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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 | import OAI.Computability.DegreeRigidity.SetModels.CollapseWithoutChoice
namespace OAI
namespace TuringRigidity.BoundedSetTheory.InternalCollapse
open TransitiveNameModel
universe u
theorem prefix_mem_without_choice (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hω : ZFSet.omega.{u} ∈ M)
{A p : ZFSet.{u}} (hA : A ∈ M) (hp : Prefix A p) : p ∈ M := by
classical
obtain ⟨n,hp⟩ := hp
have h0 : (∅ : ZFSet.{u}) ∈ M := hM _ hω _ ZFSet.omega_zero
cases n with
| zero =>
have eq : p = ∅ := by
apply ZFSet.ext; intro z
constructor
· intro hz
obtain ⟨i,hi,_,_,_⟩ := hp.1 z hz
exact False.elim (ZFSet.notMem_empty i hi)
· exact fun hz => False.elim (ZFSet.notMem_empty z hz)
exact eq ▸ h0
| succ n =>
obtain ⟨v,hv,hva⟩ := InternalWellOrder.FunctionGraph.exists_presentation hp
let f : ℕ → ZFSet.{u} := fun i => if i ≤ n then v (natSet i) else ∅
have hf : ∀ i, f i ∈ M := by
intro i
dsimp only [f]
split_ifs with hi
· exact hM A hA _ (hva _ ((natSet_mem_natSet i (n+1)).mpr (by omega)))
· exact h0
have eq : p = finiteModelGraph f n := by
apply ZFSet.ext; intro z
rw [mem_finiteModelGraph]
constructor
· intro hz
obtain ⟨j,hj,y,_,rfl⟩ := hp.1 z hz
obtain ⟨i,hi,rfl⟩ := (mem_natSet (n+1) j).mp hj
have hin : i ≤ n := by omega
have hy := ((hv.pair_iff _ _).mp hz).2
exact ⟨i,hin,by rw [hy]; simp only [f,ite_eq_left hin]⟩
· rintro ⟨i,hi,rfl⟩
apply (hv.pair_iff _ _).mpr
exact ⟨(natSet_mem_natSet i (n+1)).mpr (by omega),by simp only [f,ite_eq_left hi]⟩
rw [eq]
exact finiteModelGraph_mem M hM hP hU hω f hf n
theorem conditions_exist_absolute_without_choice (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : Separation M) (hω : ZFSet.omega.{u} ∈ M) {A : ZFSet.{u}} (hA : A ∈ M) :
∃ c ∈ M, ∀ p, p ∈ c ↔ Prefix A p := by
obtain ⟨c,hc,hcdef⟩ := conditions_exist_without_choice M hM hP hU hPow hS hω hA
exact ⟨c,hc,fun p => (hcdef p).trans
⟨And.right,fun hp => ⟨prefix_mem_without_choice M hM hP hU hω hA hp,hp⟩⟩⟩
end TuringRigidity.BoundedSetTheory.InternalCollapse
end OAI
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