| import OAI.SetTheory.PartitionPrinciple.Forcing.GroundRowGenerics |
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| namespace OAI |
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| section |
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| namespace PartitionPilot.Forcing |
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| universe u |
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| 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 |
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| namespace TransitiveGround |
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| open SetRealization SetFormula |
| variable (M : TransitiveGround.{u}) |
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| 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 |
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| 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 |
|
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| 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 |
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| 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 |
|
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| 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 |
|
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| end TransitiveGround |
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| end PartitionPilot.Forcing |
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| end |
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| end OAI |
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