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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 | import OAI.Computability.DegreeRigidity.SetModels.CollapseDenseWithoutChoice
import OAI.Computability.DegreeRigidity.SetModels.InternalCollapseCounting
namespace OAI
namespace TuringRigidity.BoundedSetTheory.InternalCollapse
open TransitiveNameModel CountableForcing
universe u
attribute [local instance] order collapsePreorder
theorem unionGraph_total_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M)
(hS : Separation M) (hω : ZFSet.omega.{u} ∈ M)
{A c : ZFSet.{u}} (hA : A ∈ M) (hcM : c ∈ M)
(hc : ∀ p, p ∈ c ↔ p ∈ M ∧ Prefix A p) (hne : ∃ x, x ∈ A)
(G : GenericFilter (Conditions c)) (hG : AtomicForcing.GroundGeneric M G) (n : ℕ) :
∃ x ∈ A, ZFSet.pair (natSet n) x ∈ unionGraph G := by
obtain ⟨p,hp,hpn⟩ := hG (definedAt c A n) (definedAt_mem_without_choice M hM hS hω hcM hA n)
(definedAt_dense_without_choice M hM hP hU hω hA hc hne n)
obtain ⟨_,x,hx,hnx⟩ := ZFSet.mem_sep.mp hpn
exact ⟨x,hx,(mem_unionGraph G _).mpr ⟨p,hp,hnx⟩⟩
theorem unionGraph_onto_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M)
(hS : Separation M) (hω : ZFSet.omega.{u} ∈ M)
{A c : ZFSet.{u}} (hA : A ∈ M) (hcM : c ∈ M)
(hc : ∀ p, p ∈ c ↔ p ∈ M ∧ Prefix A p)
(G : GenericFilter (Conditions c)) (hG : AtomicForcing.GroundGeneric M G) :
∀ x ∈ A, ∃ n ∈ ZFSet.omega, ZFSet.pair n x ∈ unionGraph G := by
intro x hx
obtain ⟨p,hp,hpx⟩ := hG (covers c x) (covers_mem_without_choice M hM hS hω hcM (hM _ hA _ hx))
(covers_dense_without_choice M hM hP hU hω hA hc hx)
obtain ⟨_,n,hn,hnx⟩ := ZFSet.mem_sep.mp hpx
exact ⟨n,hn,(mem_unionGraph G _).mpr ⟨p,hp,hnx⟩⟩
theorem unionGraph_function_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M)
(hS : Separation M) (hω : ZFSet.omega.{u} ∈ M)
{A c : ZFSet.{u}} (hA : A ∈ M) (hcM : c ∈ M)
(hc : ∀ p, p ∈ c ↔ p ∈ M ∧ Prefix A p) (hne : ∃ x, x ∈ A)
(G : GenericFilter (Conditions c)) (hG : AtomicForcing.GroundGeneric M G) :
TransitiveNameModel.FunctionGraph ZFSet.omega A (unionGraph G) := by
have hcP : ∀ p ∈ c, Prefix A p := fun p hp => ((hc p).mp hp).2
constructor
· intro z hz
obtain ⟨p,_,hzp⟩ := (mem_unionGraph G z).mp hz
exact ZFSet.mem_prod.mp (prefix_subset (hcP _ (label_mem c p)) hzp)
· intro i hi
obtain ⟨n,rfl⟩ := (mem_omega i).mp hi
obtain ⟨x,hx,hnx⟩ := unionGraph_total_without_choice M hM hP hU hS hω hA hcM hc hne G hG n
exact ⟨x,hx,hnx,fun y _ hny => unionGraph_functional hcP G hny hnx⟩
end TuringRigidity.BoundedSetTheory.InternalCollapse
end OAI
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