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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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.CanonicalOmegaSyntax
import OAI.Computability.DegreeRigidity.Representation.AmbientDegreeSyntax
import OAI.Computability.DegreeRigidity.Constructibility.ConstructibleGroundReals
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel
universe u
noncomputable def canonicalReals (r : ℕ) : SentenceForm :=
.ex (.conj (canonicalOmega 0) (FullSetForcing.ownPower 0 (r+1)))
theorem canonicalReals_spec (M : ZFSet.{u}) (hM : Transitive M)
(hw : ZFSet.omega.{u} ∈ M)
(r : ℕ) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) :
(canonicalReals r).Sat (M : Set ZFSet) e ↔ e r = groundReals M := by
have hc (w : ZFSet.{u}) (hwM : w ∈ M) : ∀ i, cons w e i ∈ M := by
intro i; cases i; exact hwM; exact he _
have hpower : (FullSetForcing.ownPower 0 (r+1)).Sat (M : Set ZFSet) (cons ZFSet.omega e) ↔
e r = groundReals M := by
rw [FullSetForcing.ownPower_spec M hM 0 (r+1) _ (hc _ hw)]
simp only [cons_succ,cons_zero]
constructor
· intro h
exact ZFSet.ext fun x => (h x).trans (mem_groundReals M x).symm
· intro h x
rw [h,mem_groundReals]
change (∃ w ∈ M, (canonicalOmega 0).Sat (M : Set ZFSet) (cons w e) ∧
(FullSetForcing.ownPower 0 (r+1)).Sat (M : Set ZFSet) (cons w e)) ↔ _
constructor
· rintro ⟨w,hwM,hwS,hp⟩
obtain rfl := (canonicalOmega_spec M hM hw 0 _ (hc w hwM)).mp hwS
exact hpower.mp hp
· intro h
exact ⟨_,hw,(canonicalOmega_spec M hM hw 0 _ (hc _ hw)).mpr rfl,hpower.mpr h⟩
theorem groundReals_between (M N : ZFSet.{u}) (hN : Transitive N)
(hNM : N ⊆ M) (hR : groundReals M ∈ N) : groundReals N = groundReals M := by
apply ZFSet.ext; intro x
rw [mem_groundReals,mem_groundReals]
exact ⟨fun hx => ⟨hNM hx.1,hx.2⟩,
fun hx => ⟨hN _ hR x ((mem_groundReals M x).mpr hx),hx.2⟩⟩
end TuringRigidity.RelativeConstructible
end OAI
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