File size: 3,309 Bytes
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 59 60 61 62 | import OAI.SetTheory.PartitionPrinciple.Models.GroundSupportedEqualityChoices
namespace OAI
namespace PartitionPilot.Forcing.TransitiveGround
open SetFormula SetRealization
universe u
variable (M : TransitiveGround.{u})
theorem countableFunction_ordinaryZF
(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) (infinite : Realize M.sets infinity Fin.elim0)
{X Y : ZFSet.{u}} (hX : X ∈ M.sets) (hY : Y ∈ 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 (formulaFamily (M.nameDomain s)) →
ZFRealization (M.ordinaryExtension s H).sets ∧
CollectionSchema (M.ordinaryExtension s H).sets := 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)
have carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite hX hY
have order := M.countableFunctionOrder_mem pairing unions powers separation infinite hX hY
dsimp only
intro H generic
constructor
· exact M.ordinaryExtension_ZF s pairing unions powers separation collection carrier order infinite H generic
· intro n
exact M.ordinaryExtension_collection s pairing unions powers separation collection carrier order H generic
theorem countableFunction_ordinaryZFC
(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) (infinite : Realize M.sets infinity Fin.elim0)
(choice : Realize M.sets SetFormula.choice Fin.elim0)
{X Y : ZFSet.{u}} (hX : X ∈ M.sets) (hY : Y ∈ 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 (formulaFamily (M.nameDomain s)) →
ZFRealization (M.ordinaryExtension s H).sets ∧
CollectionSchema (M.ordinaryExtension s H).sets ∧
Realize (M.ordinaryExtension s H).sets SetFormula.choice Fin.elim0 := 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)
have carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite hX hY
have order := M.countableFunctionOrder_mem pairing unions powers separation infinite hX hY
dsimp only
intro H generic
have axioms := M.countableFunction_ordinaryZF pairing unions powers separation collection infinite hX hY H generic
exact ⟨axioms.1, axioms.2,
M.ordinaryExtension_choice s pairing unions powers separation collection infinite choice carrier order H generic⟩
end PartitionPilot.Forcing.TransitiveGround
end OAI
|