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e3e0a0d | 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.SetTheory.PartitionPrinciple.Definability.CountableSetFormulas
namespace OAI
section
namespace PartitionPilot.Forcing.Name
open SetRealization
universe u
variable {P : Type u} [Preorder P] [Countable P] {s : Presentation P}
theorem atomicFamily_countable (D : NameDomain s) (small : D.names.Countable) :
(atomicFamily D).Countable := by
let : Countable D.names := small
let memSet (z : P × D.names × D.names) := Localized z.1 (MemWitness z.2.1.val z.2.2.val)
let eqSet (z : D.names × D.names) := EqDecisionWitness z.1.val z.2.val
apply ((Set.countable_range memSet).union (Set.countable_range eqSet)).mono
intro E hE
rcases hE with ⟨p, σ, τ, hσ, hτ, rfl⟩ | ⟨σ, τ, hσ, hτ, rfl⟩
· exact Or.inl ⟨(p, ⟨σ, hσ⟩, ⟨τ, hτ⟩), rfl⟩
· exact Or.inr ⟨(⟨σ, hσ⟩, ⟨τ, hτ⟩), rfl⟩
end PartitionPilot.Forcing.Name
namespace PartitionPilot.Forcing.SetFormula
open SetRealization
universe u
variable {P : Type u} [Preorder P] [Countable P] {s : Presentation P}
theorem formulaFamily_countable (D : Name.NameDomain s) (small : D.names.Countable) :
(formulaFamily D).Countable := by
let : Countable D.names := small
let decisions (z : Σ n, SetFormula n × Assignment D n) := DecisionWitness D z.2.1 z.2.2
let universals (z : Σ n, SetFormula (n + 1) × Assignment D n) := AllDecisionWitness D z.2.1 z.2.2
apply ((Name.atomicFamily_countable D small).union
((Set.countable_range decisions).union (Set.countable_range universals))).mono
intro E hE
rcases hE with atomic | (⟨n, φ, env, rfl⟩ | ⟨n, φ, env, rfl⟩)
· exact Or.inl atomic
· exact Or.inr (Or.inl ⟨⟨n, φ, env⟩, rfl⟩)
· exact Or.inr (Or.inr ⟨⟨n, φ, env⟩, rfl⟩)
theorem exists_generic_formulaFamily (D : Name.NameDomain s) (small : D.names.Countable) (p : P) :
∃ H : ConditionFilter P, H.mem p ∧ H.GenericFor (formulaFamily D) :=
exists_generic_for_countable_family p _ (formulaFamily_countable D small)
theorem exists_interpretation_truth (D : Name.NameDomain s) (small : D.names.Countable) (p : P) :
∃ H : ConditionFilter P, H.mem p ∧
∀ {n} (φ : SetFormula n) (env : Assignment D n),
Realize (D.values {q | H.mem q}) φ (evalAssignment {q | H.mem q} env) ↔
∃ q, H.mem q ∧ Forces D φ env q := by
obtain ⟨H, hp, generic⟩ := exists_generic_formulaFamily D small p
exact ⟨H, hp, fun φ env => forcing_truth D H generic φ env⟩
end PartitionPilot.Forcing.SetFormula
end
end OAI
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