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3277efa | 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.SetTheory.PartitionPrinciple.Transfinite.GroundOrdinalEnumerationFamilies
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem exists_choiceEnumerationTrace
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
(collection : CollectionSchema M.sets)
{A c T : ZFSet.{u}} (hA : A ∈ M.sets) (hc : c ∈ M.sets)
(hT : T ∈ M.sets) (ordinal : ordinalSetProperty T)
(code : ChoiceFunctionCode (M.nonemptyGroundSubsets A) c) :
∃ f ∈ M.sets,
OrdinalRecursionTrace (ChoiceEnumerationStep A (M.nonemptyGroundSubsets A) c) T f := by
let F := M.nonemptyGroundSubsets A
have hF : F ∈ M.sets := M.nonemptyGroundSubsets_mem powers separation hA
let params := extend A (extend F (extend c Fin.elim0))
have members : ∀ i, params i ∈ M.sets :=
Fin.cases hA (Fin.cases hF (Fin.cases hc (fun i => Fin.elim0 i)))
let formula : BoundedFormula 6 := BoundedFormula.choiceEnumerationStepFormula 3 4 5 0 1 2
have recognize (a p y : ZFSet.{u}) (ha : a ∈ M.sets) (hp : p ∈ M.sets)
(hy : y ∈ M.sets) :
Realize M.sets formula.toFormula (extend a (extend p (extend y params))) ↔
ChoiceEnumerationStep A F c a p y :=
(BoundedFormula.realize_toFormula M formula _
(Fin.cases ha (Fin.cases hp (Fin.cases hy members)))).trans
(BoundedFormula.truth_choiceEnumerationStepFormula 3 4 5 0 1 2 _)
have closure : GroundOrdinalStepClosure M T formula.toFormula params := by
intro a ha p hp _
have haM := M.transitive hT ha
obtain ⟨y, hy, step, unique⟩ :=
M.choiceEnumerationStep_exists_unique separation hA haM hp code
exact ⟨y, hy, (recognize a p y haM hp hy).2 step,
fun z hz relation => unique z ((recognize a p z haM hp hz).1 relation)⟩
obtain ⟨f, hf, trace⟩ := M.exists_ordinalRecursionTrace pairing unions separation collection
hT ordinal formula.toFormula params members closure
refine ⟨f, hf, trace.1, ?_⟩
intro a ha y entry
have step := trace.2 a ha y entry
exact (recognize a _ y (M.transitive hT ha) step.1 step.2.1).1 step.2.2
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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