| import OAI.Computability.DegreeRigidity.OrdinalCodes.NumericHierarchy |
| import OAI.Computability.DegreeRigidity.SetModels.BindCanonicalArithmetic |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.NumericSyntax |
| open ElementaryModel BoundedSetTheory TransitiveNameModel BoundedDefinability RelativeConstructible |
| universe u |
|
|
| noncomputable def bindNumbers (p : SentenceForm) : SentenceForm := |
| .ex (.conj (canonicalOmega 0) |
| (.ex (.conj (fromBounded (squareFormula 1 0)) |
| (.ex (.conj (FullSetForcing.ownPower 1 0) |
| (.ex (.conj (fromBounded (.empty 0)) p))))))) |
|
|
| theorem bindNumbers_spec (N : ZFSet.{u}) (hN : Transitive N) |
| (hw : ZFSet.omega.{u} ∈ N) (hs : ZFSet.prod ZFSet.omega ZFSet.omega ∈ N) |
| (p : SentenceForm) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ N) : |
| (bindNumbers p).Sat (N : Set ZFSet) e ↔ |
| ∃ B ∈ N, PowerBound N (ZFSet.prod ZFSet.omega ZFSet.omega) B ∧ |
| p.Sat (N : Set ZFSet) (cons (natSet 0) (cons B |
| (cons (ZFSet.prod ZFSet.omega ZFSet.omega) (cons ZFSet.omega e)))) := by |
| have hc {a : ZFSet.{u}} {d : ℕ → ZFSet.{u}} (ha : a ∈ N) (hd : ∀ i, d i ∈ N) : |
| ∀ i, cons a d i ∈ N := by intro i; cases i; exact ha; exact hd _ |
| have hz : natSet.{u} 0 ∈ N := hN _ hw _ ((mem_omega _).mpr ⟨0,rfl⟩) |
| have hwf (w : ZFSet.{u}) (hwN : w ∈ N) := canonicalOmega_spec N hN hw 0 (cons w e) (hc hwN he) |
| have hsf (s : ZFSet.{u}) (hsN : s ∈ N) : |
| (fromBounded (squareFormula 1 0)).Sat (N : Set ZFSet) (cons s (cons ZFSet.omega e)) ↔ |
| s = ZFSet.prod ZFSet.omega ZFSet.omega := by |
| rw [bounded_sat,Formula.absolute _ N hN _ (hc hsN (hc hw he)),squareFormula_spec] |
| rfl |
| have hbf (B : ZFSet.{u}) (hBN : B ∈ N) : |
| (FullSetForcing.ownPower 1 0).Sat (N : Set ZFSet) |
| (cons B (cons (ZFSet.prod ZFSet.omega ZFSet.omega) (cons ZFSet.omega e))) ↔ |
| PowerBound N (ZFSet.prod ZFSet.omega ZFSet.omega) B := by |
| rw [FullSetForcing.ownPower_spec N hN _ _ _ (hc hBN (hc hs (hc hw he)))] |
| rfl |
| have hzf (B z : ZFSet.{u}) (hBN : B ∈ N) (hzN : z ∈ N) : |
| (fromBounded (.empty 0)).Sat (N : Set ZFSet) |
| (cons z (cons B (cons (ZFSet.prod ZFSet.omega ZFSet.omega) (cons ZFSet.omega e)))) ↔ |
| z = natSet 0 := by |
| rw [bounded_sat,Formula.absolute _ N hN _ (hc hzN (hc hBN (hc hs (hc hw he)))),Formula.eval_empty] |
| rfl |
| change (∃ w ∈ N, (canonicalOmega 0).Sat _ (cons w e) ∧ |
| ∃ s ∈ N, (fromBounded (squareFormula 1 0)).Sat _ (cons s (cons w e)) ∧ |
| ∃ B ∈ N, (FullSetForcing.ownPower 1 0).Sat _ (cons B (cons s (cons w e))) ∧ |
| ∃ z ∈ N, (fromBounded (.empty 0)).Sat _ (cons z (cons B (cons s (cons w e)))) ∧ |
| p.Sat _ (cons z (cons B (cons s (cons w e))))) ↔ _ |
| constructor |
| · rintro ⟨w,hwN,hwS,s,hsN,hsS,B,hBN,hBS,z,hzN,hzS,hp⟩ |
| obtain rfl := (hwf w hwN).mp hwS |
| obtain rfl := (hsf s hsN).mp hsS |
| obtain rfl := (hzf B z hBN hzN).mp hzS |
| exact ⟨B,hBN,(hbf B hBN).mp hBS,hp⟩ |
| · rintro ⟨B,hBN,hBS,hp⟩ |
| exact ⟨_,hw,(hwf _ hw).mpr rfl,_,hs,(hsf _ hs).mpr rfl,B,hBN,(hbf B hBN).mpr hBS, |
| _,hz,(hzf B _ hBN hz).mpr rfl,hp⟩ |
|
|
| theorem finiteGraph_subset (f : ℕ → ℕ) (k : ℕ) : |
| finiteNaturalGraph.{u} f k ⊆ ZFSet.prod ZFSet.omega ZFSet.omega := by |
| intro z hz |
| obtain ⟨i,_,rfl⟩ := (mem_finiteNaturalGraph f k z).mp hz |
| exact ZFSet.mem_prod.mpr ⟨_,(mem_omega _).mpr ⟨i,rfl⟩,_,(mem_omega _).mpr ⟨f i,rfl⟩,rfl⟩ |
|
|
| theorem covers_of_power {N B : ZFSet.{u}} |
| (hfin : ∀ f k, finiteNaturalGraph f k ∈ N) |
| (hB : PowerBound N (ZFSet.prod ZFSet.omega ZFSet.omega) B) : Covers B := |
| fun f k => (hB _).mpr ⟨hfin f k,finiteGraph_subset f k⟩ |
|
|
| end TuringRigidity.NumericSyntax |
|
|
| end OAI |
|
|
