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e3e0a0d | 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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 | import Mathlib
import OAI.SetTheory.PartitionConsistency.Logic.Proofs
import OAI.SetTheory.PartitionConsistency.Logic.SetLanguage
import OAI.SetTheory.PartitionConsistency.Forcing.ForcingQuotient
import OAI.SetTheory.PartitionConsistency.Logic.CountableModels
import OAI.SetTheory.PartitionConsistency.Logic.Completeness
import OAI.SetTheory.PartitionConsistency.Algebra.DiagramOrder
import OAI.SetTheory.PartitionConsistency.Algebra.Placements
import OAI.SetTheory.PartitionConsistency.Forcing.Diagrams
import OAI.SetTheory.PartitionConsistency.Forcing.CohenForcing
import OAI.SetTheory.PartitionConsistency.Forcing.SupportFilter
import OAI.SetTheory.PartitionConsistency.Forcing.ProductForcing
import OAI.SetTheory.PartitionConsistency.Algebra.PlacementExtensions
import OAI.SetTheory.PartitionConsistency.Algebra.MergerWords
import OAI.SetTheory.PartitionConsistency.Algebra.MergerCones
import OAI.SetTheory.PartitionConsistency.Algebra.ProbeWords
import OAI.SetTheory.PartitionConsistency.Algebra.ProbeGates
import OAI.SetTheory.PartitionConsistency.Algebra.ProbeMargins
import OAI.SetTheory.PartitionConsistency.Algebra.ConservativeExtensions
import OAI.SetTheory.PartitionConsistency.Algebra.ProbePresentations
import OAI.SetTheory.PartitionConsistency.Algebra.PersistentProfiles
import OAI.SetTheory.PartitionConsistency.Algebra.DirectLimits
import OAI.SetTheory.PartitionConsistency.Algebra.GateTowers
import OAI.SetTheory.PartitionConsistency.Logic.GroundModels
import OAI.SetTheory.PartitionConsistency.Forcing.ForcingNames
import OAI.SetTheory.PartitionConsistency.Forcing.NameActions
import OAI.SetTheory.PartitionConsistency.Forcing.AtomicForcing
import OAI.SetTheory.PartitionConsistency.Forcing.ForcingTruth
import OAI.SetTheory.PartitionConsistency.Forcing.SymmetricNames
import OAI.SetTheory.PartitionConsistency.Forcing.SymmetricSets
import OAI.SetTheory.PartitionConsistency.Forcing.SymmetricChoice
import OAI.SetTheory.PartitionConsistency.Forcing.EquivariantMixing
import OAI.SetTheory.PartitionConsistency.Forcing.ProductNames
import OAI.SetTheory.PartitionConsistency.Forcing.ChoiceFailure
import OAI.SetTheory.PartitionConsistency.Construction.MonoidBase
import OAI.SetTheory.PartitionConsistency.Construction.MonoidTowers
import OAI.SetTheory.PartitionConsistency.Construction.MonoidRounds
import OAI.SetTheory.PartitionConsistency.Construction.CompleteRounds
import OAI.SetTheory.PartitionConsistency.Construction.SaturatedMonoids
import OAI.SetTheory.PartitionConsistency.Construction.DiagramPlacements
import OAI.SetTheory.PartitionConsistency.Models.OrbitNames
import OAI.SetTheory.PartitionConsistency.Models.LocalQuotients
import OAI.SetTheory.PartitionConsistency.Models.LocalRecords
import OAI.SetTheory.PartitionConsistency.Models.AvailableRecords
import OAI.SetTheory.PartitionConsistency.Models.ComponentGluing
import OAI.SetTheory.PartitionConsistency.Models.ConcreteSystem
import OAI.SetTheory.PartitionConsistency.Models.QuotientTransport
import OAI.SetTheory.PartitionConsistency.Models.TailGraphs
import OAI.SetTheory.PartitionConsistency.Models.GraphFusion
import OAI.SetTheory.PartitionConsistency.Models.PersistentCodes
import OAI.SetTheory.PartitionConsistency.Models.OrdinalCodes
import OAI.SetTheory.PartitionConsistency.Models.CodeLinks
import OAI.SetTheory.PartitionConsistency.Models.UniformNames
import OAI.SetTheory.PartitionConsistency.Models.OrdinalPresentation
import OAI.SetTheory.PartitionConsistency.Models.ClassFibers
import OAI.SetTheory.PartitionConsistency.Models.OrdinalChoice
import OAI.SetTheory.PartitionConsistency.Models.QuotientActions
import OAI.SetTheory.PartitionConsistency.Models.GermSelectors
import OAI.SetTheory.PartitionConsistency.Models.StableSelectors
import OAI.SetTheory.PartitionConsistency.Models.OrbitAssignments
import OAI.SetTheory.PartitionConsistency.Models.AssignedGraphs
import OAI.SetTheory.PartitionConsistency.Models.CodeCardinals
import OAI.SetTheory.PartitionConsistency.Construction.MergerPushouts
import OAI.SetTheory.PartitionConsistency.Construction.ConservativeHulls
import OAI.SetTheory.PartitionConsistency.Construction.ConservativeCells
import OAI.SetTheory.PartitionConsistency.Construction.CellTowers
import OAI.SetTheory.PartitionConsistency.Construction.CellCuts
import OAI.SetTheory.PartitionConsistency.Construction.CellReorder
import OAI.SetTheory.PartitionConsistency.Construction.BlockTowers
import OAI.SetTheory.PartitionConsistency.Models.ConcreteQuotients
import OAI.SetTheory.PartitionConsistency.Forcing.RelationCalculus
import OAI.SetTheory.PartitionConsistency.Models.UniformImages
import OAI.SetTheory.PartitionConsistency.Models.OrderedSections
import OAI.SetTheory.PartitionConsistency.Models.SliceGeometry
import OAI.SetTheory.PartitionConsistency.Models.SliceAssembly
namespace OAI
/-! ## PartitionPrinciple -/
namespace PartitionConsistency.SetForcingNames.SymmetricSystem
universe u v w
noncomputable section
variable {P : Type w} [PartialOrder P] [OrderTop P] {Γ : Type v} [Group Γ]
variable (S : SymmetricSystem.{u} P Γ)
omit [OrderTop P] in
/-- Purely logical conversion of reverse-injection witnesses to the full
Partition Principle sentence. Its quantifiers range over all HS names. -/
theorem forces_PP_of_dense
(h : ∀ (p : P) (f X Y : S.Name),
S.Forces ![f,X,Y] p (SetLanguage.surjection 0 1 2) →
∃ q, q ≤ p ∧ ∃ j : S.Name, S.Forces ![j,Y,X] q (SetLanguage.injection 0 1 2))
(p : P) : S.Forces Fin.elim0 p SetLanguage.PP := by
change S.atomic.ForceFormula Fin.elim0 p SetLanguage.PP
rw [SetLanguage.PP,S.atomic.forceFormula_all S.atomic_regular]
intro X
have hX : Fin.snoc (Fin.elim0 : Fin 0 → S.Name) X = ![X] := by
ext k
fin_cases k; rfl
rw [hX,S.atomic.forceFormula_all S.atomic_regular]
intro Y
have hY : Fin.snoc ![X] Y = ![X,Y] := by
ext k
fin_cases k <;> rfl
rw [hY,S.atomic.forceFormula_imp S.atomic_regular]
intro q hqp hf r hrq
obtain ⟨s,hsr,f,hf⟩ := hf r hrq
have hv : Fin.snoc ![X,Y] f = ![X,Y,f] := by
ext k
fin_cases k <;> rfl
rw [hv] at hf
have hf' : S.Forces ![f,X,Y] s (SetLanguage.surjection 0 1 2) :=
(S.forces_surjection _ s 0 1 2).mpr ((S.forces_surjection _ s 2 0 1).mp hf)
obtain ⟨t,hts,j,hj⟩ := h s f X Y hf'
refine ⟨t,hts.trans hsr,j,?_⟩
have hvj : Fin.snoc ![X,Y] j = ![X,Y,j] := by
ext k
fin_cases k <;> rfl
rw [hvj]
exact (S.forces_injection _ t 2 1 0).mpr ((S.forces_injection _ t 0 1 2).mp hj)
end
end PartitionConsistency.SetForcingNames.SymmetricSystem
namespace PartitionConsistency.ConcreteSystem
open MonoidConstruction CardinalParameters DiagramForcing SetForcingNames
open ProductForcing SetForcingNames.SymmetricSystem
noncomputable section
/-- All-set reduction for the constructed system, including the empty-set
case. The presentation and subsequent choices are made at arbitrary ranks. -/
theorem reverse_injection_dense (f X Y : S.Name) (p : P)
(hf : S.Forces ![f,X,Y] p (SetLanguage.surjection 0 1 2)) :
∃ q, q ≤ p ∧ ∃ j : S.Name, S.Forces ![j,Y,X] q (SetLanguage.injection 0 1 2) := by
by_cases hne : ∃ q, q ≤ p ∧ ∃ x : S.Name, S.atomic.MemF q x X
· obtain ⟨q,hqp,x,hx⟩ := hne
obtain ⟨η,s,hs⟩ := ordinal_presentation aleph0_le_μ cardinal_T size_lt T.e complement_cardinal
(posetCode κ) (posetCode_injective κ) X x q hx
let D : SliceData S := ⟨s,f,A,X,Y,η.toZFSet⟩
obtain ⟨r,hrq,j,hj⟩ := presentation_reverse_injection D rfl q hs (S.atomic.forceFormula_mono hqp hf)
exact ⟨r,hrq.trans hqp,j,hj⟩
· have he : ∀ y q, q ≤ p → ¬ S.atomic.MemF q y Y := by
intro y q hqp hy
obtain ⟨r,hrq,x,hx,_⟩ := S.surjection_onto hf hqp hy q le_rfl
exact hne ⟨r,hrq.trans hqp,x,hx⟩
obtain ⟨j,hj⟩ := S.empty_domain_injection Y X p he
exact ⟨p,le_rfl,j,hj⟩
/-- The Partition Principle for arbitrary internal sets and surjections. -/
theorem forces_PP (p : P) : S.Forces Fin.elim0 p SetLanguage.PP :=
S.forces_PP_of_dense (fun p f X Y => reverse_injection_dense f X Y p) p
/-- The actual symmetric forcing interpretation supplies every axiom of the
full target theory, with no inaccessible-cardinal or model-existence premise. -/
theorem forces_target (p : P) : ∀ φ ∈ SetLanguage.targetTheory, S.Forces Fin.elim0 p φ := by
intro φ hφ
rcases hφ with hφ | hφ
· exact forces_ZF p φ hφ
· rcases hφ with rfl | rfl | rfl | hfalse
· exact forces_PP p
· exact forces_ACwo p
· exact forces_not_AC p
/-- Soundness of the nontrivial regular forcing interpretation rules out an
ordinary finite first-order proof of contradiction from the target theory. -/
theorem consistent_target : Consistent SetLanguage.targetTheory :=
S.consistent_of_forces (⊤ : P) (forces_target ⊤)
end
end PartitionConsistency.ConcreteSystem
namespace PartitionConsistency
/-- Relative consistency for pure first-order set theories, with syntactic
consistency and no transitive-ground or inaccessible-cardinal hypothesis. -/
theorem main_consistency :
Consistent SetLanguage.ZF → Consistent SetLanguage.targetTheory := by
intro _
exact ConcreteSystem.consistent_target
end PartitionConsistency
end OAI
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