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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 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 | import OAI.Computability.DegreeRigidity.SetModels.SetModelArithmeticDischarge
import OAI.Computability.DegreeRigidity.SetModels.CountableSetReal
namespace OAI
namespace TuringRigidity.SetModelCountability
open BoundedSetTheory TransitiveNameModel SetModelReals SetModelFunctions
open SetModelSequences SetDegreeDecoding PersistentRestrictions PersistentPresentation
open EncodedForcing SetModelArithmetic
universe u
theorem counted_reals_columns {M R : ZFSet.{u}} (C : Context M) (hR : R ∈ M)
(hreal : ∀ x ∈ R, ∃ B : Oracle, realSet B = x)
(hcount : BoundedSetTheory.InternallyCountable M R) :
∃ A ∈ reals M, ∀ B : Oracle, (∃ n, columns A n = B) ↔ realSet B ∈ R := by
classical
obtain ⟨q,hq,hqf,hqo⟩ := hcount
have hex (n : ℕ) : ∃ B : Oracle, realSet B ∈ R ∧
ZFSet.pair (natSet n) (realSet B) ∈ q := by
obtain ⟨x,hx,hnx,_⟩ := hqf.2 (natSet n) ((mem_omega _).mpr ⟨n,rfl⟩)
obtain ⟨B,rfl⟩ := hreal x hx
exact ⟨B,hx,hnx⟩
choose B hB using hex
have hseq : q = sequenceSet B := by
apply ZFSet.ext; intro z
constructor
· intro hz
obtain ⟨x,hx,y,_,rfl⟩ := hqf.1 z hz
obtain ⟨n,rfl⟩ := (mem_omega x).mp hx
have hy := hqf.functional ((mem_omega _).mpr ⟨n,rfl⟩) hz (hB n).2
exact (mem_sequenceSet B _).mpr ⟨n,by rw [hy]⟩
· intro hz
obtain ⟨n,rfl⟩ := (mem_sequenceSet B z).mp hz
exact (hB n).2
have hBM (n : ℕ) : B n ∈ reals M := C.transitive R hR _ (hB n).1
obtain ⟨r,hr,hrc⟩ := internal_reals M C.transitive C.power C.separation C.infinity
let A : Oracle := fun v => B (Nat.unpair v).1 (Nat.unpair v).2
have hAM : A ∈ reals M := flatten_mem M C.transitive C.pairing C.union C.power
C.separation C.infinity hr hrc C.pairing_mem pairingSet_code B hBM (hseq ▸ hq)
have hcol (n : ℕ) : columns A n = B n := by
funext k; simp only [columns,A,Nat.unpair_pair]
refine ⟨A,hAM,fun X => ?_⟩
constructor
· rintro ⟨n,rfl⟩; rw [hcol]; exact (hB n).1
· intro hX
obtain ⟨n,hn,hnX⟩ := hqo _ hX
obtain ⟨n,rfl⟩ := (mem_omega n).mp hn
have he := hqf.functional ((mem_omega _).mpr ⟨n,rfl⟩) (hB n).2 hnX
refine ⟨n,?_⟩
rw [hcol]
exact realSet_injective he
theorem presentation_of_counted_reals {M R : ZFSet.{u}} (C : Context M) (hR : R ∈ M)
(hreal : ∀ x ∈ R, ∃ B : Oracle, realSet B = x)
(hcount : BoundedSetTheory.InternallyCountable M R) (I : CountableIdeal)
(hI : ∀ d, d ∈ I.carrier ↔ ∃ B : Oracle, realSet B ∈ R ∧ degree B = d) :
∃ A ∈ reals M, Presented I A := by
obtain ⟨A,hA,hcols⟩ := counted_reals_columns C hR hreal hcount
refine ⟨A,hA,fun d => ?_⟩
rw [hI]
constructor
· rintro ⟨B,hB,hBd⟩
obtain ⟨n,hn⟩ := (hcols B).mpr hB
exact ⟨n,by rw [hn,hBd]⟩
· rintro ⟨n,hn⟩
exact ⟨columns A n,(hcols _).mp ⟨n,rfl⟩,hn⟩
theorem persistent_mem_of_counted_reals {M R : ZFSet.{u}} (C : Context M) (hR : R ∈ M)
(hreal : ∀ x ∈ R, ∃ B : Oracle, realSet B = x)
(hcount : BoundedSetTheory.InternallyCountable M R) (I : CountableIdeal)
(hI : ∀ d, d ∈ I.carrier ↔ ∃ B : Oracle, realSet B ∈ R ∧ degree B = d)
(ρ : I ≃o I) (hρ : Persistent I ρ)
(hz : degree (OracleJump.jump FixedArithmetic.zero) ∈ I.carrier) :
automorphismSet ρ ∈ M := by
obtain ⟨A,hAM,hA⟩ := presentation_of_counted_reals C hR hreal hcount I hI
exact (SetModelSyntax.persistent_sets C hA hAM ρ hρ hz).2
end TuringRigidity.SetModelCountability
end OAI
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