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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 | import OAI.Computability.DegreeRigidity.Representation.PrescribedArithmeticGenericProgram
import OAI.Computability.DegreeRigidity.Representation.PrescribedFirstUncountable
import OAI.Computability.DegreeRigidity.CohenForcing.CohenGroundGeneric
namespace OAI
namespace TuringRigidity.RelativeConstructible
open TransitiveNameModel BoundedSetTheory CountableForcing ElementaryModel PersistentRestrictions
open GenericIdentity OracleJump CohenColumnRealName
attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder
theorem global_arithmetic_generic_program (π : Degree ≃o Degree) :
∃ (R : Oracle) (p : OracleCode),
degree R = π.symm (degree (jump FixedArithmetic.zero)) ∧
∀ Y : Oracle, ArithmeticPrefixForcing.Generic (fun _ => iterate R 5) Y →
RepresentsAt π p (iterate R 5) Y := by
classical
by_contra hnone
push Not at hnone
let d := π.symm (degree (jump FixedArithmetic.zero))
let C := {R : Oracle // degree R = d} × OracleCode
let : Countable {R : Oracle // degree R = d} := (countable_degree_fiber d).to_subtype
obtain ⟨R₀,hR₀⟩ := degree_surjective d
let : Nonempty C := ⟨(⟨R₀,hR₀⟩,OracleCode.zero)⟩
have hw : ∀ c : C, ∃ Y : Oracle,
ArithmeticPrefixForcing.Generic (fun _ => iterate c.1.val 5) Y ∧
¬ RepresentsAt π c.2 (iterate c.1.val 5) Y := by
intro c
exact hnone c.1.val c.2 c.1.property
choose W hWg hWbad using hw
obtain ⟨e,he⟩ := exists_surjective_nat C
obtain ⟨M,hM,hT,hcount,_,⟨K,hK⟩,hWM,himages,ρ,hρ,_,_,_⟩ :=
PrescribedFirstUncountable.prescribed_covering_model_with_omega_one π (fun n => W (e n))
let := hcount
have hA := product_mem M hM hT.pairing hT.union hT.powerSet hT.separation.finitePrefix.bounded
hK.2.1 (sourceT_omega_mem M hM hT)
obtain ⟨G,_,hG,hME,_,_,_⟩ := CohenGroundGeneric.exists_extension M (ZFSet.prod K ZFSet.omega)
hM hT hA ⊤
obtain ⟨R,p,_,hR,_,_,hrep,_⟩ :=
prescribed_arithmetic_generic_program π M K hM hT hK himages ρ hρ G hG
let c : C := (⟨R,hR⟩,p)
obtain ⟨n,hn⟩ := he c
have hW : W c ∈ modelReals M := by simpa only [hn] using hWM n
exact hWbad c (hrep (W c) (hME hW) (hWg c))
end TuringRigidity.RelativeConstructible
end OAI
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