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 postulate -- mimimul and x∋minimal is an Axiom of choice minimal : (x : OD ) → ¬ (x == od∅ )→ OD -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) ) -- minimality (may proved by ε-induction ) minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) -- -- Axiom of choice in intutionistic logic implies the exclude middle -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog -- ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p ppp {p} {a} d = d p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice p∨¬p p with is-o∅ ( od→ord ( record { def = λ x → p } )) p∨¬p p | yes eq = case2 (¬p eq) where ps = record { def = λ x → p } lemma : ps == od∅ → p → ⊥ lemma eq p0 = ¬x<0 {od→ord ps} (eq→ eq p0 ) ¬p : (od→ord ps ≡ o∅) → p → ⊥ ¬p eq = lemma ( subst₂ (λ j k → j == k ) oiso o∅≡od∅ ( o≡→== eq )) p∨¬p p | no ¬p = case1 (ppp {p} {minimal ps (λ eq → ¬p (eqo∅ eq))} lemma) where ps = record { def = λ x → p } eqo∅ : ps == od∅ → od→ord ps ≡ o∅ eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq)) lemma : ps ∋ minimal ps (λ eq → ¬p (eqo∅ eq)) lemma = x∋minimal ps (λ eq → ¬p (eqo∅ eq)) decp : ( p : Set n ) → Dec p -- assuming axiom of choice decp p with p∨¬p p decp p | case1 x = yes x decp p | case2 x = no x double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic double-neg-eilm {A} notnot with decp A -- assuming axiom of choice ... | yes p = p ... | no ¬p = ⊥-elim ( notnot ¬p ) OrdP : ( x : Ordinal ) ( y : OD ) → Dec ( Ord x ∋ y ) OrdP x y with trio< x (od→ord y) OrdP x y | tri< a ¬b ¬c = no ¬c OrdP x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) OrdP x y | tri> ¬a ¬b c = yes c open import zfc OD→ZFC : ZFC OD→ZFC = record { ZFSet = OD ; _∋_ = _∋_ ; _≈_ = _==_ ; ∅ = od∅ ; Select = Select ; isZFC = isZFC } where -- infixr 200 _∈_ -- infixr 230 _∩_ _∪_ isZFC : IsZFC (OD ) _∋_ _==_ 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 : OD ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD choice-func X {x} not X∋x = minimal x not choice : (X : OD ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A choice X {A} X∋A not = x∋minimal A not