open import Level open import Ordinals module ODC {n : Level } (O : Ordinals {n} ) where open import zf open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) open import Relation.Binary.PropositionalEquality open import Data.Nat.Properties open import Data.Empty open import Relation.Nullary open import Relation.Binary open import Relation.Binary.Core open import logic open import nat import OD open inOrdinal O open OD O open OD.OD open OD._==_ open ODAxiom odAxiom open HOD open _∧_ _=h=_ : (x y : HOD) → Set n x =h= y = od x == od y postulate -- mimimul and x∋minimal is an Axiom of choice minimal : (x : HOD ) → ¬ (x =h= od∅ )→ HOD -- this should be ¬ (x =h= od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) x∋minimal : (x : HOD ) → ( ne : ¬ (x =h= od∅ ) ) → odef x ( od→ord ( minimal x ne ) ) -- minimality (may proved by ε-induction with LEM) minimal-1 : (x : HOD ) → ( ne : ¬ (x =h= od∅ ) ) → (y : HOD ) → ¬ ( odef (minimal x ne) (od→ord y)) ∧ (odef x (od→ord y) ) -- -- Axiom of choice in intutionistic logic implies the exclude middle -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog -- pred-od : ( p : Set n ) → HOD pred-od p = record { od = record { def = λ x → (x ≡ o∅) ∧ p } ; odmax = osuc o∅; ¬a ¬b c = yes c open import zfc HOD→ZFC : ZFC HOD→ZFC = record { ZFSet = HOD ; _∋_ = _∋_ ; _≈_ = _=h=_ ; ∅ = od∅ ; Select = Select ; isZFC = isZFC } where -- infixr 200 _∈_ -- infixr 230 _∩_ _∪_ isZFC : IsZFC (HOD ) _∋_ _=h=_ od∅ Select isZFC = record { choice-func = choice-func ; choice = choice } where -- Axiom of choice ( is equivalent to the existence of minimal in our case ) -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] choice-func : (X : HOD ) → {x : HOD } → ¬ ( x =h= od∅ ) → ( X ∋ x ) → HOD choice-func X {x} not X∋x = minimal x not choice : (X : HOD ) → {A : HOD } → ( X∋A : X ∋ A ) → (not : ¬ ( A =h= od∅ )) → A ∋ choice-func X not X∋A choice X {A} X∋A not = x∋minimal A not