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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 55 56 57 58 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundOrdinarySubmodelReflection
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetFormula SetRealization
universe u
variable (M : TransitiveGround.{u})
variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0)
variable (unions : Realize M.sets SetFormula.union Fin.elim0)
variable (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
variable (collection : CollectionSchema M.sets) (infinite : Realize M.sets infinity Fin.elim0)
variable (choice : Realize M.sets SetFormula.choice Fin.elim0) (small : M.sets.Countable)
include pairing unions powers separation collection infinite choice small in
theorem countableFunction_sequenceReflection
{X Y U : ZFSet.{u}} (hX : X ∈ M.sets) (hY : Y ∈ M.sets) (hU : U ∈ M.sets) :
letI := M.smallFunctionOrder X Y ZFSet.omega
let s := SetMembers.presentation (M.smallPartialFunctions X Y ZFSet.omega)
∀ H : ConditionFilter (SetMembers (M.smallPartialFunctions X Y ZFSet.omega)),
H.GenericFor (M.codedFamily s) → H.GenericFor (formulaFamily (M.nameDomain s)) →
GroundSequenceReflection M (M.ordinaryExtension s H) U := by
let := M.smallFunctionOrder X Y ZFSet.omega
let := M.smallFunctionTop X Y ZFSet.omega ⟨X, hX⟩
let s := SetMembers.presentation (M.smallPartialFunctions X Y ZFSet.omega)
dsimp only
intro H genericCoded genericFormula
exact M.ordinaryExtension_sequenceReflection s pairing unions powers separation collection infinite
choice (M.countablePartialFunctions_mem pairing unions powers separation infinite hX hY)
(M.countableFunctionOrder_mem pairing unions powers separation infinite hX hY) small hU
(M.countable_function_internally_closed pairing unions powers separation infinite choice hX hY)
H genericCoded genericFormula
include pairing unions powers separation collection infinite choice small in
theorem exists_countableFunction_generic_sequenceReflection
{X Y : ZFSet.{u}} (hX : X ∈ M.sets) (hY : Y ∈ M.sets)
(p : SetMembers (M.smallPartialFunctions X Y ZFSet.omega)) :
letI := M.smallFunctionOrder X Y ZFSet.omega
let s := SetMembers.presentation (M.smallPartialFunctions X Y ZFSet.omega)
∃ H : ConditionFilter (SetMembers (M.smallPartialFunctions X Y ZFSet.omega)),
H.mem p ∧ H.GenericFor (M.codedFamily s) ∧
H.GenericFor (formulaFamily (M.nameDomain s)) ∧
∀ U ∈ M.sets, GroundSequenceReflection M (M.ordinaryExtension s H) U := by
let := M.smallFunctionOrder X Y ZFSet.omega
obtain ⟨H, hp, genericCoded, genericFormula⟩ :=
M.exists_countableFunction_generic small pairing unions powers separation infinite hX hY p
exact ⟨H, hp, genericCoded, genericFormula, fun _ hU =>
M.countableFunction_sequenceReflection pairing unions powers separation collection infinite choice
small hX hY hU H genericCoded genericFormula⟩
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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