File size: 2,751 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 | import OAI.Computability.DegreeRigidity.Constructibility.RelativePersistentExtension
import OAI.Computability.DegreeRigidity.Representation.CommonGenericModels
namespace OAI
namespace TuringRigidity.FullSetForcing
open TransitiveNameModel BoundedSetTheory ElementaryModel SetDegreeDecoding
open PersistentRestrictions RecursiveNames CountableForcing AtomicForcing
universe u
theorem persistent_extension_name (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 ∧
∃ t : Name (Conditions c), t.encode (label c) ∈ M ∧ t.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,hσE⟩ := RelativeConstructible.persistent_extension_in_model E hE hTE I ρ hρ hz hIJ
obtain ⟨t,ht,hval⟩ := (mem_extensionSet M c G.carrier _).mp hσE
exact ⟨hIJ,σ,he,hσ,t,ht,hval⟩
theorem model_automorphism_output_name (M c : ZFSet.{u}) [Countable (Conditions M)]
(G : Set (Conditions c))
(hE : Transitive (genericExtensionSet M c G)) (hTE : SourceT (genericExtensionSet M c G))
(σ : modelIdeal (genericExtensionSet M c G) hE hTE ≃o
modelIdeal (genericExtensionSet M c G) hE hTE)
(A : Oracle) (hA : A ∈ modelReals (genericExtensionSet M c G)) :
∃ B : Oracle, ∃ t : Name (Conditions c),
t.encode (label c) ∈ M ∧ t.val G = realCode B ∧
degree B = (σ ⟨degree A,⟨A,hA,rfl⟩⟩).val := by
obtain ⟨B,hB,hdegree⟩ := (σ ⟨degree A,⟨A,hA,rfl⟩⟩).property
obtain ⟨t,ht,hval⟩ := (mem_extensionSet M c G (realCode B)).mp hB
exact ⟨B,t,ht,hval,hdegree⟩
end TuringRigidity.FullSetForcing
end OAI
|