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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 53 54 55 56 57 58 59 60 61 62 | import OAI.SetTheory.PartitionPrinciple.Forcing.GroundOutsideRowNames
import OAI.SetTheory.PartitionPrinciple.Forcing.GroundPrincipalSupportZF
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u}) {X Y I G : Type u}
variable [Group G] [PartialOrder X] [PartialOrder I]
theorem exists_countableFunction_ZF
(a : PosetAction G X) (d : PosetAction G I)
(s : Presentation X) (t : Presentation Y) (b : Presentation G) (k : Presentation I)
(C : I) (directed : ∀ C D : I, ∃ E, C ≤ E ∧ D ≤ E)
(small : M.sets.Countable)
(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)
(hX : s.carrier ∈ M.sets) (hY : t.carrier ∈ M.sets) (values : t.carrier.Nonempty)
(hG : b.carrier ∈ M.sets) (hI : k.carrier ∈ M.sets)
(coordinate_graph : a.actionGraph s b ∈ M.sets)
(support_order : k.conditionOrder ∈ M.sets) (support_graph : d.actionGraph k b ∈ M.sets)
(p : SetMembers (M.smallPartialFunctions s.carrier t.carrier ZFSet.omega)) :
letI := M.smallFunctionOrder s.carrier t.carrier ZFSet.omega
let act := M.smallFunctionAction a s t b pairing unions powers separation hX hY hG
(M.omega_mem separation infinite) coordinate_graph
let c := SetMembers.presentation (M.smallPartialFunctions s.carrier t.carrier ZFSet.omega)
let F := d.principalSupportFamily C directed
∃ H : ConditionFilter (SetMembers (M.smallPartialFunctions s.carrier t.carrier ZFSet.omega)),
H.mem p ∧ H.GenericFor (M.codedFamily c) ∧
H.GenericFor (formulaFamily (M.nameDomain c)) ∧
H.GenericFor (formulaFamily (act.symmetricDomain c (M.nameDomain c) F)) ∧
M.sets ⊆ (act.symmetricDomain c (M.nameDomain c) F).values {q | H.mem q} ∧
ZFRealization ((act.symmetricDomain c (M.nameDomain c) F).values {q | H.mem q}) ∧
ZFSet.IsFunc s.carrier t.carrier (M.countableGenericGraph s.carrier t.carrier H) := by
let := M.smallFunctionOrder s.carrier t.carrier ZFSet.omega
let := M.smallFunctionTop s.carrier t.carrier ZFSet.omega ⟨s.carrier, hX⟩
let act := M.smallFunctionAction a s t b pairing unions powers separation hX hY hG
(M.omega_mem separation infinite) coordinate_graph
let c := SetMembers.presentation (M.smallPartialFunctions s.carrier t.carrier ZFSet.omega)
obtain ⟨H, hp, ground, ordinary, symmetric, includes, zf⟩ :=
M.exists_ZF_from_principalSupports d act k c b C directed small pairing unions powers separation collection infinite
(M.countablePartialFunctions_mem pairing unions powers separation infinite hX hY)
(M.countableFunctionOrder_mem pairing unions powers separation infinite hX hY)
hG hI support_order support_graph
(M.smallFunctionActionGraph_mem a s t b pairing unions powers separation hX hY hG
(M.omega_mem separation infinite) coordinate_graph) p
exact ⟨H, hp, ground, ordinary, symmetric, includes, zf,
M.countableGenericGraph_isFunc pairing unions powers separation infinite hX hY values H ground⟩
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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