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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 | import OAI.Computability.DegreeRigidity.SetModels.InternalAtomicTruth
namespace OAI
namespace TuringRigidity.AtomicForcing
open TransitiveNameModel BoundedSetTheory CountableForcing
universe u
theorem countable_ground_generic (M : ZFSet.{u}) [Countable (Conditions M)]
(hne : ∃ x, x ∈ M) {c : ZFSet.{u}} [Preorder (Conditions c)] (p : Conditions c) :
∃ G : GenericFilter (Conditions c), p ∈ G.carrier ∧ GroundGeneric M G := by
classical
obtain ⟨x,hx⟩ := hne
obtain ⟨m,_⟩ := label_surjective M hx
let _ : Nonempty (Conditions M) := ⟨m⟩
obtain ⟨e,he⟩ := exists_surjective_nat (Conditions M)
let D : ℕ → Set (Conditions c) := fun n =>
if Dense {q : Conditions c | label c q ∈ label M (e n)} then
{q | label c q ∈ label M (e n)} else Set.univ
have hD : ∀ n, Dense (D n) := by
intro n
dsimp only [D]
split
· assumption
· exact fun q => ⟨q,le_rfl,Set.mem_univ _⟩
obtain ⟨G,hp,hG⟩ := exists_generic D hD p
refine ⟨G,hp,?_⟩
intro E hEM hE
obtain ⟨m,hm⟩ := label_surjective M hEM
obtain ⟨n,rfl⟩ := he m
have hd : Dense {q : Conditions c | label c q ∈ label M (e n)} := by
rw [hm]; exact hE
obtain ⟨q,hq,hqn⟩ := hG n
refine ⟨q,hq,?_⟩
simpa only [D,ite_eq_left hd,Set.mem_ofPred_eq,hm] using hqn
end TuringRigidity.AtomicForcing
end OAI
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