Cardinalities of sets

Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, Elif Uskuplu, Julian KG, Victor Blanchi, fernabnor and louismntnu.

Created on 2022-05-25.
Last modified on 2023-11-24.

module set-theory.cardinalities where
Imports 
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

Idea

The cardinality of a set is its isomorphism class. We take isomorphism classes of sets by set truncating the universe of sets of any given universe level. Note that this definition takes advantage of the univalence axiom: By the univalence axiom isomorphic sets are equal, and will be mapped to the same element in the set truncation of the universe of all sets.

Definition

Cardinalities

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)

cardinality : {l : Level} → Set l → cardinal l
cardinality A = unit-trunc-Set A

Inequality of cardinalities

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 a<b b<c →
        unit-leq-cardinality
          ( a)
          ( c)
          ( transitive-mere-emb
            ( inv-unit-leq-cardinality b c b<c)
            ( inv-unit-leq-cardinality a b a<b)))
      ( Z))
    ( Y))
  ( X)

Properties

For sets, the type # X ＝ # Y is equivalent to the type of mere equivalences from X to Y

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)

Assuming excluded middle we can show that leq-cardinality is a partial order

Using the previous result and assuming excluded middle, we can conclude leq-cardinality is a partial order by showing that it is antisymmetric.

antisymmetric-leq-cardinality :
  {l1 : Level} (X Y : cardinal l1) → (LEM l1) →
  leq-cardinality X Y → leq-cardinality Y X → X ＝ Y
antisymmetric-leq-cardinality {l1} X Y lem =
  apply-dependent-universal-property-trunc-Set'
  ( λ u →
    set-Prop
      ( function-Prop
        ( leq-cardinality u Y)
        ( function-Prop
          ( leq-cardinality Y u)
          ( Id-Prop (cardinal-Set l1) u Y))))
  ( λ a →
    apply-dependent-universal-property-trunc-Set'
    ( λ v →
      set-Prop
        ( function-Prop
          ( leq-cardinality (cardinality a) v)
          ( function-Prop
            ( leq-cardinality v (cardinality a))
            ( Id-Prop (cardinal-Set l1) (cardinality a) v))))
    ( λ b a<b b<a →
      map-inv-equiv (is-effective-cardinality a b)
        (antisymmetric-mere-emb lem
        (inv-unit-leq-cardinality _ _ a<b)
        (inv-unit-leq-cardinality _ _ b<a)))
    ( Y))
  ( X)

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