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namespace OAI
section
namespace PartitionPilot.Forcing
universe u
theorem mem_two_iff (x : ZFSet.{u}) :
x ∈ natSetCode 2 ↔ x = natSetCode 0 ∨ x = natSetCode 1 := by
rw [mem_natSetCode]
constructor
· rintro ⟨n, hn, rfl⟩
have cases : n = 0 ∨ n = 1 := by omega
rcases cases with rfl | rfl
· exact Or.inl rfl
· exact Or.inr rfl
· rintro (rfl | rfl)
· exact ⟨0, by omega, rfl⟩
· exact ⟨1, by omega, rfl⟩
theorem two_distinct_values :
∃ a ∈ natSetCode 2, ∃ b ∈ natSetCode 2, a ≠ (b : ZFSet.{u}) := by
refine ⟨natSetCode 0, (mem_two_iff _).2 (Or.inl rfl),
natSetCode 1, (mem_two_iff _).2 (Or.inr rfl), ?_⟩
intro eq
have impossible := natSetCode_injective eq
omega
namespace TransitiveGround
open SetRealization SetFormula
variable (M : TransitiveGround.{u})
noncomputable def binaryGenericRow (I C : ZFSet.{u}) :
letI := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
ConditionFilter (SetMembers (M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)) →
ZFSet.{u} → ZFSet.{u} := by
letI := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
exact fun H i => ZFSet.sep (fun c => ZFSet.pair (ZFSet.pair i c) (natSetCode 1) ∈
M.countableGenericGraph (ZFSet.prod I C) (natSetCode 2) H) C
theorem mem_binaryGenericRow (I C : ZFSet.{u}) :
letI := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
∀ (H : ConditionFilter (SetMembers
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))) (i c : ZFSet.{u}),
c ∈ M.binaryGenericRow I C H i ↔ c ∈ C ∧
ZFSet.pair (ZFSet.pair i c) (natSetCode 1) ∈
M.countableGenericGraph (ZFSet.prod I C) (natSetCode 2) H := by
let := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
intro H i c
exact ZFSet.mem_sep
theorem binaryGenericRow_subset (I C : ZFSet.{u}) :
letI := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
∀ (H : ConditionFilter (SetMembers
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))) (i : ZFSet.{u}),
M.binaryGenericRow I C H i ⊆ C := by
let := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
intro H i c hc
exact ((M.mem_binaryGenericRow I C H i c).1 hc).1
theorem binaryGenericRow_ne
(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)
(choice : Realize M.sets SetFormula.choice Fin.elim0)
{I C i j : ZFSet.{u}} (hI : I ∈ M.sets) (hC : C ∈ M.sets)
(hi : i ∈ I) (hj : j ∈ I) (distinct : i ≠ j)
(uncountable : ¬ M.groundEmbeddable C ZFSet.omega) :
letI := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
∀ H : ConditionFilter (SetMembers
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)),
H.GenericFor (M.codedFamily (SetMembers.presentation
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))) →
M.binaryGenericRow I C H i ≠ M.binaryGenericRow I C H j := by
let := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
intro H generic same
obtain ⟨c, hc, a, ha, b, hb, ne, first, second, absent⟩ :=
M.countableGenericGraph_rows_separated pairing unions powers separation infinite choice
hI hC (M.natSetCode_mem separation infinite 2) hi hj distinct
two_distinct_values uncountable H generic
rcases (mem_two_iff a).1 ha with rfl | rfl
· rcases (mem_two_iff b).1 hb with rfl | rfl
· exact ne rfl
· have right := (M.mem_binaryGenericRow I C H j c).2 ⟨hc, second⟩
have left := (M.mem_binaryGenericRow I C H i c).1 (same ▸ right)
have impossible := natSetCode_injective
((M.countableGenericGraph_partial (ZFSet.prod I C) (natSetCode 2) H).unique
(ZFSet.pair i c) (natSetCode 1) (natSetCode 0) left.2 first)
omega
· have left := (M.mem_binaryGenericRow I C H i c).2 ⟨hc, first⟩
exact absent ((M.mem_binaryGenericRow I C H j c).1 (same ▸ left)).2
theorem binaryGenericRows_injective
(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)
(choice : Realize M.sets SetFormula.choice Fin.elim0)
{I C : ZFSet.{u}} (hI : I ∈ M.sets) (hC : C ∈ M.sets)
(uncountable : ¬ M.groundEmbeddable C ZFSet.omega) :
letI := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
∀ H : ConditionFilter (SetMembers
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega)),
H.GenericFor (M.codedFamily (SetMembers.presentation
(M.smallPartialFunctions (ZFSet.prod I C) (natSetCode 2) ZFSet.omega))) →
Function.Injective (fun i : SetMembers I => M.binaryGenericRow I C H (SetMembers.raw i)) := by
let := M.smallFunctionOrder (ZFSet.prod I C) (natSetCode 2) ZFSet.omega
intro H generic i j same
apply SetMembers.raw_injective I
by_contra distinct
exact M.binaryGenericRow_ne pairing unions powers separation infinite choice hI hC
(SetMembers.raw_mem i) (SetMembers.raw_mem j) distinct uncountable H generic same
end TransitiveGround
end PartitionPilot.Forcing
end
end OAI
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