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Hebrew University of Jerusalem

arXiv:math/9209210v1 [math.LO] 25 Sep 1992∗FORCINGGarvin Melles∗Hebrew University of JerusalemFebruary 9, 2018AbstractLet Mbe a transitive model of ZFCand let B be a M -complete Boolean algebra in M. (Possibly a proper class.) We definea generalized notion of forcing with such Boolean algebras,∗forcing.We prove that1.

If G is a∗forcing complete ultrafilter on B, then M[G] |=ZFC.2. Let H ⊆M.

If there is a least transitive model N such thatH ∈M,OrdM = OrdN , and N |= ZFC, then we denote Nby M[H]. We show that all models of ZFC of the form M[H]are∗forcing extensions of M. As an immediate corollary weget that L[0#] is a∗forcing extension of L.1IntroductionIn this paper we introduce a generalization of forcing,∗forcing.

If M isa transitive model of ZFC and B is a M -complete Boolean algebra inM, then a ∗forcing extension of M is a transitive class of the form M[G]where G is an M -complete ultrafilter on B .If B is a set Booleanalgebra, then∗forcing and forcing coincide, so the interesting case is whenB is a proper class. One indication of the importance of∗forcing is givenby the fact that L[0#] is a ∗forcing extension of L. (In fact we show muchmore, as in the abstract.) In what follows we review Boolean valued models,and show how Boolean valued models on names for sets lead to pre-∗forcing,∗Would like to thank Ehud Hrushovski for supporting him with funds from NSF GrantDMS 89595111

∗forcing, and forcing with Boolean values for formulas. We prove that ifM[G] is a∗forcing extension of M and G is∗forcing complete, thenM[G] |= ZFC.

In the final section we show that if M[H] is the smallesttransitive model of ZFC such that OrdM = OrdM[H] and H ∈M, thenthere is a∗forcing complete ultrafilter H on the Boolean algebra Bform(a proper class in M ) such that M[H] = M[H].2Boolean Valued ModelsDefinition 2.1 A Boolean valued model M(B) (of the language of set the-ory) is a triple (U, I, E) where U is the universe and I and E are binaryfunctions with values in some Boolean algebra. Keeping with tradition wewrite I(x, y) as ||x = y|| and E(x, y) as ||x ∈y||.Iand E mustsatisfy the following laws:1.

||x = x|| = 12. ||x = y|| = ||y = x||3.

||x = y|| • ||y = z|| ≤||x = z||4. ||x ∈y|| • ||v = x|| • ||w = y|| ≤||v ∈w||For each formula and a1, .

. .

, an from U we define the Boolean value ofas follows:1. ||¬ψ(a1, .

. .

, an)|| = −||ψ(a1, . .

. , an)||2.

||ϕ ∧ψ(a1, . .

. , an)|| = ||ϕ(a1, .

. .

, an)|| • ||ψ(a1, . .

. , an)||3.

||∃xϕ(x, a1, . .

. , an)|| =Pa∈M(B)||ϕ(a, a1, .

. .

, an)||If M(B) is a Boolean valued model (or even if || || is defined only for theatomic formulas) we can with any ultrafilter Fon B construct a modelM(B)/F, as follows: Define an equivalence relation over M(B) by lettingx ≡yiff||x = y|| ∈FDefine the relation E on the equivalence classes by[x] E [y]iff||x ∈y|| ∈FLet M(B)/F = (M(B)/ ≡, E).2

Lemma 2.2 If M(B) is a Boolean valued model and F is an ultrafilteron B such that for every formula ϕ = ∃xψ(x) and a1, . .

. , an from theuniverse of M(B), if||∃xψ(x, a1, .

. .

, an)|| ∈Fimples there is an a in the universe of M(B) such that||ψ(a, a1, . .

. , an)|| ∈Fthen for every formula θ, and b1, .

. .

, bm from the universe of M(B),M(B)/F |= θ([b1], . .

. , [bm]) iff||θ(b1, .

. .

, bm)|| ∈FproofLike the proof of Lemma 18.1 in [Jech].3∗Forcing and∗Forcing Complete UltrafiltersDefinition 3.1 If Mis a transitive model of ZFC and B is a classBoolean algebra in M then we define the Boolean valued names for M, MBas[α∈OrdMMBαwhere Bα = {b ∈B | rank(b) ≤α}.Definition 3.2 Let Mbe a transitive model of ZFC and let N be aclass in M. We call N in the rest of this definition the class of names.Let P be a class of unary relation symbols. We define the class of formulasForm(M, N, P) as the smallest class C such that1.

Each symbol in P is in C, together with each symbol of P with aname substituted for the variable2. x = y and x ∈y are in C, together with substitutions of variablesin x = y and x ∈y by names3.

If ϕ(x0, x1, . .

. , xn, A) is in C then so are¬ϕ(x0, x1, .

. .

, xn, A)∃x0ϕ(x0, x1, . .

. , xn, A)3

4. Let I be a set.

If for each i ∈I, ϕ(x0, x1, . .

. , xn, Ai) ∈C, then^i∈Iϕ(x0, x1, .

. .

, xn, Ai)∈C_i∈Iϕ(x0, x1, . .

. , xn, Ai)∈CLet Sent(M, N, P) be the elements of Form(M, N, P) without free vari-ables.

Let Sent(M, N, P)(q.f.) be the sentences in Sent(M, MN, P) with-out quantifiers and similarly for Form(M, N, P)(q.f.

).Definition 3.3 A pre- ∗forcing notion is a quadruple (M, MB, ||||∗, G)such that1. M is a transitive model of ZFC2.

MB is the class of Boolean valued names for M3. || ||∗is a function with domain the elements of Sent(M, MB)(q.f.

)such that(a) ||x = y||∗= 0 if x ̸= y and ||x = y||∗= 1 if x = y(b) ||x ∈y||∗= b if x ∈dom y and y(x) = b and 0 otherwise(c) If I is a set and for each i ∈I,Ai is a set of names, theni. ||¬ψ(A0)||∗= −||ψ(A0)||∗ii.

|| Wi∈Iϕ(Ai)||∗= Pi∈I||ϕ(Ai)||∗iii. || Vi∈Iϕ(Ai)||∗= Qi∈I||ϕ(Ai)||∗4.

G is a M -complete ultrafilter on B, i.e., it has the property that ifX ⊆B andXX ∈Gthen there is an x ∈X such that x ∈G.We denote (MB, || ||∗) by M(B).Theorem 3.4 If (M, MB, ||||∗, G) is a pre- ∗forcing notion, then forevery ϕ(A) ∈Sent(M, MB)(q.f. ),M(B)/G |= ϕ(A)iff||ϕ(A)||∗∈G4

proofBy induction on the complexity of ϕ and the definition of pre-∗forcing notion.So we have a notion of a condition forcing a formula, at least for quantifierfree elements of Sent(M, MB). The disadvantage with pre-∗forcing is thatthe model M(B)/G is not in general a model of extensionality.

But thefunction || ||∗induces a function || || so that if we denote the ∗Booleanvalued model (M, MB, || ||) by MB, then extensionality holds in MB/G . ( (MB, || ||) is just the usual notion of Boolean valued model for forcing.

)Definition 3.5 If (M, MB, ||||∗, G) is a pre- ∗forcing notion then wedefine by induction on Γ(ρ(x), ρ(y)) where ρ(x) is the rank of x and Γis the canonical well ordering of Ord × Ord,||x ∈y|| =Xt∈dom y||x = t|| • ||t ∈y||∗||x = y|| =Yt∈dom x(−||t ∈x||∗+ ||t ∈y||) •Yt∈dom y(−||t ∈y||∗+ ||t ∈x||)We call (M, MB, || ||, G) a∗forcing notion.Theorem 3.6 If (M, MB, ||||∗, G) is a pre- ∗forcing notion then ||||and MB/G with respect to || || are defined andMB/G ∼= M[G]by the isomorphism h[x] = iG(x) whereM[G] = {iG(x) | x ∈MB}with ∈on M[G] interpreted as the real ∈relation and iG is defined inthe usual manner.proofSee lemmata 18.2, 18.3, 18.9 and exercise 18.6 of [Jech].Theorem 3.7 (The forcing theorem for ∗forcing) If ϕ(A) ∈Sent(M, MB)(q.f. ),thenM[G] |= ϕ(A)iff||ϕ(A)|| ∈G5

proofFor every sentence ϕ(A) ∈Sent(M, MB) there is a correspondingstatement ϕ′(A′) such that ||ϕ(A)|| = ||ϕ′(A′)||∗andMB/G |= ϕ(A) ↔M(B)/G |= ϕ′(A′)Furthermore, ϕ(A) ∈Sent(M, MB)(q.f.) ↔ϕ′(A′) ∈Sent(M, MB)(q.f.

)If (M, MB, || ||∗, G) is a pre- ∗forcing notion and if ϕ′(A′) ∈Sent(M, MB)(q.f. )we know thatM(B)/G |= ϕ′(A′) ↔||ϕ′(A′)||∗∈GThis lifts toMB/G |= ϕ(A) ↔||ϕ(A)|| ∈GSince M[G] and MB/G are isomorphic we get the statement in the the-orem.Definition 3.8 Let (M, MB, || ||, G) be a∗forcing notion.

Suppose thatfor every first order formula ϕ(x0, x1, . .

. , xn) and set of names ˙a1, .

. .

, ˙anthere is a set Bϕ(x0,˙a1,...,˙an) of names such that for every C ⊃Bϕ(x0,˙a1,...,˙an),X˙b∈Bϕ(x0, ˙a1,..., ˙an)||ϕ(˙b, ˙a1, . .

. , ˙an)|| =X˙c∈C||ϕ(˙c, ˙a1, .

. .

, ˙an)||where the definition of || || is extended to all first order fomulas by induc-tion on their complexity as follows:1. ||¬ϕ(˙a1, .

. .

, ˙an)|| = −||ϕ(˙a1, . .

. , ˙an)||2.

||ϕ(˙a1, . .

. , ˙an) ∧ψ(˙a1, .

. .

, ˙an)|| = ||ϕ(˙a1, . .

. , ˙an)|| • ||ψ(˙a1, .

. .

, ˙an)||3.||∃x0ϕ(x0, ˙a1, . .

. , ˙an)|| =X˙b∈Bϕ(x0, ˙a1,..., ˙an)||ϕ(˙b, ˙a1, .

. .

, ˙an)||Then we say (M, MB, || ||, G) is a forcing notion with Boolean values forformulas.Theorem 3.9 If (M, MB, || ||, G) is a forcing notion with Boolean valuesfor formulas, thenM[G] |= ZFC −P6

proofFirst we show that the forcing theorem holds for all first orderformulas by induction on the complexity of the formulas, and then the usualproof that a forcing extension (using a set Boolean algebra) satisfies ZFCfor all the axioms except power set goes through.Definition 3.10 Let S be a class. A formula ϕ(v1, .

. .

, vn) is said to be aformula of set theory enriched by S if it is a first order formula built fromthe usual atomic formulas of set theory together with atomic formulas of theform s(x) for s ∈S where the interpretation of s(x) is x ∈s.Definition 3.11 If α < OrdM, let MBαbe the names of rank ≤α, andlet MBααbe the elements of MBα of rank ≤α.Definition 3.12 Let α < OrdM and let (M, MB, || ||, G) be a∗forcingnotion.Let ϕ(x1, . .

. , xn) be a formula and let˙a1, .

. .

, ˙an be names inMBαα . By induction on the complexity of ϕ we define||ϕ(˙a1, .

. .

, ˙an)||αas follows:1. ||ϕ(˙a1, .

. .

, ˙an)||α = ||ϕ(˙a1, . .

. , ˙an)|| if ϕ is quantifier free.2.

||ϕ(˙a1, . .

. , ˙an) ∧ψ(˙a1, .

. .

, ˙an)||α = ||ϕ(˙a1, . .

. , ˙an)||α•||ψ(˙a1, .

. .

, ˙an)||α3. ||¬ϕ(˙a1, .

. .

, ˙an)||α = −||ϕ(˙a1, . .

. , ˙an)||α4.

||∃xϕ(˙a1, . .

. , ˙an)||α =Pb∈MBαα||ψ(b, ˙a1, .

. .

, ˙an)||αDefinition 3.13 Let α < OrdM and let (M, MB, || ||, G) be a∗forcingnotion. Let ϕ(x1, .

. .

, xn) be a formula. We define by induction on thecomplexity of ϕ the meaning of ϕ(x1, .

. .

, xn) reflects in MBαα .1. If ϕ(x1, .

. .

, xn) is quantifier free then ϕ(x1, . .

. , xn) reflects in everyMBαα2.

ϕ(x1, . .

. , xn) reflects in MBααiff¬ϕ(x1, .

. .

, xn) reflects in MBαα3. ϕ(x1, .

. .

, xn) ∧ψ(x1, . .

. , xn) reflects in MBααiffϕ(x1, .

. .

, xn) andψ(x1, . .

. , xn) reflect in MBαα7

4. If ϕ(x1, .

. .

, xn) = ∃x0ψ(x0, x1, . .

. , xn) then ϕ(x1, .

. .

, xn) reflectsin MBααiff(a) ψ(x0, x1, . .

. , xn) reflects in MBαα(b) ∀β > α∃γ > β such that ψ(x0, x1, .

. .

, xn) reflects in MBγγandif b ∈MBγγand ||ψ(b, ˙a1, . .

. , ˙an)||γ ∈G, for some ˙a1, .

. .

, ˙anfrom MBααthen there is an a ∈MBααsuch that||ψ(a, ˙a1, . .

. , ˙an)||α ∈GDefinition 3.14 If (M, MB, ||||, G) is a∗forcing notion then we saythat G is∗forcing complete if1.

For every α < OrdM, there is an ordinal β such that if ˙a is a namewith domain MBαα , then for some name ˙b in MBββ||˙a = ˙b|| ∈G2. For every formula of set theory ϕ(x1, .

. .

, xn) and for every α α such that ϕ(x1, . .

. , xn) reflects inMBββTheorem 3.15 If (M, MB, || ||, G) is a ∗forcing notion and G is ∗forcingcomplete, thenM[G] |= ZFCproofThe only nontrivial axioms to prove are choice, union, power set,replacement and separation.

Separation follows from replacement and fromseparation it is not hard to prove union.Replacement First we prove by induction on the complexity of ϕ that if ϕreflects in MBααthen ∀˙a1, . .

. , ˙an ∈MBααM[G] |= ϕ(iG(˙a1), .

. .

, iG(˙an)) ↔iG[MBαα ] |= ϕ(iG(˙a1), . .

. , iG(˙an))↔||ϕ(˙a1, .

. .

, ˙an)||α ∈G(The second ↔follows from the definition of || ||α. ) If ϕ is quantifierfree then it follows immediately.

The only nontrivial step is if ϕ = ∃xψ(x).8

Suppose M[G] |= ϕ(iG(˙a1), . .

. , iG(˙an)).

Then for some β < OrdM thereis a ˙b ∈MBββsuch thatM[G] |= ψ(iG(˙b), iG(˙a1), . .

. , iG(˙an))Since G is ∗forcing complete there is a γ > β such that ψ(x0, x1, .

. .

, xn)reflects in MBγγwhich implies by the induction hypothesis thatiG[MBγγ ] |= ψ(iG(˙b), iG(˙a1), . .

. , iG(˙an))and ||ψ(˙b, ˙a1, .

. .

, ˙an)||γ ∈G. Since G is ∗forcing complete and ϕ(x1, .

. .

, xn)reflects in MBαα ,ψ(x0, x1, . .

. , xn) reflects in MBααand for some˙a ∈MBαα ,||ψ(˙a, ˙a1, .

. .

, ˙an)||α ∈Gand iG[MBαα ] |= ψ(iG(˙a), iG(˙a1), . .

. , iG(˙an)) which implies||ϕ(˙a1, .

. .

, ˙an)||α ∈Gand iG[MBαα ] |= ϕ(iG(˙a1), . .

. , iG(˙an)).

Now if ||ϕ(˙a1, . .

. , ˙an)||α ∈G thenwe know that since G is M -complete that for some a ∈MBαα ,||ψ(˙a, ˙a1, .

. .

, ˙an)||α ∈GNow by the induction hypothesis we getM[G] |= ψ(iG(˙a), iG(˙a1), . .

. , iG(˙an))i.e., M[G] |= ϕ(iG(˙a1), .

. .

, iG(˙an)). Now let F(x) = y be a function de-fined by the formula ϕ(x, y, iG(˙a1), .

. .

, iG(˙an)) with the ˙ai from MBαα .Let X be a set and let iG( ˙X) = X. Let β > α such that ϕ(x, y, ˙a1, .

. .

, ˙an)reflects in MBββand˙X ∈MBββ . Let˙Ybe the name with domain MBββsuch that ∀˙a ∈MBββ ,˙Y (˙a) =Xx∈dom ˙X||ϕ(x, ˙a, ˙a1, .

. .

, ˙an)||β • ||x ∈˙X||Then F[X] = iG( ˙Y ).Power Set By point 1. of the definition of∗forcing complete and by sep-aration we know that for eachα < OrdMthatP(iG(MBαα ))exists.9

Now applying separation and the fact that if X ∈M[G], then for someα < OrdM, X ⊆iG[MBαα ] we get the power set axiom holds in M[G].Choice If X is a name a set in M[G] and f is the name with domain= {(ˇx, x)B | x ∈dom X} such that f((ˇx, x)B) is uniformly 1, then iG(f)is a function from dom X onto iG(X). Since dom X is well ordered (asit is in M ) iG(f) induces a well order on iG(X).Corollary 3.16 Let T be a theory extending ZFC.

If (M, MB, || ||, G)is a∗forcing notion and G is∗forcing complete, thenM[G] |= Tiff∀ϕ ∈T, ∃α < OrdM such that MBααreflects ϕ and ||ϕ||α ∈G.We would like necessary and sufficient conditions that M[G] |= ZFC butthe proof does not seem to go in the other direction unless G is a class inM[G]. But if G is a class in M[G] then the {iG(MBαα ) | α < OrdM}form a cumulative hierarchy in M[G] and we can prove:Theorem 3.17 If M[G] is a∗forcing extension of M,G is a class inM[G] and M[G] |= ZFC then G is∗forcing complete.proofIf G is a class in M[G] then the {iG(MBαα ) | α < OrdM} forma cumulative hierarchy in M[G] so that for every formula ϕ and for everyordinal α < OrdM, there is a β > α such that ϕ reflects (in the usualsense) in iG(MBββ ).

Now we can show by induction on the complexity ofϕ that ϕ reflects in MBββ . Since the power set axiom holds in M[G] weknow that condition 1. in the definition of∗forcing complete must alsohold, otherwise for some α the collection of subsets of iG(MBαα ) form aclass in M[G].If we consider models of set theory with an addition predicate G(x) meantas an interpretation for G we can get necessary and sufficient conditionsthat M[G] |= ZFCG(x) with the interpretation of G(x) being G wherewe define ZFCG(x) as ZFC except that we allow in the separation andreplacement schemata the formulas to be enriched by G(x).

So for the restof this section if we write M[G] we mean the model in the language (=, ∈, G(x)) with the interpretation of G(x) being G. If (M, MB, || ||, G)is a∗forcing notion then we can extend the definition of |||| so that10

(M, MB, || ||) is a∗Boolean valued model in the language (=, ∈, G(x))so that if ˙a is a name theniG(˙a) ∈GiffM[G] |= G(iG(˙a)iff||G(˙a)|| ∈GIf b ∈B, letG(ˇb) = bOtherwise, if ˙a is a name in MBα , let||G(˙a)|| =Xb∈Bα||˙a = ˇb|| • ||G(ˇb)||Now we can extend the definition of || || to sentences in Sent(M, MB, {G(x)})(q.f. )in the natural way.Definition 3.18 If (M, MB, ||||, G) is a∗forcing notion then we saythat G is∗forcing complete relative to formulas enriched by G(x) if1.

For every α < OrdM, there is an ordinal β such that if ˙a is a namewith domain MBαα , then for some name ˙b in MBββ||˙a = ˙b|| ∈G2. For every formula of set theory enriched by G(x),ϕ(x1, .

. .

, xn) andfor every α < OrdM, there is an ordinal β > α such that ϕ(x1, . .

. , xn)reflects in MBββTheorem 3.19 If (M, MB, || ||, G) is a∗forcing notion then M[G] |=ZFCG(x)iffG is∗forcing complete relative to formulas of set theoryenriched by G(x).proofSimilar.4∗Forcing and models of ZFC of the form M[H]Definition 4.1 If Uis a transitive set, then DefA(U) is the set of allsubsets of Udefinable by formulas of set theory enriched by A(x) withquantifiers restricted to U and parameters from U.11

Definition 4.2 If M is a transitive model of ZFC and H ⊂M, thenif there is a least transitive model Nof ZFC such that H ∈MandOrdM = OrdN, then we denote N by M[H].Theorem 4.3 Let Mbe a transitive model of ZFC. If H ⊆MandM[H] exists, thenM[H] =[α∈OrdMLα(M ∪{H})where1.

L0(M ∪{H}) = ∅2. Lα(M ∪{H}) = Sβ<αLβ(M ∪{H}) if α is a limit3.

Lα+1(M ∪{H}) =SA∈M∪{H}DefALα(M ∪{H})proofSee [Jech] exercise 15.13.The above theorem induces a natural notion of names and of interpretationfunction iH.Definition 4.4 The class names isSα∈OrdM namesα where we define nameαby induction on the ordinals in M as follows:1. names0 = ∅2. namesα = Sβ<αnamesβ if α is a limit3.˜y ∈namesα+1 iff˜y is a function with domain namesα and ˜y(˜z) =ϕα(˜z, ˜z1, .

. .

, ˜zn) where ϕ(v0, v1, . .

. , vn) is a formula of set theoryenriched by M ∪{H} and the ˜zi are in namesαDefinition 4.5 If M[H] exists we define by induction on α < OrdM,iHas follows:1. iH(∅) = ∅12

2. If ˜y is a name in namesα+1 with the form as in the definition ofnames above, theniH(˜y) = {iH(˜z) | iH[namesα] |= ϕ(iH(˜z), iH(˜z1), .

. .

, iH(˜zn))}where iH[namesα] = {iH(˜z) | ˜z ∈namesα} = Lα(M ∪{H})Definition 4.6 Bform is the class Sent(M, names, M∪{H})(q.f.) madeinto a M -complete Boolean algebra by equating sentences logically equivalentin propositional logic.

We have1. 0 = [∅= ∅∧∅̸= ∅]2.

1 = [∅= ∅∨∅̸= ∅]3. −[ϕ] = [¬ϕ]4.

If {Ai | i ∈I} is a set of sets of names, then(a)Q {[ϕ(Ai)] | i ∈I} = [ Vi∈Iϕi(Ai)](b)P {[ϕ(Ai)] | i ∈I} = [ Wi∈Iϕ(Ai)]Definition 4.7 We define for every α < OrdM a function Asnα and aclass function I such that1. dom Asnα = symbols of the form ϕα(x1, .

. .

, xm, ˜a1, . .

. , ˜an) where˜a1, .

. .

, ˜an are in namesα2. range Asnα ⊆Form(M, names, M ∪{H})(q.f.)3.

dom I = names4. range I ⊆MBformWe define Asnα by induction on the complexity of ϕ.1.

If ϕ is quantifier free, thenAsnα(ϕα) = [ϕ]2. Asnα(¬ϕα) = −[Asnα(ϕα)]13

3. Asnα(ϕα ∧ψα) = [Asnα(ϕα)] • [Asnα(ψα)]4.

Asnα(∃xψ(x)α) = [W˜a∈namesαAsnα(ψα(˜a))]We let I =Sα∈OrdM Iα where we define Iα by induction on α as follows:Let I0(∅) = ∅. Suppose Iα has been defined.

Now if ˜y ∈nameα+1 withelements in the range of ˜y of the form ϕα(˜a, ˜a1, . .

. , ˜an) then Iα+1(˜y) hasdomain{Iα(˜a) | ˜a ∈namesα}andIα+1(˜y)(Iα(˜a)) = Asnα(ϕ(˜a, ˜a1, .

. .

, ˜an))Theorem 4.8 Every model of the form M[H] is a∗forcing extension ofM via Bform by a∗forcing complete ultrafilter.proofLet H = {[ϕ(A)] | ϕ(A) ∈Sent(M, names, M ∪{H})(q.f.) ∧M[H] |= ϕ(iH[A])}.

We leave to the reader the proof that H is an ultrafilterand H is M -complete. Since H is definable inside M[H] (If ϕ(A) ∈Sent(M, names, M ∪{H})(q.f.) then M[H] |= ϕ(iH[A])} iff∃α(A ⊆namesα ∧iH[namesα] |= ϕ(iH[A]) ) we know that M[H] = iH[MBform] ⊆M[H].

To prove the other direction we first prove by induction on the rankof ˜y thatiH(˜y) = iH(I(˜y))For˜y = ∅it follows from the definitions.So let α = β + 1 and let˜y ∈namesα −namesβ. Suppose ˜y(˜a) = ϕα(˜a, ˜a1, .

. .

, ˜an). ThenLα(M ∪{H}) |= ϕ(iH(˜a), iH(˜a1), .

. .

, iH(˜an)) iffLα(M ∪{H}) |= Asnαϕ(iH(˜a), iH(˜a1), . .

. , iH(˜an)) iffM[H] |= Asnαϕ(iH(˜a), iH(˜a1), .

. .

, iH(˜an)) iffAsnαϕ(˜a, ˜a1, . .

. , ˜an) ∈HAs a result,iH(˜y) = {iH(˜a) | Lα(M ∪{H}) |= ϕ(iH(˜a), iH(˜a1), .

. .

, iH(˜an))}= {iH(˜a) | Asnαϕ(˜a, ˜a1, . .

. , ˜an) ∈H}14

= {iH(I(˜a)) | Asnαϕ(˜a, ˜a1, . .

. , ˜an) ∈H}= iH(I(˜y))So we have M[H] = iH[names] ⊆iH[MBform] = M[H].

(As M[H] |=ZFC and H is a class in M[H] we know H must be ∗forcing complete. )As a corollary, we get the following.Corollary 4.9 L[0#] = L[G] where G is a∗forcing complete ultrafilteron BLform.REFERENCES[Jech] T. Jech, Set Theory, Academic Press, 1978.15

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출처: arXiv:9209.210 • 원문 보기