File size: 2,961 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 63 64 65 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundNaturalNumbers
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem countablePartialFunctions_mem
(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)
(infinite : Realize M.sets infinity Fin.elim0)
{X Y : ZFSet.{u}} (hX : X ∈ M.sets) (hY : Y ∈ M.sets) :
M.smallPartialFunctions X Y ZFSet.omega ∈ M.sets :=
M.smallPartialFunctions_mem pairing unions powers separation hX hY
(M.omega_mem separation infinite)
theorem countablePartialFunction_countable {X Y p : ZFSet.{u}}
(hp : p ∈ M.smallPartialFunctions X Y ZFSet.omega) : Countable (SetMembers p) :=
M.smallPartialFunction_countable hp
theorem countableFunctionOrder_mem
(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)
(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
(SetMembers.presentation (M.smallPartialFunctions X Y ZFSet.omega)).conditionOrder ∈ M.sets :=
M.smallFunctionOrder_mem pairing unions powers separation hX hY
(M.omega_mem separation infinite)
theorem countablePartialFunctions_restrict (separation : SeparationSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
{X Y p q : ZFSet.{u}} (hp : p ∈ M.smallPartialFunctions X Y ZFSet.omega)
(hq : q ∈ M.sets) (subset : q ⊆ p) : q ∈ M.smallPartialFunctions X Y ZFSet.omega :=
M.smallPartialFunctions_restrict separation hp hq (M.omega_mem separation infinite) subset
theorem exists_countableFunction_generic
(small : M.sets.Countable)
(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)
(infinite : Realize M.sets infinity Fin.elim0)
{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
∃ H : ConditionFilter (SetMembers (M.smallPartialFunctions X Y ZFSet.omega)), H.mem p ∧
H.GenericFor (M.codedFamily (SetMembers.presentation (M.smallPartialFunctions X Y ZFSet.omega))) ∧
H.GenericFor (formulaFamily (M.nameDomain
(SetMembers.presentation (M.smallPartialFunctions X Y ZFSet.omega)))) :=
M.exists_smallFunction_generic small pairing unions powers separation hX hY
(M.omega_mem separation infinite) p
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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