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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 | import OAI.Computability.DegreeRigidity.SetModels.CollapseCountWithoutChoice
namespace OAI
namespace TuringRigidity.BoundedSetTheory.InternalCollapse
open TransitiveNameModel CountableForcing
universe u
attribute [local instance] order collapsePreorder
theorem unionGraph_mem_extension_without_choice (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : Separation M) (hR : SigmaReplacement M) (hI : Infinity M)
{c : ZFSet.{u}} (hcM : c ∈ M) (h0 : (∅ : ZFSet.{u}) ∈ c)
(G : GenericFilter (Conditions c)) :
unionGraph G ∈ genericExtensionSet M c G.carrier := by
let _ := top c h0
have htop : ⊤ ∈ G.carrier := by
obtain ⟨p,hp⟩ := G.nonempty
exact G.upper le_top hp
have hEU := extension_union M c (orderSet c) hM hP hU hPow hS hcM
(orderSet_mem_without_choice M hM hP hU hPow hS hcM) (orderSet_pair c) G
apply union_mem _ (genericExtensionSet_transitive M c hM G.carrier) hEU
exact genericFilterSet_mem_extension M c hM hP hU hPow hS hR hI hcM G.carrier htop
theorem internal_counting_without_choice (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : Separation M) (hR : SigmaReplacement M) (hI : Infinity 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) :
InternallyCountable (genericExtensionSet M c G.carrier) A := by
have hω := omega_mem M hM hS hI
have h0 : (∅ : ZFSet.{u}) ∈ c :=
(hc ∅).mpr ⟨hM _ hω _ ZFSet.omega_zero,prefix_empty A⟩
exact ⟨unionGraph G,unionGraph_mem_extension_without_choice M hM hP hU hPow hS hR hI hcM h0 G,
unionGraph_function_without_choice M hM hP hU hS hω hA hcM hc hne G hG,
unionGraph_onto_without_choice M hM hP hU hS hω hA hcM hc G hG⟩
end TuringRigidity.BoundedSetTheory.InternalCollapse
end OAI
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