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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 83 84 | import OAI.Computability.DegreeRigidity.SetModels.NameDifferenceDense
import OAI.Computability.DegreeRigidity.Representation.GroundGenericRequirements
namespace OAI
namespace TuringRigidity.FullSetForcing
open RecursiveNames TransitiveNameModel BoundedSetTheory AtomicForcing CountableForcing
universe u
variable {c : ZFSet.{u}} [Preorder (Conditions c)] [OrderTop (Conditions c)]
omit [Preorder (Conditions c)] [OrderTop (Conditions c)] in
theorem ground_name_sequence (M N : ZFSet.{u}) [Countable (Conditions N)] (hMN : M ⊆ N)
(t₀ : Name (Conditions c)) (ht₀ : t₀.encode (label c) ∈ M) :
∃ ts : ℕ → Name (Conditions c), (∀ n, (ts n).encode (label c) ∈ M) ∧
∀ t : Name (Conditions c), t.encode (label c) ∈ M →
∃ n, (ts n).encode (label c) = t.encode (label c) := by
classical
obtain ⟨m,_⟩ := label_surjective N (hMN ht₀)
let _ : Nonempty (Conditions N) := ⟨m⟩
obtain ⟨e,he⟩ := exists_surjective_nat (Conditions N)
have pick (n : ℕ) : ∃ s : Name (Conditions c), s.encode (label c) ∈ M ∧
∀ t : Name (Conditions c), t.encode (label c) ∈ M →
t.encode (label c) = label N (e n) → s.encode (label c) = t.encode (label c) := by
by_cases h : ∃ t : Name (Conditions c), t.encode (label c) ∈ M ∧
t.encode (label c) = label N (e n)
· obtain ⟨t,ht,heq⟩ := h
exact ⟨t,ht,fun s _ hs => heq.trans hs.symm⟩
· exact ⟨t₀,ht₀,fun t ht heq => False.elim (h ⟨t,ht,heq⟩)⟩
choose ts hts using pick
refine ⟨ts,fun n => (hts n).1,?_⟩
intro t ht
obtain ⟨m,hm⟩ := label_surjective N (hMN ht)
obtain ⟨n,rfl⟩ := he m
exact ⟨n,(hts n).2 t ht hm.symm⟩
theorem exists_generic_omitting_subset (M N : ZFSet.{u}) [Countable (Conditions N)]
(hM : Transitive M) (hT : SourceT M) (hMN : M ⊆ N)
(hc : c ∈ M) {o a S : ZFSet.{u}} (hoM : o ∈ M) (ha : a ∈ M)
(hSa : S ⊆ a) (hSM : S ∉ M)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(p : Conditions c) :
∃ G : GenericFilter (Conditions c), p ∈ G.carrier ∧ GroundGeneric N G ∧
S ∉ genericExtensionSet M c G.carrier := by
have ht₀ := encoded_check_mem M c hM hT.pairing hT.union hT.powerSet
hT.separation.finitePrefix.bounded hT.replacement.finitePrefix hT.infinity hc hc
obtain ⟨ts,hts,hcover⟩ := ground_name_sequence M N hMN (Name.check c) ht₀
let E : ℕ → Set (Conditions c) := fun n => NameDifference a S (ts n)
have hE : ∀ n, Dense (E n) := fun n =>
nameDifference_dense M hM hT hc hoM ha hSa hSM ho (ts n) (hts n)
obtain ⟨G,hp,hG,hGE⟩ := ground_generic_with_requirements N ⟨c,hMN hc⟩ E hE p
refine ⟨G,hp,hG,?_⟩
intro hSG
obtain ⟨t,ht,hval⟩ := (mem_extensionSet M c G.carrier S).mp hSG
obtain ⟨n,hn⟩ := hcover t ht
obtain ⟨q,hq,hqn⟩ := hGE n
have hGM : GroundGeneric M G := fun D hD hd => hG D (hMN hD) hd
have hne := nameDifference_ne_value M hM hT hc hoM ha ho (ts n) (hts n) G hGM hq hqn
apply hne
exact (Name.val_eq_of_encode_eq (label c) (label_injective c) G.carrier (ts n) t hn).trans hval
theorem subset_mem_iff_all_generic_extensions (M N : ZFSet.{u}) [Countable (Conditions N)]
(hM : Transitive M) (hT : SourceT M) (hMN : M ⊆ N)
(hc : c ∈ M) {o a S : ZFSet.{u}} (hoM : o ∈ M) (ha : a ∈ M) (hSa : S ⊆ a)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(p : Conditions c) :
S ∈ M ↔ ∀ G : GenericFilter (Conditions c), GroundGeneric N G → p ∈ G.carrier →
S ∈ genericExtensionSet M c G.carrier := by
classical
constructor
· intro hSM G _ hp
exact ground_inclusion_set M c hM hT.pairing hT.union hT.powerSet
hT.separation.finitePrefix.bounded hT.replacement.finitePrefix hT.infinity hc
G.carrier (G.upper le_top hp) hSM
· intro h
by_contra hSM
obtain ⟨G,hp,hG,hSG⟩ := exists_generic_omitting_subset M N hM hT hMN hc hoM ha hSa hSM ho p
exact hSG (h G hG hp)
end TuringRigidity.FullSetForcing
end OAI
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