ -7,10 +7,13 @@ module foundation.mere-decidable-embeddings where
Imports ```agda +open import foundation.cantor-schroder-bernstein-decidable-embeddings open import foundation.decidable-embeddings open import foundation.functoriality-propositional-truncation +open import foundation.mere-equivalences open import foundation.propositional-truncations open import foundation.universe-levels +open import foundation.weak-limited-principle-of-omniscience open import foundation-core.propositions @@ -64,12 +67,10 @@ transitive-leq-Large-Preorder mere-decidable-emb-Large-Preorder X Y Z = ### Assuming WLPO, then types equipped with mere decidable embeddings form a partial ordering -> This remains to be formalized. - -```text +```agda antisymmetric-mere-decidable-emb : - {l1 l2 : Level} {X : UU l1} {Y : UU l2} → WLPO (l1 ⊔ l2) → + {l1 l2 : Level} {X : UU l1} {Y : UU l2} → level-WLPO (l1 ⊔ l2) → mere-decidable-emb X Y → mere-decidable-emb Y X → mere-equiv X Y antisymmetric-mere-decidable-emb wlpo = - map-binary-trunc-Prop (decidable-emb-Cantor-Schröder-Bernstein wlpo) + map-binary-trunc-Prop (Cantor-Schröder-Bernstein-WLPO wlpo) ``` diff --git a/src/set-theory/complemented-inequality-cardinals.lagda.md b/src/set-theory/complemented-inequality-cardinals.lagda.md index f8f3b7c85f..bf592f501c 100644 --- a/src/set-theory/complemented-inequality-cardinals.lagda.md +++ b/src/set-theory/complemented-inequality-cardinals.lagda.md @@ -20,8 +20,10 @@ open import foundation.set-truncations open import foundation.sets open import foundation.univalence open import foundation.universe-levels +open import foundation.weak-limited-principle-of-omniscience open import set-theory.cardinals +open import set-theory.equality-cardinals ```
@@ -210,11 +212,9 @@ Using the previous result and assuming the weak limited principle of omniscience, we can conclude `leq-complemented-Cardinal` is a partial order by showing that it is antisymmetric. -> This remains to be formalized. - -```text +```agda module _ - {l : Level} (wlpo : WLPO l) + {l : Level} (wlpo : level-WLPO l) where antisymmetric-leq-complemented-cardinality : From d815eae69c51d5803769afeb17e6472e827409cb Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Wed, 29 Oct 2025 14:46:27 +0100 Subject: [PATCH 39/50] poset of cardinals --- .../law-of-excluded-middle.lagda.md | 3 ++ ...complemented-inequality-cardinals.lagda.md | 35 +++++++++++++++---- src/set-theory/inequality-cardinals.lagda.md | 25 +++++++++++++ 3 files changed, 56 insertions(+), 7 deletions(-) diff --git a/src/foundation/law-of-excluded-middle.lagda.md b/src/foundation/law-of-excluded-middle.lagda.md index a633ea32d6..ba4acd50cd 100644 --- a/src/foundation/law-of-excluded-middle.lagda.md +++ b/src/foundation/law-of-excluded-middle.lagda.md @@ -33,6 +33,9 @@ asserts that any [proposition](foundation-core.propositions.md) `P` is LEM : (l : Level) → UU (lsuc l) LEM l = (P : Prop l) → is-decidable (type-Prop P) +LEMω : UUω +LEMω = {l : Level} → LEM l + prop-LEM : (l : Level) → Prop (lsuc l) prop-LEM l = Π-Prop (Prop l) (is-decidable-Prop) ``` diff --git a/src/set-theory/complemented-inequality-cardinals.lagda.md b/src/set-theory/complemented-inequality-cardinals.lagda.md index bf592f501c..16544c8713 100644 --- a/src/set-theory/complemented-inequality-cardinals.lagda.md +++ b/src/set-theory/complemented-inequality-cardinals.lagda.md @@ -22,6 +22,9 @@ open import foundation.univalence open import foundation.universe-levels open import foundation.weak-limited-principle-of-omniscience +open import order-theory.large-posets +open import order-theory.large-preorders + open import set-theory.cardinals open import set-theory.equality-cardinals ``` @@ -30,21 +33,21 @@ open import set-theory.equality-cardinals ## Idea -We may say a [cardinal of sets](set-theory.cardinals.md) `X` is -{{#concept "less than or equal to" Agda=leq-complemented-cardinality}} a -cardinal `Y` if any [set](foundation-core.sets.md) in the isomorphism class -represented by `X` [embeds](foundation-core.embeddings.md) as a +We may say a [cardinal](set-theory.cardinals.md) `X` is +{{#concept "less than or equal to" Agda=leq-complemented-Cardinal}} a cardinal +`Y` if any [set](foundation-core.sets.md) in the isomorphism class represented +by `X` [embeds](foundation-core.embeddings.md) as a [complemented subtype](foundation.decidable-subtypes.md) into any set in the isomorphism class represented by `Y`. In other words, if there is a -[decidable embedding](foundation.decidable-embeddings.md) from one to the other. -This defines the +[decidable embedding](foundation.decidable-embeddings.md) from the first to the +second. This defines the {{#concept "complemented ordering" Disambiguation="on cardinalities of sets"}} on cardinalities of sets. Under the assumption of the [weak limited principle of omniscience](foundation.weak-limited-principle-of-omniscience.md), this relation is antisymmetric and hence defines a -[partial order](order-theory.posets.md) on cardinalites. +[partial order](order-theory.posets.md) on cardinals. ## Definition @@ -243,3 +246,21 @@ module _ ( Id-Prop (Cardinal-Set l) X Y)))) ( antisymmetric-leq-complemented-cardinality) ``` + +### The large poset of cardinals under complemented inequality + +```agda +large-preorder-complemented-Cardinal : Large-Preorder lsuc (_⊔_) +large-preorder-complemented-Cardinal = + λ where + .type-Large-Preorder → Cardinal + .leq-prop-Large-Preorder → leq-complemented-prop-Cardinal + .refl-leq-Large-Preorder → refl-leq-complemented-Cardinal + .transitive-leq-Large-Preorder → transitive-leq-complemented-Cardinal + +large-poset-complemented-Cardinal : WLPO → Large-Poset lsuc (_⊔_) +large-poset-complemented-Cardinal wlpo = + λ where + .large-preorder-Large-Poset → large-preorder-complemented-Cardinal + .antisymmetric-leq-Large-Poset → antisymmetric-leq-complemented-Cardinal wlpo +``` diff --git a/src/set-theory/inequality-cardinals.lagda.md b/src/set-theory/inequality-cardinals.lagda.md index a76559edbe..a12e340c19 100644 --- a/src/set-theory/inequality-cardinals.lagda.md +++ b/src/set-theory/inequality-cardinals.lagda.md @@ -22,6 +22,9 @@ open import foundation.sets open import foundation.univalence open import foundation.universe-levels +open import order-theory.large-posets +open import order-theory.large-preorders + open import set-theory.cardinals open import set-theory.equality-cardinals ``` @@ -211,6 +214,28 @@ module _ ( antisymmetric-leq-cardinality) ``` +### The large poset of cardinals + +```agda +large-preorder-Cardinal : Large-Preorder lsuc (_⊔_) +large-preorder-Cardinal = + λ where + .type-Large-Preorder → Cardinal + .leq-prop-Large-Preorder → leq-prop-Cardinal + .refl-leq-Large-Preorder → refl-leq-Cardinal + .transitive-leq-Large-Preorder → transitive-leq-Cardinal + +large-poset-Cardinal : LEMω → Large-Poset lsuc (_⊔_) +large-poset-Cardinal lem = + λ where + .large-preorder-Large-Poset → large-preorder-Cardinal + .antisymmetric-leq-Large-Poset → antisymmetric-leq-Cardinal lem +``` + +## See also + +- [Complemented inequality of cardinals](set-theory.complemented-inequality-cardinals.md) + ## External links - [Cardinality](https://en.wikipedia.org/wiki/Cardinality) at Wikipedia From b70647848cec97862be1d4caea33b0d0ca7b3dad Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Wed, 29 Oct 2025 14:58:44 +0100 Subject: [PATCH 40/50] formatting --- src/foundation/mere-decidable-embeddings.lagda.md | 13 ++++++------- src/foundation/mere-embeddings.lagda.md | 11 ++++++----- 2 files changed, 12 insertions(+), 12 deletions(-) diff --git a/src/foundation/mere-decidable-embeddings.lagda.md b/src/foundation/mere-decidable-embeddings.lagda.md index f2a3e80d3f..abf7f4707b 100644 --- a/src/foundation/mere-decidable-embeddings.lagda.md +++ b/src/foundation/mere-decidable-embeddings.lagda.md @@ -56,13 +56,12 @@ transitive-mere-decidable-emb : transitive-mere-decidable-emb = map-binary-trunc-Prop comp-decidable-emb mere-decidable-emb-Large-Preorder : Large-Preorder lsuc (_⊔_) -type-Large-Preorder mere-decidable-emb-Large-Preorder l = UU l -leq-prop-Large-Preorder mere-decidable-emb-Large-Preorder = - mere-decidable-emb-Prop -refl-leq-Large-Preorder mere-decidable-emb-Large-Preorder = - refl-mere-decidable-emb -transitive-leq-Large-Preorder mere-decidable-emb-Large-Preorder X Y Z = - transitive-mere-decidable-emb +mere-decidable-emb-Large-Preorder = + λ where + .type-Large-Preorder l → UU l + .leq-prop-Large-Preorder → mere-decidable-emb-Prop + .refl-leq-Large-Preorder → refl-mere-decidable-emb + .transitive-leq-Large-Preorder X Y Z → transitive-mere-decidable-emb ``` ### Assuming WLPO, then types equipped with mere decidable embeddings form a partial ordering diff --git a/src/foundation/mere-embeddings.lagda.md b/src/foundation/mere-embeddings.lagda.md index 68bc5beb44..ce425aa632 100644 --- a/src/foundation/mere-embeddings.lagda.md +++ b/src/foundation/mere-embeddings.lagda.md @@ -59,11 +59,12 @@ transitive-mere-emb : transitive-mere-emb = map-binary-trunc-Prop comp-emb mere-emb-Large-Preorder : Large-Preorder lsuc (_⊔_) -type-Large-Preorder mere-emb-Large-Preorder l = UU l -leq-prop-Large-Preorder mere-emb-Large-Preorder = mere-emb-Prop -refl-leq-Large-Preorder mere-emb-Large-Preorder = refl-mere-emb -transitive-leq-Large-Preorder mere-emb-Large-Preorder X Y Z = - transitive-mere-emb +mere-emb-Large-Preorder = + λ where + .type-Large-Preorder l → UU l + .leq-prop-Large-Preorder → mere-emb-Prop + .refl-leq-Large-Preorder → refl-mere-emb + .transitive-leq-Large-Preorder X Y Z → transitive-mere-emb ``` ### Assuming excluded middle, if there are mere embeddings between `A` and `B` in both directions, then there is a mere equivalence between them From 6ba4c964e73c25a8e837bd2a78f0f7deed0d29c6 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Wed, 29 Oct 2025 15:03:21 +0100 Subject: [PATCH 41/50] edit --- src/set-theory/cardinals.lagda.md | 12 ++++++++---- 1 file changed, 8 insertions(+), 4 deletions(-) diff --git a/src/set-theory/cardinals.lagda.md b/src/set-theory/cardinals.lagda.md index 8fbe61cb9a..33fc3b3f05 100644 --- a/src/set-theory/cardinals.lagda.md +++ b/src/set-theory/cardinals.lagda.md @@ -26,11 +26,15 @@ of a [set](foundation-core.sets.md) is its [isomorphism](category-theory.isomorphisms-in-categories.md) class. We take isomorphism classes of sets by [set truncating](foundation.set-truncations.md) the universe of sets of any given -[universe level](foundation.universe-levels.md). Note that this definition takes -advantage of the [univalence axiom](foundation.univalence.md): By the univalence -axiom [isomorphic sets](foundation.isomorphisms-of-sets.md) are +[universe level](foundation.universe-levels.md), and such an isomorphism class +is called a +{{#concept "cardinal" Disambiguation="sets" WD="cardinal number" WDID=Q163875}}. + +Note that this definition takes advantage of the +[univalence axiom](foundation.univalence.md): By the univalence axiom +[isomorphic sets](foundation.isomorphisms-of-sets.md) are [equal](foundation-core.identity-types.md), and will be mapped to the same -element in the set truncation of the universe of all sets. +element in the set truncation of the universe of all sets. S ## Definition From c6a57037647ed617d0adf5e873973f8e79933cc9 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Wed, 29 Oct 2025 15:08:22 +0100 Subject: [PATCH 42/50] fixes --- src/foundation/mere-decidable-embeddings.lagda.md | 2 +- src/foundation/mere-embeddings.lagda.md | 2 +- src/set-theory/complemented-inequality-cardinals.lagda.md | 6 +----- 3 files changed, 3 insertions(+), 7 deletions(-) diff --git a/src/foundation/mere-decidable-embeddings.lagda.md b/src/foundation/mere-decidable-embeddings.lagda.md index abf7f4707b..7153fe862d 100644 --- a/src/foundation/mere-decidable-embeddings.lagda.md +++ b/src/foundation/mere-decidable-embeddings.lagda.md @@ -60,7 +60,7 @@ mere-decidable-emb-Large-Preorder = λ where .type-Large-Preorder l → UU l .leq-prop-Large-Preorder → mere-decidable-emb-Prop - .refl-leq-Large-Preorder → refl-mere-decidable-emb + .refl-leq-Large-Preorder → refl-mere-decidable-emb .transitive-leq-Large-Preorder X Y Z → transitive-mere-decidable-emb ``` diff --git a/src/foundation/mere-embeddings.lagda.md b/src/foundation/mere-embeddings.lagda.md index ce425aa632..cb9ef4ac91 100644 --- a/src/foundation/mere-embeddings.lagda.md +++ b/src/foundation/mere-embeddings.lagda.md @@ -63,7 +63,7 @@ mere-emb-Large-Preorder = λ where .type-Large-Preorder l → UU l .leq-prop-Large-Preorder → mere-emb-Prop - .refl-leq-Large-Preorder → refl-mere-emb + .refl-leq-Large-Preorder → refl-mere-emb .transitive-leq-Large-Preorder X Y Z → transitive-mere-emb ``` diff --git a/src/set-theory/complemented-inequality-cardinals.lagda.md b/src/set-theory/complemented-inequality-cardinals.lagda.md index 16544c8713..cc4bfe3aba 100644 --- a/src/set-theory/complemented-inequality-cardinals.lagda.md +++ b/src/set-theory/complemented-inequality-cardinals.lagda.md @@ -209,11 +209,7 @@ module _ ## Properties -### Assuming the weak limited principle of omniscience, then complemented inequality is antisymmetric - -Using the previous result and assuming the weak limited principle of -omniscience, we can conclude `leq-complemented-Cardinal` is a partial order by -showing that it is antisymmetric. +### Assuming the weak limited principle of omniscience, then complemented inequality forms a partial order ```agda module _ From 406f1798a8580a4fc119168beee10893d2b98b1e Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 21:57:04 +0000 Subject: [PATCH 43/50] fixes --- .../cantor-schroder-bernstein-escardo.lagda.md | 4 ++-- src/foundation/diaconescus-theorem.lagda.md | 2 +- src/foundation/law-of-excluded-middle.lagda.md | 14 +++++++------- src/foundation/mere-embeddings.lagda.md | 2 +- src/foundation/set-truncations.lagda.md | 10 +++++----- .../standard-apartness-relations.lagda.md | 2 +- src/logic/dirk-gentlys-principle.lagda.md | 4 ++-- src/order-theory/zorns-lemma.lagda.md | 2 +- src/set-theory/cardinals.lagda.md | 4 ++-- .../complemented-inequality-cardinals.lagda.md | 9 ++++++++- src/set-theory/equality-cardinals.lagda.md | 13 ++++--------- src/set-theory/inequality-cardinals.lagda.md | 9 ++------- 12 files changed, 36 insertions(+), 39 deletions(-) diff --git a/src/foundation/cantor-schroder-bernstein-escardo.lagda.md b/src/foundation/cantor-schroder-bernstein-escardo.lagda.md index e5e25491a8..abe291dd5b 100644 --- a/src/foundation/cantor-schroder-bernstein-escardo.lagda.md +++ b/src/foundation/cantor-schroder-bernstein-escardo.lagda.md @@ -43,7 +43,7 @@ Escardó proved that a Cantor–Schröder–Bernstein theorem also holds for ```agda module _ - {l1 l2 : Level} (lem : LEM (l1 ⊔ l2)) + {l1 l2 : Level} (lem : level-LEM (l1 ⊔ l2)) {A : UU l1} {B : UU l2} where abstract @@ -68,7 +68,7 @@ module _ ```agda module _ - {l1 l2 : Level} (lem : LEM (l1 ⊔ l2)) + {l1 l2 : Level} (lem : level-LEM (l1 ⊔ l2)) (A : Set l1) (B : Set l2) where abstract diff --git a/src/foundation/diaconescus-theorem.lagda.md b/src/foundation/diaconescus-theorem.lagda.md index adcacce8cc..f0141f9366 100644 --- a/src/foundation/diaconescus-theorem.lagda.md +++ b/src/foundation/diaconescus-theorem.lagda.md @@ -80,7 +80,7 @@ instance-theorem-Diaconescu P ac-P = ( ac-P is-surjective-map-surjection-bool-suspension) theorem-Diaconescu : - {l : Level} → level-AC0 l l → LEM l + {l : Level} → level-AC0 l l → level-LEM l theorem-Diaconescu ac P = instance-theorem-Diaconescu P ( ac (suspension-set-Prop P) (fiber map-surjection-bool-suspension)) diff --git a/src/foundation/law-of-excluded-middle.lagda.md b/src/foundation/law-of-excluded-middle.lagda.md index ba4acd50cd..5c74fd8ce2 100644 --- a/src/foundation/law-of-excluded-middle.lagda.md +++ b/src/foundation/law-of-excluded-middle.lagda.md @@ -30,14 +30,14 @@ asserts that any [proposition](foundation-core.propositions.md) `P` is ## Definition ```agda -LEM : (l : Level) → UU (lsuc l) -LEM l = (P : Prop l) → is-decidable (type-Prop P) +level-LEM : (l : Level) → UU (lsuc l) +level-LEM l = (P : Prop l) → is-decidable (type-Prop P) -LEMω : UUω -LEMω = {l : Level} → LEM l +LEM : UUω +LEM = {l : Level} → level-LEM l -prop-LEM : (l : Level) → Prop (lsuc l) -prop-LEM l = Π-Prop (Prop l) (is-decidable-Prop) +prop-level-LEM : (l : Level) → Prop (lsuc l) +prop-level-LEM l = Π-Prop (Prop l) (is-decidable-Prop) ``` ## Properties @@ -46,7 +46,7 @@ prop-LEM l = Π-Prop (Prop l) (is-decidable-Prop) ```agda decidable-prop-Prop : - {l : Level} → LEM l → Prop l → Decidable-Prop l + {l : Level} → level-LEM l → Prop l → Decidable-Prop l pr1 (decidable-prop-Prop lem P) = type-Prop P pr1 (pr2 (decidable-prop-Prop lem P)) = is-prop-type-Prop P pr2 (pr2 (decidable-prop-Prop lem P)) = lem P diff --git a/src/foundation/mere-embeddings.lagda.md b/src/foundation/mere-embeddings.lagda.md index cb9ef4ac91..2cb411aae4 100644 --- a/src/foundation/mere-embeddings.lagda.md +++ b/src/foundation/mere-embeddings.lagda.md @@ -72,7 +72,7 @@ mere-emb-Large-Preorder = ```agda antisymmetric-mere-emb : {l1 l2 : Level} {X : UU l1} {Y : UU l2} → - LEM (l1 ⊔ l2) → mere-emb X Y → mere-emb Y X → mere-equiv X Y + level-LEM (l1 ⊔ l2) → mere-emb X Y → mere-emb Y X → mere-equiv X Y antisymmetric-mere-emb lem = map-binary-trunc-Prop (Cantor-Schröder-Bernstein-Escardó lem) ``` diff --git a/src/foundation/set-truncations.lagda.md b/src/foundation/set-truncations.lagda.md index ce22a33fb3..5326e8c1ff 100644 --- a/src/foundation/set-truncations.lagda.md +++ b/src/foundation/set-truncations.lagda.md @@ -137,11 +137,11 @@ module _ (a : type-trunc-Set A) (b : type-trunc-Set (B a)) → type-Set (C a b) apply-twice-dependent-universal-property-trunc-Set' = apply-dependent-universal-property-trunc-Set' - ( λ x → Π-Set (trunc-Set (B x)) (C x)) - ( λ x → - apply-dependent-universal-property-trunc-Set' - ( C (unit-trunc-Set x)) - ( f x)) + ( λ x → Π-Set (trunc-Set (B x)) (C x)) + ( λ x → + apply-dependent-universal-property-trunc-Set' + ( C (unit-trunc-Set x)) + ( f x)) module _ {l1 l2 l3 l4 : Level} {A : UU l1} {B : type-trunc-Set A → UU l2} diff --git a/src/foundation/standard-apartness-relations.lagda.md b/src/foundation/standard-apartness-relations.lagda.md index 73ece0d862..1a6c9f5ca7 100644 --- a/src/foundation/standard-apartness-relations.lagda.md +++ b/src/foundation/standard-apartness-relations.lagda.md @@ -37,7 +37,7 @@ is-standard-Apartness-Relation : {l1 l2 : Level} (l3 : Level) {A : UU l1} (R : Apartness-Relation l2 A) → UU (l1 ⊔ l2 ⊔ lsuc l3) is-standard-Apartness-Relation {l1} {l2} l3 {A} R = - LEM l3 → (x y : A) → (x ≠ y) ↔ apart-Apartness-Relation R x y + level-LEM l3 → (x y : A) → (x ≠ y) ↔ apart-Apartness-Relation R x y ``` ## Properties diff --git a/src/logic/dirk-gentlys-principle.lagda.md b/src/logic/dirk-gentlys-principle.lagda.md index a0a110c912..64800258c8 100644 --- a/src/logic/dirk-gentlys-principle.lagda.md +++ b/src/logic/dirk-gentlys-principle.lagda.md @@ -74,12 +74,12 @@ instance-Dirk-Gently-Principle-LEM' P Q (inr np) = inl-disjunction (ex-falso ∘ np) level-Dirk-Gently-Principle-LEM : - (l1 l2 : Level) → LEM l1 → level-Dirk-Gently-Principle l1 l2 + (l1 l2 : Level) → level-LEM l1 → level-Dirk-Gently-Principle l1 l2 level-Dirk-Gently-Principle-LEM l1 l2 lem P Q = instance-Dirk-Gently-Principle-LEM' P Q (lem P) level-Dirk-Gently-Principle-LEM' : - (l1 l2 : Level) → LEM l2 → level-Dirk-Gently-Principle l1 l2 + (l1 l2 : Level) → level-LEM l2 → level-Dirk-Gently-Principle l1 l2 level-Dirk-Gently-Principle-LEM' l1 l2 lem P Q = swap-disjunction (level-Dirk-Gently-Principle-LEM l2 l1 lem Q P) ``` diff --git a/src/order-theory/zorns-lemma.lagda.md b/src/order-theory/zorns-lemma.lagda.md index a2e403817a..53dd5cad61 100644 --- a/src/order-theory/zorns-lemma.lagda.md +++ b/src/order-theory/zorns-lemma.lagda.md @@ -105,7 +105,7 @@ module _ ```agda module _ - {l1 l2 l3 : Level} (lem : LEM (l1 ⊔ l3)) + {l1 l2 l3 : Level} (lem : level-LEM (l1 ⊔ l3)) where inhabited-zorns-lemma-zorns-lemma : diff --git a/src/set-theory/cardinals.lagda.md b/src/set-theory/cardinals.lagda.md index 33fc3b3f05..7c930b915e 100644 --- a/src/set-theory/cardinals.lagda.md +++ b/src/set-theory/cardinals.lagda.md @@ -28,13 +28,13 @@ isomorphism classes of sets by [set truncating](foundation.set-truncations.md) the universe of sets of any given [universe level](foundation.universe-levels.md), and such an isomorphism class is called a -{{#concept "cardinal" Disambiguation="sets" WD="cardinal number" WDID=Q163875}}. +{{#concept "cardinal" Disambiguation="of sets" WD="cardinal number" WDID=Q163875}}. Note that this definition takes advantage of the [univalence axiom](foundation.univalence.md): By the univalence axiom [isomorphic sets](foundation.isomorphisms-of-sets.md) are [equal](foundation-core.identity-types.md), and will be mapped to the same -element in the set truncation of the universe of all sets. S +element in the set truncation of the universe of all sets. ## Definition diff --git a/src/set-theory/complemented-inequality-cardinals.lagda.md b/src/set-theory/complemented-inequality-cardinals.lagda.md index cc4bfe3aba..0de213282c 100644 --- a/src/set-theory/complemented-inequality-cardinals.lagda.md +++ b/src/set-theory/complemented-inequality-cardinals.lagda.md @@ -37,7 +37,7 @@ We may say a [cardinal](set-theory.cardinals.md) `X` is {{#concept "less than or equal to" Agda=leq-complemented-Cardinal}} a cardinal `Y` if any [set](foundation-core.sets.md) in the isomorphism class represented by `X` [embeds](foundation-core.embeddings.md) as a -[complemented subtype](foundation.decidable-subtypes.md) into any set in the +[decidable subtype](foundation.decidable-subtypes.md) into any set in the isomorphism class represented by `Y`. In other words, if there is a [decidable embedding](foundation.decidable-embeddings.md) from the first to the second. This defines the @@ -211,6 +211,9 @@ module _ ### Assuming the weak limited principle of omniscience, then complemented inequality forms a partial order +This is a direct consequence of the +[Cantor–Schröder–Bernstein theorem for decidable embeddings](foundation.cantor-schroder-bernstein-decidable-embeddings.md). + ```agda module _ {l : Level} (wlpo : level-WLPO l) @@ -260,3 +263,7 @@ large-poset-complemented-Cardinal wlpo = .large-preorder-Large-Poset → large-preorder-complemented-Cardinal .antisymmetric-leq-Large-Poset → antisymmetric-leq-complemented-Cardinal wlpo ``` + +## See also + +- [Inequality of cardinals](set-theory.inequality-cardinals.md) diff --git a/src/set-theory/equality-cardinals.lagda.md b/src/set-theory/equality-cardinals.lagda.md index 8eea5937e5..8e1f4227d5 100644 --- a/src/set-theory/equality-cardinals.lagda.md +++ b/src/set-theory/equality-cardinals.lagda.md @@ -32,10 +32,10 @@ open import set-theory.cardinals Two [cardinals of sets](set-theory.cardinals.md) `X` and `Y` are {{#concept "similar" Disambiguation="cardinals" Agda=sim-Cardinal}} if there -[merely](foundation.inhabited-types.md) is an -[equivalence](foundation-core.equivalences.md) between any two representing two -types. This characterizes [equality](foundation-core.identity-types.md) of -cardinals. +[merely exists](foundation.inhabited-types.md) an +[equivalence](foundation-core.equivalences.md) between any two representing +[sets](foundation-core.sets.md). This characterizes +[equality](foundation-core.identity-types.md) of cardinals. ## Definition @@ -311,8 +311,3 @@ large-similarity-relation-Cardinal = .eq-sim-Large-Similarity-Relation → eq-sim-Cardinal ``` - -## External links - -- [Cardinality](https://en.wikipedia.org/wiki/Cardinality) at Wikipedia -- [cardinal number](https://ncatlab.org/nlab/show/cardinal+number) at $n$Lab diff --git a/src/set-theory/inequality-cardinals.lagda.md b/src/set-theory/inequality-cardinals.lagda.md index a12e340c19..717d4df0b2 100644 --- a/src/set-theory/inequality-cardinals.lagda.md +++ b/src/set-theory/inequality-cardinals.lagda.md @@ -186,7 +186,7 @@ that `leq-Cardinal` is antisymmetric and hence a partial order. ```agda module _ - {l : Level} (lem : LEM l) + {l : Level} (lem : level-LEM l) where antisymmetric-leq-cardinality : @@ -225,7 +225,7 @@ large-preorder-Cardinal = .refl-leq-Large-Preorder → refl-leq-Cardinal .transitive-leq-Large-Preorder → transitive-leq-Cardinal -large-poset-Cardinal : LEMω → Large-Poset lsuc (_⊔_) +large-poset-Cardinal : LEM → Large-Poset lsuc (_⊔_) large-poset-Cardinal lem = λ where .large-preorder-Large-Poset → large-preorder-Cardinal @@ -235,8 +235,3 @@ large-poset-Cardinal lem = ## See also - [Complemented inequality of cardinals](set-theory.complemented-inequality-cardinals.md) - -## External links - -- [Cardinality](https://en.wikipedia.org/wiki/Cardinality) at Wikipedia -- [cardinal number](https://ncatlab.org/nlab/show/cardinal+number) at $n$Lab From 2e3064d215475748a568f9c3a8ed12a86742ac63 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 22:54:09 +0000 Subject: [PATCH 44/50] fixes --- .../dependent-products-cardinals.lagda.md | 22 ---------- .../dependent-sums-cardinals.lagda.md | 42 +------------------ src/set-theory/zero-cardinal.lagda.md | 40 +----------------- 3 files changed, 2 insertions(+), 102 deletions(-) diff --git a/src/set-theory/dependent-products-cardinals.lagda.md b/src/set-theory/dependent-products-cardinals.lagda.md index 3dc5cbc232..ba06b02794 100644 --- a/src/set-theory/dependent-products-cardinals.lagda.md +++ b/src/set-theory/dependent-products-cardinals.lagda.md @@ -7,40 +7,18 @@ module set-theory.dependent-products-cardinals where
Imports ```agda -open import elementary-number-theory.natural-numbers - open import foundation.action-on-identifications-functions -open import foundation.coproduct-types -open import foundation.decidable-propositions -open import foundation.decidable-subtypes -open import foundation.decidable-types open import foundation.dependent-pair-types -open import foundation.discrete-types -open import foundation.double-negation -open import foundation.function-extensionality open import foundation.function-types open import foundation.functoriality-set-truncation open import foundation.identity-types -open import foundation.inhabited-types -open import foundation.isolated-elements -open import foundation.logical-equivalences -open import foundation.negated-equality -open import foundation.negation -open import foundation.powersets -open import foundation.propositional-truncations -open import foundation.sections open import foundation.set-truncations open import foundation.sets -open import foundation.surjective-maps open import foundation.transport-along-identifications open import foundation.universe-levels -open import foundation-core.empty-types -open import foundation-core.fibers-of-maps open import foundation-core.propositions -open import logic.propositionally-decidable-types - open import set-theory.cardinality-projective-sets open import set-theory.cardinality-recursive-sets open import set-theory.cardinals diff --git a/src/set-theory/dependent-sums-cardinals.lagda.md b/src/set-theory/dependent-sums-cardinals.lagda.md index 0d5e7486f0..ac7a7f1f34 100644 --- a/src/set-theory/dependent-sums-cardinals.lagda.md +++ b/src/set-theory/dependent-sums-cardinals.lagda.md @@ -7,44 +7,21 @@ module set-theory.dependent-sums-cardinals where
Imports ```agda -open import elementary-number-theory.natural-numbers - open import foundation.action-on-identifications-functions -open import foundation.connected-maps -open import foundation.coproduct-types -open import foundation.decidable-propositions -open import foundation.decidable-subtypes -open import foundation.decidable-types open import foundation.dependent-pair-types -open import foundation.discrete-types -open import foundation.double-negation -open import foundation.function-extensionality open import foundation.function-types -open import foundation.functoriality-propositional-truncation open import foundation.functoriality-set-truncation open import foundation.identity-types open import foundation.inhabited-types -open import foundation.isolated-elements -open import foundation.logical-equivalences open import foundation.mere-embeddings -open import foundation.negated-equality -open import foundation.negation -open import foundation.powersets open import foundation.projective-types -open import foundation.propositional-truncations -open import foundation.sections open import foundation.set-truncations open import foundation.sets -open import foundation.surjective-maps open import foundation.transport-along-identifications open import foundation.universe-levels -open import foundation-core.empty-types -open import foundation-core.fibers-of-maps open import foundation-core.propositions -open import logic.propositionally-decidable-types - open import set-theory.cardinality-projective-sets open import set-theory.cardinality-recursive-sets open import set-theory.cardinals @@ -163,7 +140,7 @@ module _ ( Σ-Cardinal X (pr1 ∘ K) , is-inhabited-Σ-Cardinal (pr1 ∘ K) (pr2 ∘ K)) ``` -### Inequality is preserved under dependent sums over projective types +### Inequality of cardinalities is preserved under dependent sums over projective types ```agda module _ @@ -183,20 +160,3 @@ module _ ( is-projective-X) ( λ x → inv-unit-leq-cardinality (K x) (P x) (f x))) ``` - -TODO - -```text -module _ - {l1 l2 : Level} (X : Cardinality-Recursive-Set l1 l2) - (let type-X = type-Cardinality-Recursive-Set X) - (is-projective-X : is-projective-Level' l2 (type-Cardinality-Recursive-Set X)) - where - - leq-Σ-Cardinal' : - (K P : type-X → Cardinal l2) → - ((i : type-X) → leq-Cardinal (K i) (P i)) → - leq-Cardinal (Σ-Cardinal' X K) (Σ-Cardinal' X P) - leq-Σ-Cardinal' K P f = {! !} - -- proof somehow proceeds by using that since `X` is cardinality-recursive, it suffices to show this for families of sets, and then it's just an easy fact of dependent sums. -``` diff --git a/src/set-theory/zero-cardinal.lagda.md b/src/set-theory/zero-cardinal.lagda.md index 9e2923e3c1..bdcde8a939 100644 --- a/src/set-theory/zero-cardinal.lagda.md +++ b/src/set-theory/zero-cardinal.lagda.md @@ -7,42 +7,10 @@ module set-theory.zero-cardinal where
Imports ```agda -open import elementary-number-theory.natural-numbers - -open import foundation.action-on-identifications-functions -open import foundation.coproduct-types -open import foundation.decidable-propositions -open import foundation.decidable-subtypes -open import foundation.decidable-types -open import foundation.dependent-pair-types -open import foundation.discrete-types -open import foundation.double-negation -open import foundation.equality-coproduct-types -open import foundation.function-extensionality -open import foundation.function-types -open import foundation.functoriality-set-truncation -open import foundation.identity-types -open import foundation.isolated-elements -open import foundation.logical-equivalences -open import foundation.negated-equality -open import foundation.negation -open import foundation.powersets -open import foundation.propositional-truncations -open import foundation.sections -open import foundation.set-truncations -open import foundation.sets -open import foundation.surjective-maps +open import foundation.empty-types open import foundation.universe-levels -open import foundation-core.empty-types -open import foundation-core.fibers-of-maps -open import foundation-core.propositions - -open import logic.propositionally-decidable-types - -open import set-theory.cardinality-projective-sets open import set-theory.cardinals -open import set-theory.inequality-cardinals ```
@@ -59,9 +27,3 @@ the isomorphism class of [empty](foundation.empty-types.md) zero-Cardinal : Cardinal lzero zero-Cardinal = cardinality empty-Set ``` - -## Properties - -### Any cardinal less than or equal to the zero cardinal is the zero cardinal - -> This remains to be formalized. From 0901713823f6496d3a0748e1f3e7d40db7bfef65 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 22:55:00 +0000 Subject: [PATCH 45/50] remove konigs theorem formalization --- .../wikipedia-list-of-theorems.lagda.md | 9 - src/set-theory.lagda.md | 3 - ...complemented-inequality-cardinals.lagda.md | 269 ------------------ src/set-theory/konigs-theorem.lagda.md | 134 --------- ...rict-indexed-inequality-cardinals.lagda.md | 202 ------------- 5 files changed, 617 deletions(-) delete mode 100644 src/set-theory/complemented-inequality-cardinals.lagda.md delete mode 100644 src/set-theory/konigs-theorem.lagda.md delete mode 100644 src/set-theory/strict-indexed-inequality-cardinals.lagda.md diff --git a/src/literature/wikipedia-list-of-theorems.lagda.md b/src/literature/wikipedia-list-of-theorems.lagda.md index b9cf8ec25c..0c7846e7b9 100644 --- a/src/literature/wikipedia-list-of-theorems.lagda.md +++ b/src/literature/wikipedia-list-of-theorems.lagda.md @@ -140,15 +140,6 @@ open import order-theory.knaster-tarski-fixed-point-theorem using greatest-fixed-point-knaster-tarski-Suplattice) ``` -### König's theorem {#Q1077462} - -**Author:** [Fredrik Bakke](https://www.ntnu.edu/employees/fredrik.bakke) - -```agda -open import set-theory.konigs-theorem using - ( Königs-Theorem) -``` - ### Lawvere's fixed point theorem {#Q15809744} **Author:** [Egbert Rijke](https://egbertrijke.github.io) diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index 089be79c69..8e5929aeb5 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -55,7 +55,6 @@ open import set-theory.cantors-diagonal-argument public open import set-theory.cardinality-projective-sets public open import set-theory.cardinality-recursive-sets public open import set-theory.cardinals public -open import set-theory.complemented-inequality-cardinals public open import set-theory.countable-sets public open import set-theory.cumulative-hierarchy public open import set-theory.dependent-products-cardinals public @@ -68,10 +67,8 @@ open import set-theory.inequality-cardinals public open import set-theory.inequality-increasing-binary-sequences public open import set-theory.infinite-sets public open import set-theory.inhabited-cardinals public -open import set-theory.konigs-theorem public open import set-theory.positive-elements-increasing-binary-sequences public open import set-theory.russells-paradox public -open import set-theory.strict-indexed-inequality-cardinals public open import set-theory.strict-lower-bounds-increasing-binary-sequences public open import set-theory.uncountable-sets public open import set-theory.zero-cardinal public diff --git a/src/set-theory/complemented-inequality-cardinals.lagda.md b/src/set-theory/complemented-inequality-cardinals.lagda.md deleted file mode 100644 index 0de213282c..0000000000 --- a/src/set-theory/complemented-inequality-cardinals.lagda.md +++ /dev/null @@ -1,269 +0,0 @@ -# Complemented inequality on cardinals - -```agda -module set-theory.complemented-inequality-cardinals where -``` - -
Imports - -```agda -open import foundation.action-on-identifications-functions -open import foundation.dependent-pair-types -open import foundation.equivalences -open import foundation.function-extensionality -open import foundation.identity-types -open import foundation.large-binary-relations -open import foundation.mere-decidable-embeddings -open import foundation.propositional-extensionality -open import foundation.propositions -open import foundation.set-truncations -open import foundation.sets -open import foundation.univalence -open import foundation.universe-levels -open import foundation.weak-limited-principle-of-omniscience - -open import order-theory.large-posets -open import order-theory.large-preorders - -open import set-theory.cardinals -open import set-theory.equality-cardinals -``` - -
- -## Idea - -We may say a [cardinal](set-theory.cardinals.md) `X` is -{{#concept "less than or equal to" Agda=leq-complemented-Cardinal}} a cardinal -`Y` if any [set](foundation-core.sets.md) in the isomorphism class represented -by `X` [embeds](foundation-core.embeddings.md) as a -[decidable subtype](foundation.decidable-subtypes.md) into any set in the -isomorphism class represented by `Y`. In other words, if there is a -[decidable embedding](foundation.decidable-embeddings.md) from the first to the -second. This defines the -{{#concept "complemented ordering" Disambiguation="on cardinalities of sets"}} -on cardinalities of sets. - -Under the assumption of the -[weak limited principle of omniscience](foundation.weak-limited-principle-of-omniscience.md), -this relation is antisymmetric and hence defines a -[partial order](order-theory.posets.md) on cardinals. - -## Definition - -### Complemented boundedness of the cardinality of a set - -```agda -module _ - {l1 l2 : Level} (X : Set l1) - where - - leq-complemented-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) - leq-complemented-prop-Cardinal' = - map-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) - - compute-leq-complemented-prop-Cardinal' : - (Y : Set l2) → - leq-complemented-prop-Cardinal' (cardinality Y) ＝ - mere-decidable-emb-Prop (type-Set X) (type-Set Y) - compute-leq-complemented-prop-Cardinal' = - triangle-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) -``` - -### Complemented inequality of cardinals - -```agda -module _ - {l1 l2 : Level} - where - - leq-complemented-prop-Cardinal : - Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) - leq-complemented-prop-Cardinal = - map-universal-property-trunc-Set - ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( leq-complemented-prop-Cardinal') - - leq-complemented-Cardinal : - Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) - leq-complemented-Cardinal X Y = - type-Prop (leq-complemented-prop-Cardinal X Y) - - is-prop-leq-complemented-Cardinal : - {X : Cardinal l1} {Y : Cardinal l2} → - is-prop (leq-complemented-Cardinal X Y) - is-prop-leq-complemented-Cardinal {X} {Y} = - is-prop-type-Prop (leq-complemented-prop-Cardinal X Y) -``` - -### Complemented inequality of cardinalities - -```agda -module _ - {l1 l2 : Level} (X : Set l1) (Y : Set l2) - where - - leq-complemented-prop-cardinality : Prop (l1 ⊔ l2) - leq-complemented-prop-cardinality = - leq-complemented-prop-Cardinal (cardinality X) (cardinality Y) - - leq-complemented-cardinality : UU (l1 ⊔ l2) - leq-complemented-cardinality = - leq-complemented-Cardinal (cardinality X) (cardinality Y) - - is-prop-leq-complemented-cardinality : - is-prop leq-complemented-cardinality - is-prop-leq-complemented-cardinality = - is-prop-leq-complemented-Cardinal - - compute-leq-complemented-prop-cardinality' : - leq-complemented-prop-cardinality ＝ - mere-decidable-emb-Prop (type-Set X) (type-Set Y) - compute-leq-complemented-prop-cardinality' = - ( htpy-eq - ( triangle-universal-property-trunc-Set - ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( leq-complemented-prop-Cardinal') X) (cardinality Y)) ∙ - ( compute-leq-complemented-prop-Cardinal' X Y) - - compute-leq-complemented-cardinality' : - leq-complemented-cardinality ＝ - mere-decidable-emb (type-Set X) (type-Set Y) - compute-leq-complemented-cardinality' = - ap type-Prop compute-leq-complemented-prop-cardinality' - - compute-leq-complemented-cardinality : - leq-complemented-cardinality ≃ - mere-decidable-emb (type-Set X) (type-Set Y) - compute-leq-complemented-cardinality = - equiv-eq compute-leq-complemented-cardinality' - - unit-leq-complemented-cardinality : - mere-decidable-emb (type-Set X) (type-Set Y) → - leq-complemented-cardinality - unit-leq-complemented-cardinality = - map-inv-equiv compute-leq-complemented-cardinality - - inv-unit-leq-complemented-cardinality : - leq-complemented-cardinality → - mere-decidable-emb (type-Set X) (type-Set Y) - inv-unit-leq-complemented-cardinality = - pr1 compute-leq-complemented-cardinality -``` - -### Complemented inequality on cardinals is reflexive - -```agda -refl-leq-complemented-cardinality : - is-reflexive-Large-Relation Set leq-complemented-cardinality -refl-leq-complemented-cardinality A = - unit-leq-complemented-cardinality A A - ( refl-mere-decidable-emb (type-Set A)) - -refl-leq-complemented-Cardinal : - is-reflexive-Large-Relation Cardinal leq-complemented-Cardinal -refl-leq-complemented-Cardinal = - apply-dependent-universal-property-trunc-Set' - ( λ X → set-Prop (leq-complemented-prop-Cardinal X X)) - ( refl-leq-complemented-cardinality) -``` - -### Complemented inequality on cardinals is transitive - -```agda -module _ - {l1 l2 l3 : Level} - where - - transitive-leq-complemented-cardinality : - (X : Set l1) (Y : Set l2) (Z : Set l3) → - leq-complemented-cardinality Y Z → - leq-complemented-cardinality X Y → - leq-complemented-cardinality X Z - transitive-leq-complemented-cardinality X Y Z Y≤Z X≤Y = - unit-leq-complemented-cardinality X Z - ( transitive-mere-decidable-emb - ( inv-unit-leq-complemented-cardinality Y Z Y≤Z) - ( inv-unit-leq-complemented-cardinality X Y X≤Y)) - - transitive-leq-complemented-Cardinal : - (X : Cardinal l1) (Y : Cardinal l2) (Z : Cardinal l3) → - leq-complemented-Cardinal Y Z → - leq-complemented-Cardinal X Y → - leq-complemented-Cardinal X Z - transitive-leq-complemented-Cardinal = - apply-thrice-dependent-universal-property-trunc-Set' - ( λ X Y Z → - ( leq-complemented-Cardinal Y Z → - leq-complemented-Cardinal X Y → - leq-complemented-Cardinal X Z) , - ( is-set-function-type - ( is-set-function-type - ( is-set-is-prop is-prop-leq-complemented-Cardinal)))) - ( transitive-leq-complemented-cardinality) -``` - -## Properties - -### Assuming the weak limited principle of omniscience, then complemented inequality forms a partial order - -This is a direct consequence of the -[Cantor–Schröder–Bernstein theorem for decidable embeddings](foundation.cantor-schroder-bernstein-decidable-embeddings.md). - -```agda -module _ - {l : Level} (wlpo : level-WLPO l) - where - - antisymmetric-leq-complemented-cardinality : - (X Y : Set l) → - leq-complemented-cardinality X Y → - leq-complemented-cardinality Y X → - cardinality X ＝ cardinality Y - antisymmetric-leq-complemented-cardinality X Y X≤Y Y≤X = - eq-mere-equiv-cardinality X Y - ( antisymmetric-mere-decidable-emb - ( wlpo) - ( inv-unit-leq-complemented-cardinality X Y X≤Y) - ( inv-unit-leq-complemented-cardinality Y X Y≤X)) - - antisymmetric-leq-complemented-Cardinal : - (X Y : Cardinal l) → - leq-complemented-Cardinal X Y → leq-complemented-Cardinal Y X → X ＝ Y - antisymmetric-leq-complemented-Cardinal = - apply-twice-dependent-universal-property-trunc-Set' - ( λ X Y → - set-Prop - ( function-Prop - ( leq-complemented-Cardinal X Y) - ( function-Prop - ( leq-complemented-Cardinal Y X) - ( Id-Prop (Cardinal-Set l) X Y)))) - ( antisymmetric-leq-complemented-cardinality) -``` - -### The large poset of cardinals under complemented inequality - -```agda -large-preorder-complemented-Cardinal : Large-Preorder lsuc (_⊔_) -large-preorder-complemented-Cardinal = - λ where - .type-Large-Preorder → Cardinal - .leq-prop-Large-Preorder → leq-complemented-prop-Cardinal - .refl-leq-Large-Preorder → refl-leq-complemented-Cardinal - .transitive-leq-Large-Preorder → transitive-leq-complemented-Cardinal - -large-poset-complemented-Cardinal : WLPO → Large-Poset lsuc (_⊔_) -large-poset-complemented-Cardinal wlpo = - λ where - .large-preorder-Large-Poset → large-preorder-complemented-Cardinal - .antisymmetric-leq-Large-Poset → antisymmetric-leq-complemented-Cardinal wlpo -``` - -## See also - -- [Inequality of cardinals](set-theory.inequality-cardinals.md) diff --git a/src/set-theory/konigs-theorem.lagda.md b/src/set-theory/konigs-theorem.lagda.md deleted file mode 100644 index e6de008b90..0000000000 --- a/src/set-theory/konigs-theorem.lagda.md +++ /dev/null @@ -1,134 +0,0 @@ -# Kőnig's theorem - -```agda -module set-theory.konigs-theorem where -``` - -
Imports - -```agda -open import foundation.binary-transport -open import foundation.dependent-pair-types -open import foundation.function-extensionality -open import foundation.function-types -open import foundation.functoriality-propositional-truncation -open import foundation.identity-types -open import foundation.nonsurjective-maps -open import foundation.projective-types -open import foundation.propositions -open import foundation.sets -open import foundation.universe-levels - -open import set-theory.cardinality-projective-sets -open import set-theory.cardinals -open import set-theory.dependent-products-cardinals -open import set-theory.dependent-sums-cardinals -open import set-theory.strict-indexed-inequality-cardinals -``` - -
- -## Idea - -{{#concept "Kőnig's theorem" WD="König's theorem" WDID=Q1077462 Disambiguation="for cardinals" Agda=Königs-Theorem}} -states that for any pair of families of [cardinals](set-theory.cardinals.md) $A$ -and $B$ over $I$ such that $Aᵢ < Bᵢ$ for all $i$, we have that -$$∑_{i:I}Aᵢ < ∏_{i:I}Bᵢ.$$ - -In constructive mathematics we have to be more mindful of our statements. Here -$I$ is any -[cardinality-projective set](set-theory.cardinality-projective-sets.md) and by -$Aᵢ < Bᵢ$ we mean that $Bᵢ$ is [inhabited](foundation.inhabited-types.md) and -that for every map $f : Aᵢ → Bᵢ$ there -[exists](foundation.existential-quantification.md) an element of $Bᵢ$ that $f$ -does not hit. - -## Lemma - -Given a projective type $I$ and a pair of families of types $A$ and $B$ over $I$ -such that for every $i : I$ every function from $Aᵢ$ to $Bᵢ$ misses an element, -then every function from $ΣᵢAᵢ$ to $ΠᵢBᵢ$ misses an element. - -```agda -module _ - {l1 l2 l3 : Level} - {I : UU l1} (p : is-projective-Level' (l2 ⊔ l3) I) - {A : I → UU l2} {B : I → UU l3} - where - - is-nonsurjective-map-Σ-Π-is-projective-base' : - (H : (i : I) → (f : A i → B i) → is-nonsurjective f) - (g : Σ I A → ((i : I) → B i)) → is-nonsurjective g - is-nonsurjective-map-Σ-Π-is-projective-base' H g = - map-trunc-Prop - ( λ h → (pr1 ∘ h , (λ ((i , a) , r) → pr2 (h i) (a , htpy-eq r i)))) - ( p (λ i → nonim (λ a → g (i , a) i)) (λ i → H i (λ a → g (i , a) i))) -``` - -## Theorem - -```agda -module _ - {l1 l2 : Level} - (I : Set l1) - (is-projective-I : is-projective-Level' l2 (type-Set I)) - where - - cardinality-Königs-Theorem' : - (A B : type-Set I → Set l2) → - ((i : type-Set I) → le-indexed-cardinality' (A i) (B i)) → - le-indexed-cardinality' (Σ-Set I A) (Π-Set I B) - cardinality-Königs-Theorem' A B p = - ( is-projective-I (type-Set ∘ B) (pr1 ∘ p) , - is-nonsurjective-map-Σ-Π-is-projective-base' is-projective-I (pr2 ∘ p)) - - cardinality-Königs-Theorem : - (A B : type-Set I → Set l2) → - ((i : type-Set I) → le-indexed-cardinality (A i) (B i)) → - le-indexed-cardinality (Σ-Set I A) (Π-Set I B) - cardinality-Königs-Theorem A B p = - unit-le-indexed-cardinality - ( Σ-Set I A) - ( Π-Set I B) - ( cardinality-Königs-Theorem' A B - ( λ i → inv-unit-le-indexed-cardinality (A i) (B i) (p i))) - -module _ - {l1 l2 : Level} - (I : Cardinality-Projective-Set l1 l2) - (let type-I = type-Cardinality-Projective-Set I) - (let set-I = set-Cardinality-Projective-Set I) - where - - Königs-Theorem : - (A B : type-I → Cardinal l2) → - ((i : type-I) → le-indexed-Cardinal (A i) (B i)) → - le-indexed-Cardinal (Σ-Cardinal I A) (Π-Cardinal I B) - Königs-Theorem = - apply-twice-ind-Cardinality-Projective-Set I - ( λ A B → - set-Prop - ( function-Prop - ( (i : type-I) → le-indexed-Cardinal (A i) (B i)) - ( le-indexed-prop-Cardinal (Σ-Cardinal I A) (Π-Cardinal I B)))) - ( λ A B p → - binary-tr le-indexed-Cardinal - ( inv (compute-Σ-Cardinal I A)) - ( inv (compute-Π-Cardinal I B)) - ( cardinality-Königs-Theorem - ( set-I) - ( is-projective-Cardinality-Projective-Set I) - ( A) - ( B) - ( p))) -``` - -## Comments - -More generally, for every localization `L` contained in `Set` there is an -`L`-modal Kőnig's theorem. - -## External links - -- [Kőnig's theorem (set theory)]() - on Wikipedia diff --git a/src/set-theory/strict-indexed-inequality-cardinals.lagda.md b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md deleted file mode 100644 index 6ef8a6370a..0000000000 --- a/src/set-theory/strict-indexed-inequality-cardinals.lagda.md +++ /dev/null @@ -1,202 +0,0 @@ -# Strict indexed inequality on cardinals - -```agda -module set-theory.strict-indexed-inequality-cardinals where -``` - -
Imports - -```agda -open import foundation.action-on-identifications-functions -open import foundation.cartesian-product-types -open import foundation.conjunction -open import foundation.dependent-pair-types -open import foundation.empty-types -open import foundation.equivalences -open import foundation.function-extensionality -open import foundation.function-types -open import foundation.functoriality-propositional-truncation -open import foundation.identity-types -open import foundation.inhabited-types -open import foundation.large-binary-relations -open import foundation.law-of-excluded-middle -open import foundation.mere-embeddings -open import foundation.negation -open import foundation.nonsurjective-maps -open import foundation.propositional-extensionality -open import foundation.propositional-truncations -open import foundation.propositions -open import foundation.set-truncations -open import foundation.sets -open import foundation.univalence -open import foundation.universe-levels - -open import set-theory.cardinals -open import set-theory.complemented-inequality-cardinals -open import set-theory.inequality-cardinals -open import set-theory.inhabited-cardinals -``` - -
- -## Idea - -We may say a [cardinal of sets](set-theory.cardinals.md) `X` is -{{#concept "indexed less than" Disambiguation="cardinal of sets" Agda=le-indexed-Cardinal}} -a cardinal `Y` if `Y` is inhabited and any map `f` of -[sets](foundation-core.sets.md) from the isomorphism class represented by `X` -into sets in the isomorphism class represented by `Y` is -[nonsurjective](foundation.nonsurjective-maps.md) in the sense that there exists -an element in `Y` that `f` does not hit. This is a positive way of saying that -`X` is less than `Y`. This defines the -{{#concept "strict indexing ordering" Disambiguation="on cardinalities of sets"}} -on cardinalities of sets. - -## Definition - -### The underlying strict indexed inequality between cardinalities of sets - -```agda -module _ - {l1 l2 : Level} (X : Set l1) (Y : Set l2) - where - - le-indexed-cardinality' : UU (l1 ⊔ l2) - le-indexed-cardinality' = - is-inhabited (type-Set Y) × - ((f : type-Set X → type-Set Y) → is-nonsurjective f) - - le-indexed-prop-cardinality' : Prop (l1 ⊔ l2) - le-indexed-prop-cardinality' = - product-Prop - ( is-inhabited-Prop (type-Set Y)) - ( Π-Prop - ( type-Set X → type-Set Y) - ( is-nonsurjective-Prop)) - - is-prop-le-indexed-cardinality' : - is-prop le-indexed-cardinality' - is-prop-le-indexed-cardinality' = - is-prop-type-Prop le-indexed-prop-cardinality' -``` - -### Strict indexed inequality of a cardinal with respect to a set - -```agda -module _ - {l1 l2 : Level} (X : Set l1) - where - - le-indexed-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) - le-indexed-prop-Cardinal' = - map-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( le-indexed-prop-cardinality' X) - - le-indexed-Cardinal' : Cardinal l2 → UU (l1 ⊔ l2) - le-indexed-Cardinal' Y = - type-Prop (le-indexed-prop-Cardinal' Y) - - compute-le-indexed-prop-Cardinal' : - (Y : Set l2) → - le-indexed-prop-Cardinal' (cardinality Y) ＝ - le-indexed-prop-cardinality' X Y - compute-le-indexed-prop-Cardinal' = - triangle-universal-property-trunc-Set - ( Prop-Set (l1 ⊔ l2)) - ( le-indexed-prop-cardinality' X) -``` - -### Strict indexed inequality of cardinals - -```agda -module _ - {l1 l2 : Level} - where - - le-indexed-prop-Cardinal : - Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) - le-indexed-prop-Cardinal = - map-universal-property-trunc-Set - ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( le-indexed-prop-Cardinal') - - le-indexed-Cardinal : Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) - le-indexed-Cardinal X Y = type-Prop (le-indexed-prop-Cardinal X Y) -``` - -### Strict indexed inequality of cardinalities of sets - -```agda -module _ - {l1 l2 : Level} (X : Set l1) (Y : Set l2) - where - - le-indexed-prop-cardinality : Prop (l1 ⊔ l2) - le-indexed-prop-cardinality = - le-indexed-prop-Cardinal (cardinality X) (cardinality Y) - - le-indexed-cardinality : UU (l1 ⊔ l2) - le-indexed-cardinality = type-Prop le-indexed-prop-cardinality - - is-prop-le-indexed-cardinality : is-prop le-indexed-cardinality - is-prop-le-indexed-cardinality = is-prop-type-Prop le-indexed-prop-cardinality - - eq-compute-le-indexed-prop-cardinality : - le-indexed-prop-cardinality ＝ le-indexed-prop-cardinality' X Y - eq-compute-le-indexed-prop-cardinality = - ( htpy-eq - ( triangle-universal-property-trunc-Set - ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) - ( le-indexed-prop-Cardinal') - ( X)) - ( cardinality Y)) ∙ - ( compute-le-indexed-prop-Cardinal' X Y) - - eq-compute-le-indexed-cardinality : - le-indexed-cardinality ＝ le-indexed-cardinality' X Y - eq-compute-le-indexed-cardinality = - ap type-Prop eq-compute-le-indexed-prop-cardinality - - compute-le-indexed-cardinality : - le-indexed-cardinality ≃ le-indexed-cardinality' X Y - compute-le-indexed-cardinality = - equiv-eq eq-compute-le-indexed-cardinality - - unit-le-indexed-cardinality : - le-indexed-cardinality' X Y → le-indexed-cardinality - unit-le-indexed-cardinality = map-inv-equiv compute-le-indexed-cardinality - - inv-unit-le-indexed-cardinality : - le-indexed-cardinality → le-indexed-cardinality' X Y - inv-unit-le-indexed-cardinality = map-equiv compute-le-indexed-cardinality -``` - -## Properties - -## Strict indexed inequality is irreflexive - -```agda -module _ - {l : Level} - where abstract - - irreflexive-le-indexed-cardinality' : - (A : Set l) → ¬ le-indexed-cardinality' A A - irreflexive-le-indexed-cardinality' A p = - rec-trunc-Prop empty-Prop (λ (x , p) → p (x , refl)) (pr2 p id) - - irreflexive-le-indexed-cardinality : - (A : Set l) → ¬ le-indexed-cardinality A A - irreflexive-le-indexed-cardinality A = - map-neg - ( inv-unit-le-indexed-cardinality A A) - ( irreflexive-le-indexed-cardinality' A) - - irreflexive-le-indexed-Cardinal : - (A : Cardinal l) → ¬ le-indexed-Cardinal A A - irreflexive-le-indexed-Cardinal = - apply-dependent-universal-property-trunc-Set' - ( λ X → set-Prop (neg-Prop (le-indexed-prop-Cardinal X X))) - ( irreflexive-le-indexed-cardinality) -``` From a5ca0ed4f17750dc97124aef5f1624ab2727abbf Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 22:58:45 +0000 Subject: [PATCH 46/50] =?UTF-8?q?K=C5=91nig's=20theorem?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../wikipedia-list-of-theorems.lagda.md | 9 + src/set-theory.lagda.md | 3 + ...complemented-inequality-cardinals.lagda.md | 269 ++++++++++++++++++ src/set-theory/konigs-theorem.lagda.md | 134 +++++++++ ...rict-indexed-inequality-cardinals.lagda.md | 202 +++++++++++++ 5 files changed, 617 insertions(+) create mode 100644 src/set-theory/complemented-inequality-cardinals.lagda.md create mode 100644 src/set-theory/konigs-theorem.lagda.md create mode 100644 src/set-theory/strict-indexed-inequality-cardinals.lagda.md diff --git a/src/literature/wikipedia-list-of-theorems.lagda.md b/src/literature/wikipedia-list-of-theorems.lagda.md index 0c7846e7b9..b9cf8ec25c 100644 --- a/src/literature/wikipedia-list-of-theorems.lagda.md +++ b/src/literature/wikipedia-list-of-theorems.lagda.md @@ -140,6 +140,15 @@ open import order-theory.knaster-tarski-fixed-point-theorem using greatest-fixed-point-knaster-tarski-Suplattice) ``` +### König's theorem {#Q1077462} + +**Author:** [Fredrik Bakke](https://www.ntnu.edu/employees/fredrik.bakke) + +```agda +open import set-theory.konigs-theorem using + ( Königs-Theorem) +``` + ### Lawvere's fixed point theorem {#Q15809744} **Author:** [Egbert Rijke](https://egbertrijke.github.io) diff --git a/src/set-theory.lagda.md b/src/set-theory.lagda.md index 8e5929aeb5..089be79c69 100644 --- a/src/set-theory.lagda.md +++ b/src/set-theory.lagda.md @@ -55,6 +55,7 @@ open import set-theory.cantors-diagonal-argument public open import set-theory.cardinality-projective-sets public open import set-theory.cardinality-recursive-sets public open import set-theory.cardinals public +open import set-theory.complemented-inequality-cardinals public open import set-theory.countable-sets public open import set-theory.cumulative-hierarchy public open import set-theory.dependent-products-cardinals public @@ -67,8 +68,10 @@ open import set-theory.inequality-cardinals public open import set-theory.inequality-increasing-binary-sequences public open import set-theory.infinite-sets public open import set-theory.inhabited-cardinals public +open import set-theory.konigs-theorem public open import set-theory.positive-elements-increasing-binary-sequences public open import set-theory.russells-paradox public +open import set-theory.strict-indexed-inequality-cardinals public open import set-theory.strict-lower-bounds-increasing-binary-sequences public open import set-theory.uncountable-sets public open import set-theory.zero-cardinal public diff --git a/src/set-theory/complemented-inequality-cardinals.lagda.md b/src/set-theory/complemented-inequality-cardinals.lagda.md new file mode 100644 index 0000000000..0de213282c --- /dev/null +++ b/src/set-theory/complemented-inequality-cardinals.lagda.md @@ -0,0 +1,269 @@ +# Complemented inequality on cardinals + +```agda +module set-theory.complemented-inequality-cardinals where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.function-extensionality +open import foundation.identity-types +open import foundation.large-binary-relations +open import foundation.mere-decidable-embeddings +open import foundation.propositional-extensionality +open import foundation.propositions +open import foundation.set-truncations +open import foundation.sets +open import foundation.univalence +open import foundation.universe-levels +open import foundation.weak-limited-principle-of-omniscience + +open import order-theory.large-posets +open import order-theory.large-preorders + +open import set-theory.cardinals +open import set-theory.equality-cardinals +``` + +
+ +## Idea + +We may say a [cardinal](set-theory.cardinals.md) `X` is +{{#concept "less than or equal to" Agda=leq-complemented-Cardinal}} a cardinal +`Y` if any [set](foundation-core.sets.md) in the isomorphism class represented +by `X` [embeds](foundation-core.embeddings.md) as a +[decidable subtype](foundation.decidable-subtypes.md) into any set in the +isomorphism class represented by `Y`. In other words, if there is a +[decidable embedding](foundation.decidable-embeddings.md) from the first to the +second. This defines the +{{#concept "complemented ordering" Disambiguation="on cardinalities of sets"}} +on cardinalities of sets. + +Under the assumption of the +[weak limited principle of omniscience](foundation.weak-limited-principle-of-omniscience.md), +this relation is antisymmetric and hence defines a +[partial order](order-theory.posets.md) on cardinals. + +## Definition + +### Complemented boundedness of the cardinality of a set + +```agda +module _ + {l1 l2 : Level} (X : Set l1) + where + + leq-complemented-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) + leq-complemented-prop-Cardinal' = + map-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) + + compute-leq-complemented-prop-Cardinal' : + (Y : Set l2) → + leq-complemented-prop-Cardinal' (cardinality Y) ＝ + mere-decidable-emb-Prop (type-Set X) (type-Set Y) + compute-leq-complemented-prop-Cardinal' = + triangle-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( λ Y' → mere-decidable-emb-Prop (type-Set X) (type-Set Y')) +``` + +### Complemented inequality of cardinals + +```agda +module _ + {l1 l2 : Level} + where + + leq-complemented-prop-Cardinal : + Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) + leq-complemented-prop-Cardinal = + map-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-complemented-prop-Cardinal') + + leq-complemented-Cardinal : + Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) + leq-complemented-Cardinal X Y = + type-Prop (leq-complemented-prop-Cardinal X Y) + + is-prop-leq-complemented-Cardinal : + {X : Cardinal l1} {Y : Cardinal l2} → + is-prop (leq-complemented-Cardinal X Y) + is-prop-leq-complemented-Cardinal {X} {Y} = + is-prop-type-Prop (leq-complemented-prop-Cardinal X Y) +``` + +### Complemented inequality of cardinalities + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + leq-complemented-prop-cardinality : Prop (l1 ⊔ l2) + leq-complemented-prop-cardinality = + leq-complemented-prop-Cardinal (cardinality X) (cardinality Y) + + leq-complemented-cardinality : UU (l1 ⊔ l2) + leq-complemented-cardinality = + leq-complemented-Cardinal (cardinality X) (cardinality Y) + + is-prop-leq-complemented-cardinality : + is-prop leq-complemented-cardinality + is-prop-leq-complemented-cardinality = + is-prop-leq-complemented-Cardinal + + compute-leq-complemented-prop-cardinality' : + leq-complemented-prop-cardinality ＝ + mere-decidable-emb-Prop (type-Set X) (type-Set Y) + compute-leq-complemented-prop-cardinality' = + ( htpy-eq + ( triangle-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( leq-complemented-prop-Cardinal') X) (cardinality Y)) ∙ + ( compute-leq-complemented-prop-Cardinal' X Y) + + compute-leq-complemented-cardinality' : + leq-complemented-cardinality ＝ + mere-decidable-emb (type-Set X) (type-Set Y) + compute-leq-complemented-cardinality' = + ap type-Prop compute-leq-complemented-prop-cardinality' + + compute-leq-complemented-cardinality : + leq-complemented-cardinality ≃ + mere-decidable-emb (type-Set X) (type-Set Y) + compute-leq-complemented-cardinality = + equiv-eq compute-leq-complemented-cardinality' + + unit-leq-complemented-cardinality : + mere-decidable-emb (type-Set X) (type-Set Y) → + leq-complemented-cardinality + unit-leq-complemented-cardinality = + map-inv-equiv compute-leq-complemented-cardinality + + inv-unit-leq-complemented-cardinality : + leq-complemented-cardinality → + mere-decidable-emb (type-Set X) (type-Set Y) + inv-unit-leq-complemented-cardinality = + pr1 compute-leq-complemented-cardinality +``` + +### Complemented inequality on cardinals is reflexive + +```agda +refl-leq-complemented-cardinality : + is-reflexive-Large-Relation Set leq-complemented-cardinality +refl-leq-complemented-cardinality A = + unit-leq-complemented-cardinality A A + ( refl-mere-decidable-emb (type-Set A)) + +refl-leq-complemented-Cardinal : + is-reflexive-Large-Relation Cardinal leq-complemented-Cardinal +refl-leq-complemented-Cardinal = + apply-dependent-universal-property-trunc-Set' + ( λ X → set-Prop (leq-complemented-prop-Cardinal X X)) + ( refl-leq-complemented-cardinality) +``` + +### Complemented inequality on cardinals is transitive + +```agda +module _ + {l1 l2 l3 : Level} + where + + transitive-leq-complemented-cardinality : + (X : Set l1) (Y : Set l2) (Z : Set l3) → + leq-complemented-cardinality Y Z → + leq-complemented-cardinality X Y → + leq-complemented-cardinality X Z + transitive-leq-complemented-cardinality X Y Z Y≤Z X≤Y = + unit-leq-complemented-cardinality X Z + ( transitive-mere-decidable-emb + ( inv-unit-leq-complemented-cardinality Y Z Y≤Z) + ( inv-unit-leq-complemented-cardinality X Y X≤Y)) + + transitive-leq-complemented-Cardinal : + (X : Cardinal l1) (Y : Cardinal l2) (Z : Cardinal l3) → + leq-complemented-Cardinal Y Z → + leq-complemented-Cardinal X Y → + leq-complemented-Cardinal X Z + transitive-leq-complemented-Cardinal = + apply-thrice-dependent-universal-property-trunc-Set' + ( λ X Y Z → + ( leq-complemented-Cardinal Y Z → + leq-complemented-Cardinal X Y → + leq-complemented-Cardinal X Z) , + ( is-set-function-type + ( is-set-function-type + ( is-set-is-prop is-prop-leq-complemented-Cardinal)))) + ( transitive-leq-complemented-cardinality) +``` + +## Properties + +### Assuming the weak limited principle of omniscience, then complemented inequality forms a partial order + +This is a direct consequence of the +[Cantor–Schröder–Bernstein theorem for decidable embeddings](foundation.cantor-schroder-bernstein-decidable-embeddings.md). + +```agda +module _ + {l : Level} (wlpo : level-WLPO l) + where + + antisymmetric-leq-complemented-cardinality : + (X Y : Set l) → + leq-complemented-cardinality X Y → + leq-complemented-cardinality Y X → + cardinality X ＝ cardinality Y + antisymmetric-leq-complemented-cardinality X Y X≤Y Y≤X = + eq-mere-equiv-cardinality X Y + ( antisymmetric-mere-decidable-emb + ( wlpo) + ( inv-unit-leq-complemented-cardinality X Y X≤Y) + ( inv-unit-leq-complemented-cardinality Y X Y≤X)) + + antisymmetric-leq-complemented-Cardinal : + (X Y : Cardinal l) → + leq-complemented-Cardinal X Y → leq-complemented-Cardinal Y X → X ＝ Y + antisymmetric-leq-complemented-Cardinal = + apply-twice-dependent-universal-property-trunc-Set' + ( λ X Y → + set-Prop + ( function-Prop + ( leq-complemented-Cardinal X Y) + ( function-Prop + ( leq-complemented-Cardinal Y X) + ( Id-Prop (Cardinal-Set l) X Y)))) + ( antisymmetric-leq-complemented-cardinality) +``` + +### The large poset of cardinals under complemented inequality + +```agda +large-preorder-complemented-Cardinal : Large-Preorder lsuc (_⊔_) +large-preorder-complemented-Cardinal = + λ where + .type-Large-Preorder → Cardinal + .leq-prop-Large-Preorder → leq-complemented-prop-Cardinal + .refl-leq-Large-Preorder → refl-leq-complemented-Cardinal + .transitive-leq-Large-Preorder → transitive-leq-complemented-Cardinal + +large-poset-complemented-Cardinal : WLPO → Large-Poset lsuc (_⊔_) +large-poset-complemented-Cardinal wlpo = + λ where + .large-preorder-Large-Poset → large-preorder-complemented-Cardinal + .antisymmetric-leq-Large-Poset → antisymmetric-leq-complemented-Cardinal wlpo +``` + +## See also + +- [Inequality of cardinals](set-theory.inequality-cardinals.md) diff --git a/src/set-theory/konigs-theorem.lagda.md b/src/set-theory/konigs-theorem.lagda.md new file mode 100644 index 0000000000..e6de008b90 --- /dev/null +++ b/src/set-theory/konigs-theorem.lagda.md @@ -0,0 +1,134 @@ +# Kőnig's theorem + +```agda +module set-theory.konigs-theorem where +``` + +
Imports + +```agda +open import foundation.binary-transport +open import foundation.dependent-pair-types +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.functoriality-propositional-truncation +open import foundation.identity-types +open import foundation.nonsurjective-maps +open import foundation.projective-types +open import foundation.propositions +open import foundation.sets +open import foundation.universe-levels + +open import set-theory.cardinality-projective-sets +open import set-theory.cardinals +open import set-theory.dependent-products-cardinals +open import set-theory.dependent-sums-cardinals +open import set-theory.strict-indexed-inequality-cardinals +``` + +
+ +## Idea + +{{#concept "Kőnig's theorem" WD="König's theorem" WDID=Q1077462 Disambiguation="for cardinals" Agda=Königs-Theorem}} +states that for any pair of families of [cardinals](set-theory.cardinals.md) $A$ +and $B$ over $I$ such that $Aᵢ < Bᵢ$ for all $i$, we have that +$$∑_{i:I}Aᵢ < ∏_{i:I}Bᵢ.$$ + +In constructive mathematics we have to be more mindful of our statements. Here +$I$ is any +[cardinality-projective set](set-theory.cardinality-projective-sets.md) and by +$Aᵢ < Bᵢ$ we mean that $Bᵢ$ is [inhabited](foundation.inhabited-types.md) and +that for every map $f : Aᵢ → Bᵢ$ there +[exists](foundation.existential-quantification.md) an element of $Bᵢ$ that $f$ +does not hit. + +## Lemma + +Given a projective type $I$ and a pair of families of types $A$ and $B$ over $I$ +such that for every $i : I$ every function from $Aᵢ$ to $Bᵢ$ misses an element, +then every function from $ΣᵢAᵢ$ to $ΠᵢBᵢ$ misses an element. + +```agda +module _ + {l1 l2 l3 : Level} + {I : UU l1} (p : is-projective-Level' (l2 ⊔ l3) I) + {A : I → UU l2} {B : I → UU l3} + where + + is-nonsurjective-map-Σ-Π-is-projective-base' : + (H : (i : I) → (f : A i → B i) → is-nonsurjective f) + (g : Σ I A → ((i : I) → B i)) → is-nonsurjective g + is-nonsurjective-map-Σ-Π-is-projective-base' H g = + map-trunc-Prop + ( λ h → (pr1 ∘ h , (λ ((i , a) , r) → pr2 (h i) (a , htpy-eq r i)))) + ( p (λ i → nonim (λ a → g (i , a) i)) (λ i → H i (λ a → g (i , a) i))) +``` + +## Theorem + +```agda +module _ + {l1 l2 : Level} + (I : Set l1) + (is-projective-I : is-projective-Level' l2 (type-Set I)) + where + + cardinality-Königs-Theorem' : + (A B : type-Set I → Set l2) → + ((i : type-Set I) → le-indexed-cardinality' (A i) (B i)) → + le-indexed-cardinality' (Σ-Set I A) (Π-Set I B) + cardinality-Königs-Theorem' A B p = + ( is-projective-I (type-Set ∘ B) (pr1 ∘ p) , + is-nonsurjective-map-Σ-Π-is-projective-base' is-projective-I (pr2 ∘ p)) + + cardinality-Königs-Theorem : + (A B : type-Set I → Set l2) → + ((i : type-Set I) → le-indexed-cardinality (A i) (B i)) → + le-indexed-cardinality (Σ-Set I A) (Π-Set I B) + cardinality-Königs-Theorem A B p = + unit-le-indexed-cardinality + ( Σ-Set I A) + ( Π-Set I B) + ( cardinality-Königs-Theorem' A B + ( λ i → inv-unit-le-indexed-cardinality (A i) (B i) (p i))) + +module _ + {l1 l2 : Level} + (I : Cardinality-Projective-Set l1 l2) + (let type-I = type-Cardinality-Projective-Set I) + (let set-I = set-Cardinality-Projective-Set I) + where + + Königs-Theorem : + (A B : type-I → Cardinal l2) → + ((i : type-I) → le-indexed-Cardinal (A i) (B i)) → + le-indexed-Cardinal (Σ-Cardinal I A) (Π-Cardinal I B) + Königs-Theorem = + apply-twice-ind-Cardinality-Projective-Set I + ( λ A B → + set-Prop + ( function-Prop + ( (i : type-I) → le-indexed-Cardinal (A i) (B i)) + ( le-indexed-prop-Cardinal (Σ-Cardinal I A) (Π-Cardinal I B)))) + ( λ A B p → + binary-tr le-indexed-Cardinal + ( inv (compute-Σ-Cardinal I A)) + ( inv (compute-Π-Cardinal I B)) + ( cardinality-Königs-Theorem + ( set-I) + ( is-projective-Cardinality-Projective-Set I) + ( A) + ( B) + ( p))) +``` + +## Comments + +More generally, for every localization `L` contained in `Set` there is an +`L`-modal Kőnig's theorem. + +## External links + +- [Kőnig's theorem (set theory)]() + on Wikipedia diff --git a/src/set-theory/strict-indexed-inequality-cardinals.lagda.md b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md new file mode 100644 index 0000000000..6ef8a6370a --- /dev/null +++ b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md @@ -0,0 +1,202 @@ +# Strict indexed inequality on cardinals + +```agda +module set-theory.strict-indexed-inequality-cardinals where +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.cartesian-product-types +open import foundation.conjunction +open import foundation.dependent-pair-types +open import foundation.empty-types +open import foundation.equivalences +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.functoriality-propositional-truncation +open import foundation.identity-types +open import foundation.inhabited-types +open import foundation.large-binary-relations +open import foundation.law-of-excluded-middle +open import foundation.mere-embeddings +open import foundation.negation +open import foundation.nonsurjective-maps +open import foundation.propositional-extensionality +open import foundation.propositional-truncations +open import foundation.propositions +open import foundation.set-truncations +open import foundation.sets +open import foundation.univalence +open import foundation.universe-levels + +open import set-theory.cardinals +open import set-theory.complemented-inequality-cardinals +open import set-theory.inequality-cardinals +open import set-theory.inhabited-cardinals +``` + +
+ +## Idea + +We may say a [cardinal of sets](set-theory.cardinals.md) `X` is +{{#concept "indexed less than" Disambiguation="cardinal of sets" Agda=le-indexed-Cardinal}} +a cardinal `Y` if `Y` is inhabited and any map `f` of +[sets](foundation-core.sets.md) from the isomorphism class represented by `X` +into sets in the isomorphism class represented by `Y` is +[nonsurjective](foundation.nonsurjective-maps.md) in the sense that there exists +an element in `Y` that `f` does not hit. This is a positive way of saying that +`X` is less than `Y`. This defines the +{{#concept "strict indexing ordering" Disambiguation="on cardinalities of sets"}} +on cardinalities of sets. + +## Definition + +### The underlying strict indexed inequality between cardinalities of sets + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + le-indexed-cardinality' : UU (l1 ⊔ l2) + le-indexed-cardinality' = + is-inhabited (type-Set Y) × + ((f : type-Set X → type-Set Y) → is-nonsurjective f) + + le-indexed-prop-cardinality' : Prop (l1 ⊔ l2) + le-indexed-prop-cardinality' = + product-Prop + ( is-inhabited-Prop (type-Set Y)) + ( Π-Prop + ( type-Set X → type-Set Y) + ( is-nonsurjective-Prop)) + + is-prop-le-indexed-cardinality' : + is-prop le-indexed-cardinality' + is-prop-le-indexed-cardinality' = + is-prop-type-Prop le-indexed-prop-cardinality' +``` + +### Strict indexed inequality of a cardinal with respect to a set + +```agda +module _ + {l1 l2 : Level} (X : Set l1) + where + + le-indexed-prop-Cardinal' : Cardinal l2 → Prop (l1 ⊔ l2) + le-indexed-prop-Cardinal' = + map-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( le-indexed-prop-cardinality' X) + + le-indexed-Cardinal' : Cardinal l2 → UU (l1 ⊔ l2) + le-indexed-Cardinal' Y = + type-Prop (le-indexed-prop-Cardinal' Y) + + compute-le-indexed-prop-Cardinal' : + (Y : Set l2) → + le-indexed-prop-Cardinal' (cardinality Y) ＝ + le-indexed-prop-cardinality' X Y + compute-le-indexed-prop-Cardinal' = + triangle-universal-property-trunc-Set + ( Prop-Set (l1 ⊔ l2)) + ( le-indexed-prop-cardinality' X) +``` + +### Strict indexed inequality of cardinals + +```agda +module _ + {l1 l2 : Level} + where + + le-indexed-prop-Cardinal : + Cardinal l1 → Cardinal l2 → Prop (l1 ⊔ l2) + le-indexed-prop-Cardinal = + map-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( le-indexed-prop-Cardinal') + + le-indexed-Cardinal : Cardinal l1 → Cardinal l2 → UU (l1 ⊔ l2) + le-indexed-Cardinal X Y = type-Prop (le-indexed-prop-Cardinal X Y) +``` + +### Strict indexed inequality of cardinalities of sets + +```agda +module _ + {l1 l2 : Level} (X : Set l1) (Y : Set l2) + where + + le-indexed-prop-cardinality : Prop (l1 ⊔ l2) + le-indexed-prop-cardinality = + le-indexed-prop-Cardinal (cardinality X) (cardinality Y) + + le-indexed-cardinality : UU (l1 ⊔ l2) + le-indexed-cardinality = type-Prop le-indexed-prop-cardinality + + is-prop-le-indexed-cardinality : is-prop le-indexed-cardinality + is-prop-le-indexed-cardinality = is-prop-type-Prop le-indexed-prop-cardinality + + eq-compute-le-indexed-prop-cardinality : + le-indexed-prop-cardinality ＝ le-indexed-prop-cardinality' X Y + eq-compute-le-indexed-prop-cardinality = + ( htpy-eq + ( triangle-universal-property-trunc-Set + ( hom-set-Set (Cardinal-Set l2) (Prop-Set (l1 ⊔ l2))) + ( le-indexed-prop-Cardinal') + ( X)) + ( cardinality Y)) ∙ + ( compute-le-indexed-prop-Cardinal' X Y) + + eq-compute-le-indexed-cardinality : + le-indexed-cardinality ＝ le-indexed-cardinality' X Y + eq-compute-le-indexed-cardinality = + ap type-Prop eq-compute-le-indexed-prop-cardinality + + compute-le-indexed-cardinality : + le-indexed-cardinality ≃ le-indexed-cardinality' X Y + compute-le-indexed-cardinality = + equiv-eq eq-compute-le-indexed-cardinality + + unit-le-indexed-cardinality : + le-indexed-cardinality' X Y → le-indexed-cardinality + unit-le-indexed-cardinality = map-inv-equiv compute-le-indexed-cardinality + + inv-unit-le-indexed-cardinality : + le-indexed-cardinality → le-indexed-cardinality' X Y + inv-unit-le-indexed-cardinality = map-equiv compute-le-indexed-cardinality +``` + +## Properties + +## Strict indexed inequality is irreflexive + +```agda +module _ + {l : Level} + where abstract + + irreflexive-le-indexed-cardinality' : + (A : Set l) → ¬ le-indexed-cardinality' A A + irreflexive-le-indexed-cardinality' A p = + rec-trunc-Prop empty-Prop (λ (x , p) → p (x , refl)) (pr2 p id) + + irreflexive-le-indexed-cardinality : + (A : Set l) → ¬ le-indexed-cardinality A A + irreflexive-le-indexed-cardinality A = + map-neg + ( inv-unit-le-indexed-cardinality A A) + ( irreflexive-le-indexed-cardinality' A) + + irreflexive-le-indexed-Cardinal : + (A : Cardinal l) → ¬ le-indexed-Cardinal A A + irreflexive-le-indexed-Cardinal = + apply-dependent-universal-property-trunc-Set' + ( λ X → set-Prop (neg-Prop (le-indexed-prop-Cardinal X X))) + ( irreflexive-le-indexed-cardinality) +``` From 8edb660167378b752ae65bb4c0dcbf5cc0989873 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 23:06:53 +0000 Subject: [PATCH 47/50] disambiguation --- src/literature/wikipedia-list-of-theorems.lagda.md | 2 +- src/set-theory/konigs-theorem.lagda.md | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/src/literature/wikipedia-list-of-theorems.lagda.md b/src/literature/wikipedia-list-of-theorems.lagda.md index b9cf8ec25c..d6ff543f4f 100644 --- a/src/literature/wikipedia-list-of-theorems.lagda.md +++ b/src/literature/wikipedia-list-of-theorems.lagda.md @@ -140,7 +140,7 @@ open import order-theory.knaster-tarski-fixed-point-theorem using greatest-fixed-point-knaster-tarski-Suplattice) ``` -### König's theorem {#Q1077462} +### König's theorem (set theory) {#Q1077462} **Author:** [Fredrik Bakke](https://www.ntnu.edu/employees/fredrik.bakke) diff --git a/src/set-theory/konigs-theorem.lagda.md b/src/set-theory/konigs-theorem.lagda.md index e6de008b90..5c12a19575 100644 --- a/src/set-theory/konigs-theorem.lagda.md +++ b/src/set-theory/konigs-theorem.lagda.md @@ -30,7 +30,7 @@ open import set-theory.strict-indexed-inequality-cardinals ## Idea -{{#concept "Kőnig's theorem" WD="König's theorem" WDID=Q1077462 Disambiguation="for cardinals" Agda=Königs-Theorem}} +{{#concept "Kőnig's theorem" Disambiguation="for cardinals/set theory" WD="König's theorem" WDID=Q1077462 Agda=Königs-Theorem}} states that for any pair of families of [cardinals](set-theory.cardinals.md) $A$ and $B$ over $I$ such that $Aᵢ < Bᵢ$ for all $i$, we have that $$∑_{i:I}Aᵢ < ∏_{i:I}Bᵢ.$$ From ac0d3ac3151ef8375675d87cd604691acc223005 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Mon, 17 Nov 2025 23:07:00 +0000 Subject: [PATCH 48/50] fixes --- ...rict-indexed-inequality-cardinals.lagda.md | 2 +- ...tations-complete-undirected-graph.lagda.md | 58 ++++++++++++++----- 2 files changed, 44 insertions(+), 16 deletions(-) diff --git a/src/set-theory/strict-indexed-inequality-cardinals.lagda.md b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md index 6ef8a6370a..bfb98955a9 100644 --- a/src/set-theory/strict-indexed-inequality-cardinals.lagda.md +++ b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md @@ -174,7 +174,7 @@ module _ ## Properties -## Strict indexed inequality is irreflexive +### Strict indexed inequality is irreflexive ```agda module _ diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md index 6d1ada5aae..9587db8e97 100644 --- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md +++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md @@ -899,10 +899,16 @@ module _ ( eq-pair-Σ ( eq-is-prop ( is-prop-is-decidable - ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) i))) + ( is-set-type-count + ( eX) + ( pr1 (two-elements-transposition eX Y)) + ( i)))) ( eq-is-prop ( is-prop-is-decidable - ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) j))))) ∙ + ( is-set-type-count + ( eX) + ( pr1 (two-elements-transposition eX Y)) + ( j)))))) ∙ ( r1) cases-inward-edge-left-two-elements-orientation-count i j np Y x nq nr (inr (pair r1 r2)) = @@ -928,7 +934,10 @@ module _ { y = inl r1} ( eq-is-prop ( is-prop-is-decidable - ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) i)))) ∙ + ( is-set-type-count + ( eX) + ( pr1 (two-elements-transposition eX Y)) + ( i))))) ∙ ( r2) inward-edge-left-two-elements-orientation-count : @@ -1173,10 +1182,16 @@ module _ ( eq-pair-Σ ( eq-is-prop ( is-prop-is-decidable - ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) i))) + ( is-set-type-count + ( eX) + ( pr1 (two-elements-transposition eX Y)) + ( i)))) ( eq-is-prop ( is-prop-is-decidable - ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) j))))) ∙ + ( is-set-type-count + ( eX) + ( pr1 (two-elements-transposition eX Y)) + ( j)))))) ∙ ( r1) cases-inward-edge-right-two-elements-orientation-count i j np Y x nq nr (inr (pair r1 r2)) = @@ -1205,10 +1220,15 @@ module _ ( eq-pair-Σ ( eq-is-prop ( is-prop-is-decidable - ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) i))) + ( is-set-type-count + ( eX) + ( pr1 (two-elements-transposition eX Y)) i))) ( eq-is-prop ( is-prop-is-decidable - ( is-set-type-count eX (pr1 (two-elements-transposition eX Y)) j))))) ∙ + ( is-set-type-count + ( eX) + ( pr1 (two-elements-transposition eX Y)) + ( j)))))) ∙ ( ( ap ( λ d → pr1 @@ -2707,8 +2727,10 @@ module _ ( has-decidable-equality-count eX x j)} { y = pair (inr nq) (inr nr)} ( eq-pair-Σ - ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x i))) - ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x j)))))))) ∙ + ( eq-is-prop + ( is-prop-is-decidable (is-set-type-count eX x i))) + ( eq-is-prop + ( is-prop-is-decidable (is-set-type-count eX x j)))))))) ∙ ( ( is-fixed-point-standard-transposition ( has-decidable-equality-count eX) ( np) @@ -2729,8 +2751,10 @@ module _ ( has-decidable-equality-count eX x j) ( has-decidable-equality-count eX x i)} ( eq-pair-Σ - ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x j))) - ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x i))))) ∙ + ( eq-is-prop + ( is-prop-is-decidable (is-set-type-count eX x j))) + ( eq-is-prop + ( is-prop-is-decidable (is-set-type-count eX x i))))) ∙ ( ap ( λ k → pr1 @@ -3050,8 +3074,10 @@ module _ ( has-decidable-equality-count eX y i)) { y = pair (inr nq) (inr nr)} ( eq-pair-Σ - ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x i))) - ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x j))))) ∙ + ( eq-is-prop + ( is-prop-is-decidable (is-set-type-count eX x i))) + ( eq-is-prop + ( is-prop-is-decidable (is-set-type-count eX x j))))) ∙ ( ap ( λ D → cases-orientation-two-elements-count j i @@ -3066,8 +3092,10 @@ module _ ( has-decidable-equality-count eX x j) ( has-decidable-equality-count eX x i)} ( eq-pair-Σ - ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x j))) - ( eq-is-prop (is-prop-is-decidable (is-set-type-count eX x i)))) ∙ + ( eq-is-prop + ( is-prop-is-decidable (is-set-type-count eX x j))) + ( eq-is-prop + ( is-prop-is-decidable (is-set-type-count eX x i)))) ∙ ( ap ( λ w → cases-orientation-two-elements-count j i From d6170a63fb9f7ba4a140b7ff515af7f29130eab7 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Tue, 18 Nov 2025 21:09:12 +0000 Subject: [PATCH 49/50] edit --- src/set-theory/konigs-theorem.lagda.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/src/set-theory/konigs-theorem.lagda.md b/src/set-theory/konigs-theorem.lagda.md index 5c12a19575..b04218909e 100644 --- a/src/set-theory/konigs-theorem.lagda.md +++ b/src/set-theory/konigs-theorem.lagda.md @@ -32,8 +32,7 @@ open import set-theory.strict-indexed-inequality-cardinals {{#concept "Kőnig's theorem" Disambiguation="for cardinals/set theory" WD="König's theorem" WDID=Q1077462 Agda=Königs-Theorem}} states that for any pair of families of [cardinals](set-theory.cardinals.md) $A$ -and $B$ over $I$ such that $Aᵢ < Bᵢ$ for all $i$, we have that -$$∑_{i:I}Aᵢ < ∏_{i:I}Bᵢ.$$ +and $B$ over $I$ such that $Aᵢ < Bᵢ$ for all $i$, we have that $$ΣᵢAᵢ < ΠᵢBᵢ.$$ In constructive mathematics we have to be more mindful of our statements. Here $I$ is any @@ -57,7 +56,7 @@ module _ where is-nonsurjective-map-Σ-Π-is-projective-base' : - (H : (i : I) → (f : A i → B i) → is-nonsurjective f) + (H : (i : I) (f : A i → B i) → is-nonsurjective f) (g : Σ I A → ((i : I) → B i)) → is-nonsurjective g is-nonsurjective-map-Σ-Π-is-projective-base' H g = map-trunc-Prop @@ -112,7 +111,8 @@ module _ ( (i : type-I) → le-indexed-Cardinal (A i) (B i)) ( le-indexed-prop-Cardinal (Σ-Cardinal I A) (Π-Cardinal I B)))) ( λ A B p → - binary-tr le-indexed-Cardinal + binary-tr + ( le-indexed-Cardinal) ( inv (compute-Σ-Cardinal I A)) ( inv (compute-Π-Cardinal I B)) ( cardinality-Königs-Theorem From 678939869ae8eaa7acf9439ca3b8452cf7f35e81 Mon Sep 17 00:00:00 2001 From: Fredrik Bakke Date: Sun, 23 Nov 2025 18:29:52 +0000 Subject: [PATCH 50/50] edits --- src/set-theory/cardinality-projective-sets.lagda.md | 9 +++++---- src/set-theory/cardinality-recursive-sets.lagda.md | 8 +++++--- src/set-theory/inhabited-cardinals.lagda.md | 4 ++-- src/set-theory/konigs-theorem.lagda.md | 6 +++--- .../strict-indexed-inequality-cardinals.lagda.md | 11 +++++------ 5 files changed, 20 insertions(+), 18 deletions(-) diff --git a/src/set-theory/cardinality-projective-sets.lagda.md b/src/set-theory/cardinality-projective-sets.lagda.md index 16fe26fadc..b0a79da27f 100644 --- a/src/set-theory/cardinality-projective-sets.lagda.md +++ b/src/set-theory/cardinality-projective-sets.lagda.md @@ -51,10 +51,11 @@ open import univalent-combinatorics.standard-finite-types ## Idea -A set $I$ is -{{#concept "cardinality-projective" Disamibguation="sets" Agda=Cardinality-Projective-Set}} -if it is projective and the postcomposition map -$$\mathrm{cardinality} ∘ {-} : (I → Set) → (I → \mathrm{Cardinal})$$ is +A [set](foundation-core.sets.md) $I$ is +{{#concept "cardinality-projective" Disambiguation="sets" Agda=Cardinality-Projective-Set}} +if it is [projective](foundation.projective-types.md) and the +[postcomposition map](foundation.postcomposition-functions.md) +$$\mathrm{cardinality} ∘ {-} : (I → \mathrm{Set}) → (I → \mathrm{Cardinal})$$ is 0-connected. ## Definitions diff --git a/src/set-theory/cardinality-recursive-sets.lagda.md b/src/set-theory/cardinality-recursive-sets.lagda.md index 5670b4f196..78f7da5170 100644 --- a/src/set-theory/cardinality-recursive-sets.lagda.md +++ b/src/set-theory/cardinality-recursive-sets.lagda.md @@ -32,9 +32,11 @@ open import set-theory.cardinals ## Idea -For every type $X$ there is a map $║X → Set║₀ → (X → \mathrm{Cardinal})$. We -call [sets](foundation-core.sets.md) $X$ for which this map is a retract -{{#concept "cardinality-recursive" Disamibguation="sets" Agda=Cardinality-Recursive-Set}}. +For every type $X$ there is a map +$$║X → \mathrm{Set}║₀ → (X → \mathrm{Cardinal}).$$ We call +[sets](foundation-core.sets.md) $X$ for which this map has a +[retraction](foundation.retractions.md) +{{#concept "cardinality-recursive" Disambiguation="sets" Agda=Cardinality-Recursive-Set}}. Over such sets we may form [dependent sum](set-theory.dependent-sums-cardinals.md) and [dependent product](set-theory.dependent-products-cardinals.md) diff --git a/src/set-theory/inhabited-cardinals.lagda.md b/src/set-theory/inhabited-cardinals.lagda.md index 0d0d56ae63..cf5b41c855 100644 --- a/src/set-theory/inhabited-cardinals.lagda.md +++ b/src/set-theory/inhabited-cardinals.lagda.md @@ -28,8 +28,8 @@ open import set-theory.cardinals ## Idea -A [cardinal of sets](set-theory.cardinals.md) `κ` is -{{#concept "inhabited" Disambiguation="cardinal" Agda=is-inhabited-Cardinal}}, +A [cardinal](set-theory.cardinals.md) `κ` is +{{#concept "inhabited" Disambiguation="set-cardinal" Agda=is-inhabited-Cardinal}}, if any [set](foundation-core.sets.md) in its isomorphism class is [inhabited](foundation.inhabited-types.md). diff --git a/src/set-theory/konigs-theorem.lagda.md b/src/set-theory/konigs-theorem.lagda.md index b04218909e..2325b6c663 100644 --- a/src/set-theory/konigs-theorem.lagda.md +++ b/src/set-theory/konigs-theorem.lagda.md @@ -32,10 +32,10 @@ open import set-theory.strict-indexed-inequality-cardinals {{#concept "Kőnig's theorem" Disambiguation="for cardinals/set theory" WD="König's theorem" WDID=Q1077462 Agda=Königs-Theorem}} states that for any pair of families of [cardinals](set-theory.cardinals.md) $A$ -and $B$ over $I$ such that $Aᵢ < Bᵢ$ for all $i$, we have that $$ΣᵢAᵢ < ΠᵢBᵢ.$$ +and $B$ over $I$, $Aᵢ < Bᵢ$ for all $i$ then we have that $ΣᵢAᵢ < ΠᵢBᵢ$. -In constructive mathematics we have to be more mindful of our statements. Here -$I$ is any +In constructive mathematics we have to be more mindful of our statements than +usual. Here $I$ is any [cardinality-projective set](set-theory.cardinality-projective-sets.md) and by $Aᵢ < Bᵢ$ we mean that $Bᵢ$ is [inhabited](foundation.inhabited-types.md) and that for every map $f : Aᵢ → Bᵢ$ there diff --git a/src/set-theory/strict-indexed-inequality-cardinals.lagda.md b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md index bfb98955a9..6c5ec9d7ed 100644 --- a/src/set-theory/strict-indexed-inequality-cardinals.lagda.md +++ b/src/set-theory/strict-indexed-inequality-cardinals.lagda.md @@ -41,16 +41,15 @@ open import set-theory.inhabited-cardinals ## Idea -We may say a [cardinal of sets](set-theory.cardinals.md) `X` is -{{#concept "indexed less than" Disambiguation="cardinal of sets" Agda=le-indexed-Cardinal}} +We may say a [cardinal](set-theory.cardinals.md) `X` is +{{#concept "indexed less than" Disambiguation="set-cardinal" Agda=le-indexed-Cardinal}} a cardinal `Y` if `Y` is inhabited and any map `f` of -[sets](foundation-core.sets.md) from the isomorphism class represented by `X` -into sets in the isomorphism class represented by `Y` is +[sets](foundation-core.sets.md) from the isomorphism class of `X` into sets in +the isomorphism class of `Y` is [nonsurjective](foundation.nonsurjective-maps.md) in the sense that there exists an element in `Y` that `f` does not hit. This is a positive way of saying that `X` is less than `Y`. This defines the -{{#concept "strict indexing ordering" Disambiguation="on cardinalities of sets"}} -on cardinalities of sets. +{{#concept "strict indexing ordering" Disambiguation="on set-cardinals"}}. ## Definition