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The Genericity Conjecture, as stated in Beller-Jensen-Welch [82], is the following:

arXiv:math/9211203v1 [math.LO] 24 Nov 1992THE GENERICITY CONJECTURESy D. Friedman*MITThe Genericity Conjecture, as stated in Beller-Jensen-Welch [82], is the following:(∗)If O# /∈L[R], R ⊆ω then R is generic over L.We must be precise about what is meant by “generic”.Definition. (Stated in Class Theory) A generic extension of an inner model M is an inner model M[G]such that for some forcing notion P ⊆M :(a)⟨M, P⟩is amenable and ⊩p is ⟨M, P⟩-definable for ∆∼0 sentences.

(b)G ⊆P is compatible, closed upwards and intersects every ⟨M, P⟩-definable dense D ⊆P.A set x is generic over M if it is an element of a generic extension of M. And x is strictly generic overM if M[x] is a generic extension of M.Though the above definition quantifies over classes, in the special case where M = L and O# existsthese notions are in fact first-order, as all L-amenable classes are ∆∼1 definable over L[O#]. ¿From now onassume that O# exists.Theorem A.

The Genericity Conjecture is false.The proof is based upon the fact that every real generic over L obeys a certain definability property,expressed as follows.Fact.If R is generic over L then for some L-amenable class A, Sat⟨L, A⟩is not definable over ⟨L[R], A⟩,where Sat⟨L, A⟩is the canonical satisfaction predicate for ⟨L, A⟩.Thus Theorem A is established by producing a real R s.t. O# /∈L[R] yet Sat⟨L, A⟩is definable over⟨L[R], A⟩for each L-amenable A.A weaker version of the Genericity Conjecture would state:If O# /∈L[R] then either R ∈L or R isgeneric over some inner model M not containing R. This version of the conjecture is still open.

However,this question can also be studied in contexts where O# does not exist, for example when the universe hasordinal height equal to that of the minimal transitive model of ZF. In the latter context, Mack Stanley [93]has demonstrated the consistency of the existence of a non-constructible real which belongs to every innermodel over which it is generic.

*Research Supported by the National Science Foundation, Grant #8903380–DMS.1

Section A A Non-Generic Real below O#.We first prove the Fact stated in the introduction.Lemma 1.Suppose R ⊆ω is generic over L. Then for some L-amenable class A, Sat⟨L, A⟩is not definableover ⟨L[R], A⟩with parameters.Proof. Let R ∈L[G] where G ⊆P is generic for ⟨L, P⟩-definable dense classes and P is L-amenable as in (a),(b) of the definition of generic extension.

Let A = P and suppose that Sat⟨L, P⟩were definable over ⟨L[R], P⟩with parameters. But the Truth Lemma holds for G, P for formulas mentioning G, P : ⟨L[G], G, P⟩⊨φ(G, P)iff∃p ∈G(p ⊩φ(G, P)), using the fact that ⊩in P for ∆∼0 sentences is definable over ⟨L, P⟩and the genericityof G. So Sat⟨L[G], G, P⟩is definable over ⟨L[G], G, Sat⟨L, P⟩⟩, since ⊩is definable over ⟨L, Sat⟨L, P⟩⟩forarbitrary first-order sentences.Since Sat⟨L, P⟩is definable over ⟨L[G], G, P⟩we get the definability ofsatisfaction for the latter structure over itself.

This contradicts a well-known result of Tarski.⊣The rest of this section is devoted to the construction of a real R such that R preserves L-cofinalities(cof(α) in L = cof(α) in L[R] for every α) and for every L-amenable A, Sat⟨L, A⟩is definable over ⟨L[R], A⟩. (The proof has little to do with the Sat operator; any operator from L-amenable classes to L-amenableclasses that is “reasonable” is codable by a real.

We discuss this further at the end of this section. )R will generically code a class f which is generic for a forcing of size ∞+ = least “L-cardinal” greaterthan ∞.

Since this sounds like nonsense we suggest that the reader think of ∞as some uncountable cardinalof V and then ∞+ denotes (∞+)L. Thus we will define a constructible set forcing P∞⊆L∞+ for addinga generic f ∞⊆∞such that if A ⊆∞is constructible then Sat⟨L∞, A⟩is definable over ⟨L∞[f ∞], f ∞, A⟩.Then we show how to choose the f ∞’s to “fit together” into an f ⊆ORD such that Sat⟨L, A⟩is definableover ⟨L[f], f, A⟩for each L-amenable A. Finally, we code f by a real R (using the fact that I = SilverIndiscernibles are indiscernibles for ⟨L[f], f⟩).A condition in P∞is defined as follows.

Work in L. An Easton set of ordinals is a set of ordinals Xsuch that X ∩κ is bounded in κ for every regular κ > ω. For any α ∈ORD, 2α denotes all f : α −→2and 2<α = ∪{2β|β < α}.

An Easton set of strings is a set D ⊆∪{2α|α ∈ORD} such that D ∩2<κ hascardinality less than κ for every regular κ > ω. For any X ⊆ORD let Seq(X) = ∪{2α|α ∈X}.

A conditionin P∞is (X, F, D, f) where:(a)X ⊆∞is an Easton set of ordinals(b)F : X −→P(2∞) = Power Set of 2∞such that for α ∈X, F(α) has cardinality ≤α(c)D ⊆Seq(X) is an Easton set of strings(d)f : D −→∞such that f(s) > length (s) for s ∈D.We define extension of conditions as follows. (Y, G, E, g) ≤(X, F, D, f) iff(i)Y ⊇X, E ⊇D, G(α) ⊇F(α) for α ∈X, g extends f2

(ii)If s ∈E −D then the interval (length (s) + 1, g(s)] contains no element of X, and if s ⊆S ∈F(α)for some α ≤length (s), α ∈X then g(s) /∈CS.We must define CS. For S ∈2∞let µ(S) = least p.r.

closed µ > ∞such that S ∈Lµ and then CS ={α < ∞|α = ∞∩Skolem hull (α) in Lµ(S)}. Thus CS is CUB in ∞and ⟨Lα, S ↾α⟩≺⟨L∞, S⟩forsufficiently large α ∈CS (as S ∈Skolem hull (α) in Lµ(S) for sufficiently large α < ∞).

Also note thatT /∈Lµ(S) −→CT ⊆Lim CS ∪α for some α < ∞.Our goal with this forcing is to produce a generic function fG from 2<∞into ∞such that for eachS ⊆∞, {f(S ↾α)|α < ∞} is a good approximation to the complement of CS. S ∈F(α) is a committmentthat for β > α, f(S ↾β) /∈CS (in stronger conditions).Lemma 2.

If p ∈P∞and α < ∞, S ∈2∞, s ∈2<∞then p has an extension (X, F, D, f) such that α ∈X,S ∈F(α) and s ∈D.Proof. Easy, given the fact that if s needs to be added then we can safely put f(s) = length(s) + 1.⊣Lemma 3.

P∞has the ∞+-chain condition (antichains have size ≤∞, all in L of course).Proof. Any two conditions (X, F, D, f), (X, G, D, f) are compatible, so an antichain has cardinality at mostthe number of (X, D, f)’s, which is ∞.⊣Lemma 4.

Let G be P∞-generic and write fG for ∪{f|(X, F, D, f) ∈G for some X, F, D}. If S ∈2∞thenfG(S ↾α) /∈CS for sufficiently large α < ∞.Proof.

G contains a condition (X, F, D, f) such that 0 ∈X and S ∈F(0). If s ⊆S, s /∈D then fG(s) /∈CS,by (ii) in the definition of extension.

And S ↾α /∈D for sufficiently large α < ∞.⊣Lemma 5. Let G, fG be as in Lemma 4.

If α < ∞is regular, S ∈2∞, and α /∈Lim CS then {fG(S|β)|β <α} intersects every constructible unbounded subset of α.Proof. Let A ⊆α be constructible and unbounded in α.

We show that a condition (X, F, D, f) can beextended to (X ∪{δ}, F ∗, D ∪{S ↾δ}, f ∗) for some δ, where f ∗(S ↾δ) ∈A. Choose δ < α large enoughso that S ↾δ is not an initial segment of any T ∈∪{F(β)|β ∈X ∩α} −{S}.

This is possible since X ∩αis bounded in α and F(β) has cardinality < α for each β ∈X ∩α. Then let f ∗= f ∪{⟨S ↾δ, β⟩} whereβ ∈A −CS −δ and F ∗= F ∪{⟨δ, ∅⟩}.⊣Lemma 6.

P∞preserves cofinalities (i.e., P∞⊩cof(α) = cof(α) in L for every ordinal α).Proof. For regular κ < ∞and p ∈P∞let (p)κ = “part of p below κ”, (p)κ = “part of p at or above κ” bedefined in the natural way: if p = (X, F, D, f) then(p)κ = (X ∩κ, F ↾X ∩κ, D ∩Seq κ, f ↾D ∩Seq κ) and3

(p)κ = (X −κ, F ↾X −κ, D ∩Seq(∞−κ), f ↾D ∩Seq(∞−κ)).Given p and predense ⟨∆i|i < κ⟩we find q ≤p and ⟨∆i|i < κ⟩such that ∆i ⊆∆i for all i < κ, card ∆i ≤κfor all i < κ and each ∆i is predense below q. (∆is predense if {r|r ≤some d ∈∆} is dense; it is predensebelow q if every extension of q can be extended into the afore-mentioned set.) This implies that if cof(α) ≤κin some generic extension L[G], G P∞-generic over L, then cof(α) ≤κ in L. Since P∞is ∞+-CC, this meansthat P∞preserves all cofinalities.Given p and ⟨∆i|i < κ⟩as above first extend p to p0 = (X0, F0, D0, f0) so that κ ∈X0.

Now note thatif r ≤p0 then f r(s) < κ for all s ∈Dr −D0 of length < κ (where r = (Xr, F r, Dr, f r)), by condition (ii) inthe definition of extension. Thus F = {(Xr ∩κ, Dr ∩Seq κ, f r ↾Dr ∩Seq κ)|r ≤p0} is a set of cardinalityκ.

Let ⟨(∆∗i , (Xi, Di, f i))|i < κ) be an enumeration in length κ of all pairs from {∆i|i < κ} × F.Now we extend p0 successively to p1 ≥p2 ≥. .

. in κ steps so that (pi)κ = (p0)κ for all i < κ, accordingto the following prescription:If pi has been defined, see if it has an extension ri extending some di ∈∆∗isuch that (Xri ∩κ, Dri ∩Seq κ, f ri ↾Dri ∩Seq κ) = (Xi, Di, f i).

If not then pi+1 = pi. If so, select such anri, di and define pi+1 by requiring (pi+1)κ = (p0)κ, (pi+1)κ = (ri)κ except enlarge F pi+1(κ) so as to containF ri(α) for α ∈Xri ∩κ.

For limit λ ≤κ let pλ be the greatest lower bound to ⟨pi|i < λ⟩. Finally let q = pκ.Let ∆j ⊆∆j consist of all di in the above construction that belong to ∆j, for j < κ.

The claim we mustestablish is that each ∆j is predense below q. Here’s the proof: suppose ¯q ≤q and let r ≤¯q, r extendingsome element of ∆j. Choose i < κ so that (∆∗i , (Xi, Di, f i)) = (∆j, (Xr ∩κ, Dr ∩Seq κ, f r ↾Dr ∩Seq κ)).Clearly at stage i + 1, it was possible to find ri, di as searched for in the construction.

It suffices to arguethat ri, ¯q are compatible. Now (ri)κ is extended by (pi+1)κ and hence by (r)κ.

And (ri)κ is extended by (r)κ,except possibly that F ri(α) may fail to be a subset of F r(α) for α ∈Xr ∩κ. And note that the extension(ri)κ ≥(r)κ obeys all restraint imposed by F ri(α) for α ∈Xr ∩κ since we included F ri(α) in F pi+1(κ).Thus ri and ¯q are both extended by r, provided we only enlarge F r(α) for α ∈Xr ∩κ to include F ri(α), ⊣For future reference we state:Corollary 6.1.

Suppose κ < ∞is regular and ∆⊆P∞is predense. Let P∞κ= {(p)κ|p ∈P∞}, P∞,κ ={p ∈P∞|Xp ⊆κ and Range (f p) ⊆κ} with the notion ≤of extension defined as for P∞.

Then for anyq ∈P∞κthere is q′ ≤q such that ∆q′ = {r ∈P∞,κ|r ∪q′ meets ∆, F r(α) ⊆F q′(κ) for all α ∈Xr} ispredense on P∞,κ.Proof. As in the proof of Lemma 6, successively extend q (after guaranteeing κ ∈Xq) in κ steps to q′ sothat for any (X, D, f) if r ∪q′′ meets ∆for some q′′ ≤q′, some r such that (Xr, Dr, f r) = (X, D, f) thenr ∪q′ meets ∆for some such r, where F r(α) ⊆F q′(κ) for all α ∈Xr.

Now note that if r0 ∈P∞,κ thenr0 ∪q′ has an extension meeting ∆so there is r1 such that (Xr1, Dr1, f r1) = (Xr0, Dr0, f r0) and r1 ∈∆q′.But then r0 is compatible with r1 so ∆q′ is predense on P∞,κ, as desired.⊣4

Corollary 6.2. P∞⊩GCH.Proof.

Suppose f ∞: Seq(∞) −→∞is P∞-generic. It suffices to show that if κ ≤∞is regular, A ⊆κ,A ∈L[f ∞] then A ∈L[f ∞↾Seq(κ)].

But the proof of Lemma 6 shows that given any p ⊩˙A ⊆κ there isq ≤p such that for any i < κ, {r ≤q|(r)κ = (q)κ and r decides “i ∈˙A”} is predense below q. This provesthat there is q ≤p such that q ⊩˙A ∈L[ ˙f ∞↾Seq(κ)] and so by the genericity of f ∞, A ∈L[f ∞↾Seq(κ)].

⊣Next we embark on a series of lemmas aimed at showing that P∞-generics actually exist in L[O#] when∞is any Silver indiscernible.Lemma 7. Suppose i < j are adjacent countable Silver indiscernibles.

Let π = πij denote the elementaryembedding L −→L which shifts each of the Silver indiscernibles ≥i to the next one and leaves all otherSilver indiscernibles fixed. Then there is a Pji -generic Gji such that if (X, F, D, f) belongs to Gji and S ⊆i,S ∈L then f(π(S) ↾α) /∈Cπ(S) for all π(S) ↾α ∈D.Proof.

For any k ∈ω let ℓ1 < · · · < ℓk be the first k Silver indiscernibles greater than j and let jk = j+ ∩Σ1Skolem hull of j + 1 ∪{ℓ1 . .

. ℓk} in L, ik = i+ ∩Σ1 Skolem hull of i + 1 ∪{ℓ1 .

. .

ℓk} in L. (Of course i+, j+denote the cardinal successors to i, j in L.) Let j∗k = least p.r. closed ordinal α > jk such that Lα ⊨j is thelargest cardinal.

Finally let Ck = {γ < j|γ = j ∩Σ1 Skolem hull (γ ∪{j} ∪{ℓ1 . .

. ℓk)) in L}, a CUB subsetof j.Now note that if S ⊆i, S ∈L −Lik then Cπ(S) ⊆Ck ∪γ for some γ < i.

For, µπ(S) is greater thanor equal to j∗k since otherwise π(S) belongs to Ljk and hence S belongs to Lik. Thus Cπ(S) ⊆Ck ∪γ forsome γ < j since Ck is an element of Lj∗k.

But the least such γ is definable from elements of i∪(SilverIndiscernibles ≥j), so must be less than i.Also note that the L-cofinality of jk is equal to j : Consider M =transitive collapse of Σ1 Skolem hullof j + 1 ∪{ℓ1 . .

. ℓk}.

There is a partial Σ1(M) function from a subset of j onto jk, all of whose restrictionsto ordinals γ < j have range bounded in jk. (This is why we are using Σ1 Skolem hulls rather than full ΣωSkolem hulls.) Thus the L-cofinalities of jk and j are the same, namely j.Thus we may conclude the following: The set {π(S)|S ⊆i, S ∈Lik} ∈Ljk (since it is a constructiblebounded subset of Ljk) and if S ⊆i, S ∈L−Lik then Cπ(S) is disjoint from (i, γk), where γk = least elementof Ck greater than i.Now we see how to build Gji.

We describe an ω-sequence p0 ≥p1 ≥. .

. of conditions in Pji and takeGji = {p ∈Pji |pk ≤p for some k}.

Let ⟨∆k|k ∈ω⟩be a list of all constructible dense sets on Pji so that forall k, ∆k belongs to the Σ1 Skolem hull in L of i ∪{i, j, ℓ1 . .

. ℓk+1}.

This is possible since any constructibledense set on Pji belongs to Lj++ and hence to the Σ1 Skolem hull in L of i ∪{i, j, ℓ1 . .

. ℓk} for some k. Weinductively define p0 ≥p1 ≥.

. .

so that pk belongs to the Σ1 Skolem hull in L of i+ ∪{j, ℓ1 . .

. ℓk}.

Letp0 be the weakest condition in Pji ; p0 = (∅, ∅, ∅, ∅). Suppose that k > 0 and pk−1 has been defined.

Write5

pk−1 = (X, F, D, f). First obtain ¯pk by adding i to X if necessary and defining or enlarging F(i) so as toinclude {π(S)|S ⊆i, S ∈Lik}.

Then choose pk ≤¯pk to be L-least so that pk meets ∆k−1. This completesthe construction.We show that pk ∈Σ1 Skolem hull in L of i+ ∪{j, ℓ1 .

. .

ℓk}. By induction pk−1 belongs to this hull andby choice of ⟨∆k|k ∈ω⟩, so does ∆k−1.

Now {π(S)|S ⊆i, S ∈Lik} is the range of f ↾i where f is a Σ1(L)partial function with parameters j, ℓ1 . .

. ℓk.

The latter is because Range(π ↾ik) is just jk ∩Σ1 Skolem hullin L of i ∪{j, ℓ1 . .

. ℓk}.

But given a parameter x for the domain of this Σ1(L) partial function, its rangebecomes Σ1-definable in the sense that it is in the Σ1 Skolem hull in L of {x, j, ℓ1 . .

. ℓk}.

As x can be chosenequivalently as an ordinal < i+, we get that {π(S)|S ⊆i, S ∈Lik} belongs to the Σ1 Skolem hull in L ofi+ ∪{j, ℓ1 . .

. ℓk}.

Thus so does pk. (Actually x can be chosen to be ik.

)Finally we must check that if pk = (Xk, Fk, Dk, fk) then fk(π(S) ↾α) /∈Cπ(S) for all π(S) ↾α ∈Dk,all S ⊆i in L. Assume that this is true for smaller k and we check it for k. Now if S ∈Lik then this isguaranteed by the fact that π(S) ∈F k(i), where ¯pk = (Xk, F k, Dk−1, fk−1). If S ∈L −Lik then Cπ(S) isdisjoint from (i, γk), where γk = j ∩Σ1 Skolem hull in L of γk ∪{j} ∪{ℓ1 .

. .

ℓk} and γk > i. But thenγk > i+ so Cπ(S) is disjoint from (i, ¯γk) where ¯γk = sup(j ∩Σ1 Skolem hull in L of i+ ∪{j} ∪{ℓ1 .

. .

ℓk}).Since pk ∈Σ1 Skolem hull in L of i+ ∪{j} ∪{ℓ1 . .

. ℓk}, it follows that Range(fk) ⊆¯γk and hence Range(fk)is disjoint from Cπ(S).⊣Lemma 8.

Suppose i < j are adjacent Silver indiscernibles, Gji is Pji -generic over L as in Lemma 7 andGi is Pi-generic over L. Then there exists Gj which is Pj-generic over L such that Gji = {(p)i|p ∈Gj} andq ∈Gi ←→πij(q) ∈Gj.Proof. As before, let Pj,i ⊆Pj consist of all p = (Xp, F p, Dp, f p) in Pj such that Xp ⊆i and Range(f p) ⊆i.

For any p ∈Pj,i we modify p to ¯p as follows. For S ∈F p(α), i ∈CS let ¯S = πij(S ↾i).

ForS ∈F p(α), i /∈CS let T ⊆i be L-least so that (T, CT ), (S, CS) agree through sup(CS ∩i) and let S = πij(T ).Then F ¯p(α) consists of all S for S ∈F p(α). Otherwise p, ¯p agree: (Xp, Dp, f p) = (X ¯p, D¯p, f ¯p).If p ∈Pji and i ∈Xp we let Q(p) denote {q ∈Pj,i|F q(α) ⊆F p(i) for all α ∈Xq.} Now defineGj = {p ∈Pj|(p)i ∈Gji, i ∈Xp, (p)i ∈Q((p)i) and (p)i ∈πij[Gi]}.

Note that if p0, p1 belong to Gjthen p0, p1 are compatible because (p0)i, (p1)i are compatible, the restraints from (p0)i, (p1)i are “covered”by F p0(i), F p1(i) and (p0)i, (p1)i impose at least as much restraint below i as do (p0)i, (p1)i. Note that ifGj = {p|¯p ≤p for some ¯p ∈¯Gj} then Gj is compatible, closed upwards and Gji = {(p)i|p ∈Gj}.

Alsoq ∈Gi ←→πij(q) ∈Gj, using the hypothesis that Gji satisfies Lemma 7. So it only remains to show thatGj meets all constructible predense ∆⊆Pj.The first Corollary to Lemma 6 states that it is enough to show that Gji = {(p)i|p ∈Gj} meets allconstructible predense ∆⊆Pji and that for p ∈Gji, {q ∈Q(p)|q = (r)i for some r ∈Gj} meets allconstructible ∆⊆Q(p) which are predense on ∪{Q(p∗)|p∗≤p} = Pj,i.

The former assertion is clear6

by the Pji -genericity over L of Gji = Gji. To prove the latter assertion we must show that for p ∈Gji,{q ∈Q(p)|q ∈πij[Gi]} meets every constructible ∆⊆Q(p) which is predense on Pj,i.

Given such a ∆, let∆⊆Pi be defined by ∆= {r ∈Pi|πij(r) = ¯q for some q meeting ∆}. Note that ∆is constructible becauseit equals {r ∈Pi|r = π−1ij (¯q) for some q meeting ∆} and ∆has L-cardinality ≤i.

We claim that ∆⊆Pi ispredense on Pi. Indeed, if r ∈Pi then πij(r) ∈Pj,i and therefore can be extended to some q meeting ∆.

As¯q = πij(t) for some t ≤r we have shown that r can be extended into ∆. By the Pi-genericity of Gi, chooser ∈∆∩Gi.

Then πij(r) = ¯q where q meets ∆; clearly ¯q ∈πij[Gi].⊣Lemma 9. Let i1 < i2 < .

. .

denote the first ω-many Silver indiscernibles and iω their supremum. Thenthere exist ⟨Gin|n ≥1⟩such that Gin is Pin-generic over L and whenever π :L −→L is elementary,π(iω) = iω we have p ∈Gin ←→π(p) ∈Gπ(in).Proof.

Note that any π as in the statement of the lemma restricts to an increasing map from {in|n ≥1}to itself, so Gπ(in) makes sense. We define Gin by induction on n ≥1.

Select Gi1 to be the L[O#]-leastPi1-generic (over L). Select Gi2i1 as in Lemma 7 and use Lemma 8 to define Gi2 from Gi2i1, Gi1.

Now supposethat Gin has defined, n ≥2. Then define Gin+1into be {p ∈Pin+1in|πi1in(q) ≤p for some q ∈Gi2i1} whereπi1in(im) = im+n−1 for m < ω, πi1in(j) = j for j ∈I −iω.

Then Gin+1inis Pin+1in-generic, using the ≤i1-closure of Pi2i1 and the fact that the collection of constructible dense subsets of Pin+1inis the countable unionof sets of the form πi1in(A), A of L-cardinality i1. Moreover Gin+1inobeys the condition of Lemma 7 since Gi2i1does and πi1in is elementary.

Now define Gin+1 from Gin+1in, Gin using Lemma 8.To verify p ∈Gin ←→π(p) ∈Gπ(in), note that this depends only on π ↾Liℓfor some ℓ< ω and anysuch map is the finite composition of maps of the form πm, where πm(in) = in+1 for n ≥m, πm(in) = in for1 ≤n < m. So we need only verify that for each m, n, p ∈Gin ←→πm(p) ∈Gπm(in). This is trivial unlessm ≤n as m > n −→πm(p) = p for p ∈Gin = Gπm(in).

Finally we prove the statement by induction on n ≥m. If n = m then it follows from the fact that Gin+1 was defined from Gin+1in, Gin so as to obey the conclusionof Lemma 8.

Suppose it holds for n ≥m and we wish to demonstrate the property for n + 1. But Gin+1 isdefined from Gin+1in, Gin as Gin+2 is defined from Gin+2in+1, Gin+1.

Clearly πm[Gin+1in] ⊆Gin+2in+1 and by inductionπm[Gin] ⊆Gin+1. Thus p ∈Gin+1 −→πm(p) ∈Gπm(in+1).

Conversely, p /∈Gin+1 −→p incompatible withsome q ∈Gin+1 −→πm(p) incompatible with some πm(q) ∈Gπm(in+1) −→πm(p) /∈Gπm(in+1).⊣Lemma 10. There exist ⟨Gi|i ∈I⟩such that Gi is Pi-generic over L and whenever π :L −→L iselementary, p ∈Gi ←→π(p) ∈Gπ(i).Proof.

Let t denote a Skolem term for L; thus L = {t(j1 . .

. jn)|t a Skolem term, t n-ary, j1 < · · · < jnin I}.

Now define t(j1 . .

. jn) ∈Gi ifft(σ(j1) .

. .

σ(jn)) ∈Gσ(i) where σ is the unique order-preserving mapfrom {i, j1 . .

. jn} onto an initial segment of I.

(Gi for i < iw is defined in Lemma 9. )We verify thatthis is well-defined:if t1(j1 .

. .

jn) = t2(k1 . .

. km) then let σ∗be the unique order-preserving map from7

{i, j1 . .

. jn, k1 .

. .

km} onto an initial segment of I. Then t1(σ∗(j1) .

. .

σ∗(jn)) = t2(σ∗(k1) . .

. σ∗(km)).

Butt1(σ∗(j1) . .

. σ∗(jn)) ∈Gσ∗(i) ifft1(σ1(j1) .

. .

σ1(jn)) ∈Gσ1(i) where σ1 is the unique order-preserving mapfrom {i, j1 . .

. jn} onto an initial segment of I, using Lemma 9.

The analogous statement holds for t2, so ourdefinition is well-defined. The property p ∈Gi ←→π(p) ∈Gπ(i) is clear, using our definition.⊣Now we are almost done.For any i ∈I let f i = ∪{f p|p ∈Gi}.

Thus f i :2

(a) For any L-amenable A ⊆ORD, SAT⟨L, A⟩is definable over ⟨L[f], f, A⟩. (b) I is a class of indiscernibles for ⟨L[f], f⟩.

(c) L[f] ⊨GCH.Proof. (a) We treat A as an L-amenable function A : ∞−→2.

By Lemmas 4,5 we have that for sufficientlylarge L-regular α, α ∈Lim CA ←→Range of f ↾{A ↾β|β < ∞} intersects every constructible unboundedsubset of α (where CA is defined for A to be the limit of CA↾i, i ∈I). But for α sufficiently large in CA,⟨Lα, A ↾α⟩≺⟨L, A⟩so Sat⟨L, A⟩is definable over ⟨L[f], f, A⟩.

(b) Clear by Lemma 10. (c) By Corollary 6.2.⊣Finally, using the technique of the proof of Theorem 0.2 of Beller-Jensen-Welch [82], there is a real Rsuch that f is definable over L[R] and IR = I.

Thus we conclude.Theorem 12. There is a real R ∈L[O#] such that:(a)L, L[R] have the same cofinalities(b)IR = I(c)If A is an L-amenable class then Sat⟨L, A⟩is definable over ⟨L[R], A⟩.By Lemma 1 we conclude:Theorem A.

The Genericity Conjecture is false.We close this section by mentioning a generalization of the above treatment of the SAT operator toother operators on classes. For simplicity we first state our result in terms of ω1, rather than ∞.Theorem 13.

Assume that O# exists. Suppose F is a constructible function from PL(ω1) to itself, wherePL(ω1) = all constructible subsets of (true) ω1.

Then there exists a real R

Also we may constructF ′, defined from the same parameters, so that for any A ∈PL(ω1), F(A) is definable over ⟨Lω1, A, B⟩for8

any unbounded B ⊆F ′(A). Finally note that we may assume that F ′(A) ⊆CA for all A (where A is viewedas an element of 2ω1) since CA is definable over ⟨Lω1, A, B⟩for any unbounded B ⊆CA.For any i ∈I, α ≤i ≤ω1, let F ′i be defined in L just like F ′, but with ω1 replaced by i.

Also define Pias before but with CS replaced by F ′i(S) (viewing S ∈2i as a subset of i). Then as before we can constructa generic f : 2<ω1 −→ω1 so that for any A ∈PL(ω1), F(A) is definable over ⟨Lω1[f], A⟩.

Finally code fgenerically by a real using the fact that α is countable and I ∩(α, ω1) is a set of indiscernibles for ⟨Lω1[f], f⟩.⊣To deal with operators on L-amenable classes, we have to keep track of parameters.Definition.Suppose i < j belong to I and Fi is a counstructible function from PL(i) to itself. ThenF ji : PL(j) −→PL(j) is defined as follows:Write Fi = t(α, i,⃗k) where t is a Skolem term for L, α < i and⃗k are Silver indiscernibles greater than j.

Then F ji = t(α, j,⃗k).Also define F ∞i: L-amenable classes = PL(∞) −→PL(∞) as follows: Given an L-amenable A chooset and α so that for all j ∈I greater than α, A ∩j = t(α, j,⃗k) where ⃗k are Silver indiscernibles greater thanj. Then F ∞i (A) = ∪{F ji (A ∩j)|α < j ∈I}.

An operator F : PL(∞) −→PL(∞) is countably constructibleif it is of the form F ∞ω1 where Fω1 is a constructible function from PL(ω1) to itself.Theorem 14. Assume that O# exists and F : PL(∞) −→PL(∞) is countably constructible.

Then thereexists R

The resulting real R satisfies the conclusion of the presentTheorem.⊣Remarks. (a) The definitions of F(A) over ⟨Lω1(R), A⟩, ⟨L[R], A⟩in Theorems 13, 14 respectively areindependent of A.

(b) If F : PL(ω1) −→PL(ω1) is constructible then there exists a set-generic extension of L in whichthere is a real R obeying the conclusion of Theorem 13. However we cannot expect there to be such a realin L[O#], or even compatible with the existence of O#.

The key feature of our forcing P is that not onlycan it be used to produce a real R obeying the conclusion of Theorem 12 but such a real can be found inL[O#]. If one is willing to entirely ignore compatibility with O# then there are forcings far simpler thanours which achieve the effect of Theorem 14 for any F : classes −→classes, over any model of G¨odel-Bernaysclass theory.References1.

Beller-Jensen-Welch, Coding the Universe, Cambridge University Press, 1982.2. Friedman, Minimal Universes, to appear, Advances in Mathematics, 1993.3.

M. Stanley, A Non-Generic Real Incompatible with 0#, To appear, 1993.9

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