| import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaModelCoreClosure |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.OmegaModelCore.OmegaData |
| open BoundedSetTheory |
| universe u |
| noncomputable section |
| variable {α : Type u} (S : OmegaData α) |
|
|
| def push {β : Type*} (x : β) (e : ℕ → β) : ℕ → β |
| | 0 => x |
| | n+1 => e n |
|
|
| def Eval (e : ℕ → α) : Formula → Prop |
| | .equal i j => e i = e j |
| | .member i j => S.mem (e i) (e j) |
| | .conj p q => Eval e p ∧ Eval e q |
| | .neg p => ¬ Eval e p |
| | .existsMem i p => ∃ x, S.mem x (e i) ∧ Eval (push x e) p |
|
|
| theorem collapse_eval (p : Formula) (e : ℕ → S.Core) : |
| p.Eval (fun i => S.collapse (e i)) ↔ S.Eval (fun i => (e i).val) p := by |
| induction p generalizing e with |
| | equal i j => |
| change S.collapse (e i) = S.collapse (e j) ↔ (e i).val = (e j).val |
| exact ⟨fun h => congrArg Subtype.val (S.collapse_injective h), |
| fun h => congrArg S.collapse (Subtype.ext h)⟩ |
| | member i j => exact S.collapse_mem_iff (e i) (e j) |
| | conj p q hp hq => exact and_congr (hp e) (hq e) |
| | neg p hp => exact not_congr (hp e) |
| | existsMem i p hp => |
| have hstep (y : S.Core) : p.Eval (cons (S.collapse y) (fun i => S.collapse (e i))) ↔ |
| S.Eval (push y.val (fun i => (e i).val)) p := by |
| have he : (fun i => S.collapse (push y e i)) = |
| cons (S.collapse y) (fun i => S.collapse (e i)) := by |
| funext i |
| cases i <;> rfl |
| have hv : (fun i => (push y e i).val) = push y.val (fun i => (e i).val) := by |
| funext i |
| cases i <;> rfl |
| simpa only [he,hv] using hp (push y e) |
| constructor |
| · rintro ⟨x,hx,hp'⟩ |
| obtain ⟨y,hy,rfl⟩ := (S.mem_collapse (e i) x).mp hx |
| exact ⟨y.val,hy,(hstep y).mp hp'⟩ |
| · rintro ⟨x,hx,hp'⟩ |
| let y : S.Core := ⟨x,S.core_transitive (e i).property hx⟩ |
| exact ⟨S.collapse y,(S.mem_collapse (e i) _).mpr ⟨y,hx,rfl⟩, |
| (hstep y).mpr hp'⟩ |
|
|
| def PairingAxiom : Prop := ∀ a b : α, ∃ c, ∀ x, S.mem x c ↔ x=a ∨ x=b |
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| def UnionAxiom : Prop := ∀ a : α, ∃ b, ∀ x, S.mem x b ↔ ∃ y, S.mem y a ∧ S.mem x y |
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| def SeparationAxiom : Prop := ∀ (p : Formula) (e : ℕ → α) (a : α), |
| ∃ b, ∀ x, S.mem x b ↔ S.mem x a ∧ S.Eval (push x e) p |
|
|
| end |
| end TuringRigidity.OmegaModelCore.OmegaData |
|
|
| end OAI |
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|
