File size: 2,739 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 | import OAI.SetTheory.PartitionPrinciple.Definability.PartitionPrincipleFormula
import OAI.SetTheory.PartitionPrinciple.Models.GroundMainFinalPartitionModel
namespace OAI
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
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⟩
end PartitionPilot.Forcing.TransitiveGround
end OAI
|