openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /CollapseWithoutChoice.lean
| import OAI.Computability.DegreeRigidity.SetModels.InternalCollapseConditions | |
| namespace OAI | |
| namespace TuringRigidity.BoundedSetTheory.InternalCollapse | |
| open TransitiveNameModel CountableForcing | |
| universe u | |
| attribute [local instance] order | |
| theorem append_mem_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M) | |
| (hω : ZFSet.omega.{u} ∈ M) | |
| {p x : ZFSet.{u}} (hp : p ∈ M) (hx : x ∈ M) (n : ℕ) : append p n x ∈ M := | |
| binary_union_mem M hM hP hU hp | |
| (singleton_mem M hM hP (orderedPair_mem M hM hP | |
| (hM _ (hω) _ ((mem_omega _).mpr ⟨n,rfl⟩)) hx)) | |
| theorem conditions_exist_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M) | |
| (hPow : PowerSet M) (hS : Separation M) (hω : ZFSet.omega.{u} ∈ M) | |
| {A : ZFSet.{u}} (hA : A ∈ M) : | |
| ∃ c ∈ M, ∀ p, p ∈ c ↔ p ∈ M ∧ Prefix A p := by | |
| obtain ⟨Q,hQM,hQ⟩ := internal_power M hM hPow | |
| (product_mem M hM hP hU hPow hS hω hA) | |
| let e := cons ZFSet.omega (fun _ => A) | |
| have he : ∀ i, e i ∈ M := by intro i; cases i <;> assumption | |
| let c := ZFSet.sep (fun p => (prefixFormula 1 2 0).Eval (cons p e)) Q | |
| refine ⟨c,sep_mem M hM hS _ e he hQM,fun p => ?_⟩ | |
| rw [ZFSet.mem_sep,eval_prefixFormula 1 2 0 (cons p e) rfl,hQ] | |
| exact ⟨fun h => ⟨h.1.1,h.2⟩,fun h => ⟨⟨h.1,prefix_subset h.2⟩,h.2⟩⟩ | |
| theorem orderSet_mem_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M) | |
| (hPow : PowerSet M) (hS : Separation M) | |
| {c : ZFSet.{u}} (hc : c ∈ M) : orderSet c ∈ M := by | |
| let e : ℕ → ZFSet.{u} := fun _ => c | |
| simpa only [orderSet,Formula.Eval,Formula.eval_orderedPair,Formula.eval_subset,cons_zero,cons_succ,e] using | |
| sep_mem M hM hS | |
| (.existsMem 1 (.existsMem 2 (.conj (.orderedPair 2 1 0) (.subset 0 1)))) e (fun _ => hc) | |
| (product_mem M hM hP hU hPow hS hc hc) | |
| end TuringRigidity.BoundedSetTheory.InternalCollapse | |
| end OAI | |