File size: 4,438 Bytes
9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import OAI.Computability.DegreeRigidity.Representation.ParameterCoveringModels
namespace OAI
namespace TuringRigidity.ElementaryModel
open BoundedSetTheory TransitiveNameModel RelationCollapse SentenceForm
universe u
theorem hull_real_fixed (a : ℕ → ZFSet.{u})
(hn : ∀ n, natSet.{u} n ∈ hullSet a) {A : Oracle} (hA : realCode A ∈ hullSet a) :
structureMap (hullSet a) (realCode A) = realCode A := by
apply collapse_fixed _ _ hA
intro x hx
obtain ⟨n,rfl⟩ := (mem_omega x).mp (realCode_subset A hx)
refine ⟨hn n,structureMap_fixed omega_transitive ?_ _ ((mem_omega _).mpr ⟨n,rfl⟩)⟩
intro y hy
obtain ⟨m,rfl⟩ := (mem_omega y).mp hy
exact hn m
theorem hull_omega_fixed (a : ℕ → ZFSet.{u})
(hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a) :
structureMap (hullSet a) ZFSet.omega = ZFSet.omega := by
apply collapse_fixed _ _ hω
intro x hx
obtain ⟨n,rfl⟩ := (mem_omega x).mp hx
refine ⟨hn n,structureMap_fixed omega_transitive ?_ _ ((mem_omega _).mpr ⟨n,rfl⟩)⟩
intro y hy
obtain ⟨m,rfl⟩ := (mem_omega y).mp hy
exact hn m
theorem collapsed_hull_reals (a : ℕ → ZFSet.{u})
(hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a) (A : Oracle) :
A ∈ modelReals (collapsed (hullSet a)) ↔ realCode A ∈ hullSet a := by
constructor
· intro hA
obtain ⟨x,hx,heq⟩ := (mem_collapsed (hullSet a) (realCode A)).mp hA
let e := cons x (fun _ => ZFSet.omega)
have hen : ∀ i, e i ∈ hullSet a := by intro i; cases i <;> assumption
let p := all (imp (.member 0 1) (.member 0 2))
have hs : p.Sat (collapsed (hullSet a) : Set ZFSet) (fun i => structureMap (hullSet a) (e i)) := by
simp only [p,sat_all,sat_imp,Sat,cons_zero,cons_succ,e]
intro y _ hy
rw [←heq] at hy
rw [hull_omega_fixed a hn hω]
exact realCode_subset A hy
have ht := (collapsed_hull_elementary a p e hen).mp hs
simp only [p,sat_all,sat_imp,Sat,cons_zero,cons_succ,e,Set.mem_univ,forall_const] at ht
have hxreal : realCode (decodeReal x) = x := realCode_decodeReal ht
have hfix : structureMap (hullSet a) x = x := by
rw [←hxreal]
exact hull_real_fixed a hn (hxreal.symm ▸ hx)
exact heq.symm ▸ hfix.symm ▸ hx
· intro hA
exact (mem_collapsed _ _).mpr ⟨realCode A,hA,(hull_real_fixed a hn hA).symm⟩
noncomputable def mixedSeed (a : ℕ → ZFSet.{u}) (A : ℕ → Oracle) (n : ℕ) : ZFSet.{u} :=
match n.unpair.1 with
| 0 => natSet n.unpair.2
| 1 => realCode (A n.unpair.2)
| 2 => a n.unpair.2
| _ => ZFSet.omega
theorem mixed_naturals (a : ℕ → ZFSet.{u}) (A : ℕ → Oracle) (n : ℕ) :
natSet.{u} n ∈ hullSet (mixedSeed a A) := by
apply (mem_hullSet _ _).mpr
simpa [mixedSeed] using parameter_mem (mixedSeed a A) (Nat.pair 0 n)
theorem mixed_omega (a : ℕ → ZFSet.{u}) (A : ℕ → Oracle) :
ZFSet.omega.{u} ∈ hullSet (mixedSeed a A) := by
apply (mem_hullSet _ _).mpr
simpa [mixedSeed] using parameter_mem (mixedSeed a A) (Nat.pair 3 0)
theorem parameter_models_with_elementarity (a : ℕ → ZFSet.{u}) (A : ℕ → Oracle) :
∃ d M : ZFSet.{u}, Transitive M ∧ SourceT M ∧ Countable (Conditions M) ∧
StructureExtensional d ∧ (∀ n, a n ∈ d) ∧
(∀ n, realCode (A n) ∈ d ∧ structureMap d (realCode (A n)) = realCode (A n)) ∧
(∀ n, structureMap d (a n) ∈ M) ∧
(∀ B : Oracle, B ∈ modelReals M ↔ realCode B ∈ d) ∧
(∀ p : SentenceForm, ∀ e : ℕ → ZFSet.{u}, (∀ i, e i ∈ d) →
(p.Sat (M : Set ZFSet) (fun i => structureMap d (e i)) ↔ p.Sat Set.univ e)) := by
let b := mixedSeed a A
have hn := mixed_naturals a A
have hω := mixed_omega a A
have ha (n : ℕ) : a n ∈ hullSet b := by
apply (mem_hullSet _ _).mpr
simpa [b,mixedSeed] using parameter_mem b (Nat.pair 2 n)
have hA (n : ℕ) : realCode (A n) ∈ hullSet b := by
apply (mem_hullSet _ _).mpr
simpa [b,mixedSeed] using parameter_mem b (Nat.pair 1 n)
refine ⟨hullSet b,collapsed (hullSet b),collapsed_transitive _,collapsed_hull_sourceT _,
inferInstance,hullSet_extensional _,ha,(fun n => ⟨hA n,hull_real_fixed b hn (hA n)⟩),
(fun n => (mem_collapsed _ _).mpr ⟨a n,ha n,rfl⟩),collapsed_hull_reals b hn hω,?_⟩
exact collapsed_hull_elementary b
end TuringRigidity.ElementaryModel
end OAI
|