File size: 3,383 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 | import OAI.Computability.DegreeRigidity.Constructibility.PersistentOneRealDefinition
import OAI.Computability.DegreeRigidity.Syntax.SetSatisfactionBounded
import OAI.Computability.DegreeRigidity.Constructibility.PersistentExtensionName
namespace OAI
namespace TuringRigidity.FullSetForcing
open TransitiveNameModel BoundedSetTheory ElementaryModel SetDegreeDecoding
open PersistentRestrictions RecursiveNames CountableForcing AtomicForcing RelativeConstructible
universe u
theorem persistent_extension_one_real_names
(M : ZFSet.{u}) [Countable (Conditions M)] (hM : Transitive M) (hT : SourceT M)
{c o : ZFSet.{u}} [Preorder (Conditions c)] [Top (Conditions c)]
(hc : c ∈ M) (hoM : o ∈ M)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(G : GenericFilter (Conditions c)) (hG : GroundGeneric M G) (htop : ⊤ ∈ G.carrier)
(I : CountableIdeal) (ρ : I ≃o I) (hρ : Persistent I ρ)
(hz : degree (OracleJump.jump FixedArithmetic.zero) ∈ I.carrier)
(hIM : I.carrier ⊆ (modelIdeal M hM hT).carrier) :
let E := genericExtensionSet M c G.carrier
let hE := genericExtensionSet_transitive M c hM G.carrier
let hTE := extension_sourceT M hM hT hc hoM ho G hG htop
let J := modelIdeal E hE hTE
∃ hIJ : I.carrier ⊆ J.carrier, ∃ σ : J ≃o J,
Extends hIJ ρ σ ∧ Persistent J σ ∧
∃ (δ : Ordinal.{u}) (P : Oracle) (p : SentenceForm),
δ.toZFSet ∈ E ∧ P ∈ modelReals E ∧ p.bound ≤ 2 ∧
realCode P ∈ level (groundReals E) δ ∧
automorphismSet σ = definedSubset (level (groundReals E) δ) p
(fun _ : Fin p.bound => realCode P) ∧
∃ r d a l f : Name (Conditions c),
r.encode (label c) ∈ M ∧ d.encode (label c) ∈ M ∧
a.encode (label c) ∈ M ∧ l.encode (label c) ∈ M ∧ f.encode (label c) ∈ M ∧
r.val G.carrier = groundReals E ∧ d.val G.carrier = δ.toZFSet ∧
a.val G.carrier = realCode P ∧ l.val G.carrier = level (groundReals E) δ ∧
f.val G.carrier = automorphismSet σ := by
intro E hE hTE J
have hME := ground_inclusion_set M c hM hT.pairing hT.union hT.powerSet
hT.separation.finitePrefix.bounded hT.replacement.finitePrefix hT.infinity hc G.carrier htop
have hIJ : I.carrier ⊆ J.carrier := by
intro d hd
obtain ⟨A,hA,rfl⟩ := hIM hd
refine ⟨A,?_,rfl⟩
change realCode A ∈ E
exact hME hA
obtain ⟨σ,he,hσ,δ,P,p,hδ,hP,hp,hPL,hdef⟩ :=
persistent_extension_one_real_definition E hE hTE I ρ hρ hz hIJ
have hrE := groundReals_mem E hE hTE
have hlE := level_mem E _ hE hTE hrE δ hδ
have hfE : automorphismSet σ ∈ E := by
rw [hdef]
exact definedSubset_mem E _ hE hTE.separation.finitePrefix.bounded hlE p
(fun _ => realCode P) (fun _ => hPL) (by omega)
obtain ⟨r,hr,hvr⟩ := (mem_extensionSet M c G.carrier _).mp hrE
obtain ⟨d,hd,hvd⟩ := (mem_extensionSet M c G.carrier _).mp hδ
obtain ⟨a,ha,hva⟩ := (mem_extensionSet M c G.carrier (realCode P)).mp hP
obtain ⟨l,hl,hvl⟩ := (mem_extensionSet M c G.carrier _).mp hlE
obtain ⟨f,hf,hvf⟩ := (mem_extensionSet M c G.carrier _).mp hfE
exact ⟨hIJ,σ,he,hσ,δ,P,p,hδ,hP,hp,hPL,hdef,r,d,a,l,f,
hr,hd,ha,hl,hf,hvr,hvd,hva,hvl,hvf⟩
end TuringRigidity.FullSetForcing
end OAI
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