| import OAI.SetTheory.PartitionPrinciple.Definability.PartitionPrincipleFormula |
| import OAI.SetTheory.PartitionPrinciple.Models.GroundMainFinalPartitionModel |
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| namespace OAI |
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| namespace PartitionPilot.Forcing.TransitiveGround |
| open SetFormula SetRealization |
| universe u |
| variable (M : TransitiveGround.{u}) |
| |
| theorem exists_partitionPrinciple_without_choice |
| (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) |
| (small : M.sets.Countable) |
| {K : ZFSet.{u}} (hK : K ∈ M.sets) (standard : GroundStandardInaccessible M K) : |
| ∃ N : TransitiveGround.{u}, M.sets ⊆ N.sets ∧ ZFRealization N.sets ∧ |
| (∀ X ∈ N.sets, ∀ Y ∈ N.sets, ∀ f ∈ N.sets, ZFSet.IsFunc X Y f → |
| (∀ y ∈ Y, ∃ x ∈ X, ZFSet.pair x y ∈ f) → N.groundEmbeddable Y X) ∧ |
| ¬ Realize N.sets SetFormula.choice Fin.elim0 := by |
| obtain ⟨data, ordinalC, original⟩ := |
| M.exists_groundMainFinalSupport pairing unions powers separation collection infinite choice hK standard |
| let := ordinalMainIndexPreorder ordinalC |
| obtain ⟨run, _indices, _profiles, _terminal, _extension, _unique, indexBound, _rank, |
| i0, _zero, fulfilled, _geometryProperties⟩ := original |
| exact fulfilled.exists_partitionPrinciple_not_choice collection infinite choice small hK standard |
| indexBound rfl ⟨data.seed.coordinateInjection, data.seed.injection_ground, data.seed.injection_code⟩ |
| data.seed.dense.irrefl |
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| theorem exists_model_partitionPrinciple_without_choice |
| (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) |
| (small : M.sets.Countable) |
| {K : ZFSet.{u}} (hK : K ∈ M.sets) (standard : GroundStandardInaccessible M K) : |
| ∃ N : TransitiveGround.{u}, M.sets ⊆ N.sets ∧ ZFRealization N.sets ∧ |
| Realize N.sets partitionPrinciple Fin.elim0 ∧ |
| ¬ Realize N.sets SetFormula.choice Fin.elim0 := by |
| obtain ⟨N, includes, zf, pp, notChoice⟩ := M.exists_partitionPrinciple_without_choice |
| pairing unions powers separation collection infinite choice small hK standard |
| exact ⟨N, includes, zf, (N.realize_partitionPrinciple).2 pp, notChoice⟩ |
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| end PartitionPilot.Forcing.TransitiveGround |
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| end OAI |
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