Generic Embeddings and the Failure of BoxAuthor(s): Douglas BurkeSource: Proceedings of the American Mathematical Society, Vol. 123, No. 9 (Sep., 1995), pp.2867-2871Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/2160588 .

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 123, Number 9, September 1995

GENERIC EMBEDDINGS AND THE FAILURE OF BOX

DOUGLAS BURKE

(Communicated by Andreas R. Blass)

ABSTRACT. We prove that if {a C K+ I order type of a is a cardinal} is station- ary, then Jensen's principle OK fails. We also show that VKEIK is consistent with a superstrong cardinal.

1. INTRODUCTION

The results of this paper were motivated by the question "Where can WI be sent in Woodin's non-stationary tower?" ([W88]). A set -v c 9(X) is stationary (in 3 (X)) iff Vf: X'@ -* X 3a E V (a #& X) such that a is closed under f. By "a)I can be sent to K" (in symbols a)l - K) we mean {a C KI ot(a) = w I} is stationary. More generally, a cardinal K is preserved (Pr(K)) iff {a C KI ot(a) is a cardinal} is stationary. If K is Ramsey, then c1 -+ K; Chang's conjecture is equivalent to WI -- O2 ([KM]). We show below that Pr(K+) implies -0'EK. (It was known that Chang's conjecture implies -,lE .) We also show that VK-1 Pr(K+) is consistent with a superstrong cardinal (by showing that VKEJK is consistent with a superstrong cardinal). It is easy to see that if K is supercompact, then Pr(K+).

We start with some basic well-known definitions and results. A generic em- bedding is an elementary embedding j: V -* (M, E) defined in some generic extension of V. We assume that the wellfounded part of (M, E) is collapsed to a transitive set. If P is the partial order of stationary subsets of 9(i) (or- dered by inclusion) and G C P is generic, then we get a generic embedding j: V -, (M, E). The model (M, E) is the ultrapower V-9()/G and cp(]) < A. Also, there is an A E M (A is [id]) such that {B E MIM k B E A} = j") (we abbreviate this by j") E M) (see [F] for proofs).

Lemma 1.1. Assume j: V -* (M, E) is a generic embedding with cp(j) < A and iA"-if M. Then A is in the wellfounded part of M, for all X C A (with X e V) X E M, and 3]e M such that Va E AM 23 "1 (a) =1(a)". Proof. This follows easily since j"A E M. 5

Definition 1.2. Pr(K) means that {a C Kj ot(a) is a cardinal} is stationary.

Lemma 1.3. The following are equivalent:

(1) Pr(K+).

Received by the editors September 18, 1993 and, in revised form, January 31, 1994. 1991 Mathematics Subject Classification. Primary 03E35, 03E55.

i) 1995 American Mathematical Society

2867

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2868 DOUGLAS BURKE

(2) There is a generic embedding j: V -* (M, E) such that Cp(j) < K+, J"K EM, and M k "K+ is a cardinal".

Proof. (1) -* (2). Force with the stationary subsets of 3(K+) below {a c K+I ot(a) is a cardinal}. So we just need to check that M k "K+ is cardinal". But M k "ot([id]) is a cardinal", and ot([id]) is K+ .

(2) -* (1). Let f: (K+)<(2 -, K+ (with f E V). In M, j"K+ is closed under j(f ) and ot(j"K+) = K+ is a cardinal. 5

Clearly, ce1 -* K implies Pr(K). A cardinal K is J6nsson iff {a C K ag = K}

is stationary. If C is a set of ordinals, then C' is all limit points of C. EK

means 3(Co,: a E K+) such that:

(1) Co, is club in a, (2) fi E (Co,)' implies Cfi = Cof n (3) cf(a) < K implies ot(Co,) < K.

A cardinal K is superstrong if there is an elementary embedding j: V -> M with cp(j) = K and Vj(K) c M. If JP is a partial order, then JP is a (a an ordinal) strategically closed iff

(VP I3P2 ... QPfi... )(PI _>P2>_ >_Pfi>_ )

where Q is 3 if fi is even and V if 11 is odd and the string of a many quantifiers is interpreted as a game. If M, N are models of ZFC, then M -a N means that VJM = VAN. For basic results about extenders see [MS]. For any other unexplained notions see [J].

2. FAILURE OF BOX

Theorem 2.1. Let K be a cardinal. If Pr(K+), then --'EK.

Proof. Assume that {a c K+1 ot(a) is a card} is stationary and therefore there is a generic embedding j: V -* (M, E) such that cp(j) < K+, j"K+ E M, and M l= "K+ is a card". Everywhere below K+ denotes the successor cardinal of K in V. We assume the wellfounded part of M is collapsed to a transitive set, so by Lemma 1.1, K+ e M, there is a j e M such that M k "7: K+ ,* Ord"

and Va E K+M I= "'(a) = j(a)", and for all X C K+ (with X E V) X E M. Therefore M l= "K+ is a successor card" and cp(j) < K+.

We may assume j(K+) > K+. (If j(K+) = K+, then K+ is J6nsson and so every stationary subset of K+ reflects ([T]) and therefore -EZK .)

Towards a contradiction assume (Cc,,: a E K+) is a ElK sequence. Now work in M. So (j(C),: a E j(K+)) is a Ej(K) sequence. Let y = supcEK+ j(a). So j(K) < y < j(K+) and cf(y) = K+. Let y = ot(j(C)y). So 7 < j(K) and cf(7) = K+.

Since 7 < y and cf(7) = K+, the range of I is bounded in 5y, say ab is

the bound. Choose v E (j(C)y)' n (j K+)' with ot(j(C)y n v) > ab. Let q be minimal such that j(>) > v (so sup iJ = V) . Since j(C)v = j(C)y n v,

we have that ab < ot(j(C)v) < 7. If j(q) = v, then j(C)v = j(C,) and so

ot(j(C)v) E range of j, contradiction. So j(q) > V. Since supj _ = V, we have j(C,)nv = j(C) v. Choose p such that p < q and ot(j(C)vnj(p)) > ab.

Since j(C,) n v = j(C)v and 1(p) < v, ot(j(C,,) n 1(p)) = ot(j(C)v n f(p)). But ot(j(C,) n 1(p)) = j(ot(CQ, n p)), a contradiction. E

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GENERIC EMBEDDINGS AND THE FAILURE OF BOX 2869

In [LMS] they show it is consistent (from a 2-huge) that w1 -- wc+1 . The above theorem and work of Steel, Mitchell and Schimmerling ([MS], [S], [MSS]) show that w, -- w,,,+l (or just Pr(wc,+1)) gives an inner model with a cardinal K such that O(K) = K++. (It is shown in [S] that -_1Z, gives such a model.) Schimmerling also gets an inner model of a Woodin cardinal from the failure of weaker principles; this suggests the following questions. Does 01 -- Ww+1 imply the failure of the weak square property (-_[L ) ? Does a), -- wo,,+ imply stationary reflection at wo,,+l ?

3. Box AND SUPERSTRONG CARDINALS

Theorem 3.1. There is a class forcing P such that VP = "ZFC plus V cardK EOK". If V k "d is superstrong", then VP k "3 is superstrong". Proof. P will be the Easton support iteration of the standard forcings for adding a square sequence. The main point is to check that superstrong cardinals are preserved. For any cardinal K let BK = the set of all functions p such that:

(1) dom(p) = the limit ordinals < y for some y E K+,

(2) p(a) c a is club, (3) Cf(ax) < K. X> Ot (p(a)) < K ,

(4) if f, E p(a)', then p(fl) = p(a)n n. It is easy to check that BK is K + 1 strategically closed (and so adds no new

K sequences of ordinals) and that VBK K [1K (see [J], p. 255). Given any ordinal fi define an iteration IP(,8) with Easton support (direct

limits at inaccessible cardinals, inverse limits everywhere else) by letting Q, name {1} if a < f8 or if VP(,)t [a "a is not a cardinal". Otherwise, Q. names Ba (in VP( )t). The forcing P we use is P(wl). The basic factor lemma (see [B], 5.1-5.4) gives that for any y, P(fi) 1 P(fl) r y * P(y) (where Pl(y) names P(y) in VP(fl)Y). Also, for any Mahlo cardinal y, P(fi) r y has the y-cc ([B], 2.4).

Claim. For any y, IP(y) adds no new y sequences.

Proof of Claim. We will show that Va P(y) r a is y + 1 strategically closed, and so the claim follows. We inductively define winning strategies Ta for P(y) r a such that:

(1) If Po, Pl, ... is any play according to Ta and ,6 < a, then po [ ,B P1 [fl, ... is accordingto Zfl .

(2) If Po, P1, ... is any partial play according to Ta and for some fi < a and for all i pi = pi r fl, (. ..,), then Tpo,..*) = Ta (Po, ...)

V01, ...) .

The construction at limit stages is easy (note that if we do not have an inverse limit at a, then cf(a) = a > y). For successor stages assume we have Ta and let 6a be a name such that

IkP(y)ra "a witnesses (Oa is y + 1 strategically closed".

(We may assume if I plays only l's, then II 's response with &a is 1.) Now let Ta+ I((PO, qo), *--) = (T(pa , ...), da(qO ...)). This completes the proof of the claim. O

It is easy to see that VP l= ZFC (see [J], p. 196). Also VP l= "VcardK ElK ": Let G be generic for P. Assume V[G] I= "K is a card ". So V[G [ K] I= "K is

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2870 DOUGLAS BURKE

a card". Hence QK names BK in V[G r K]. Let K+ be the successor of K in V[G r K]. So

V[G r K + 1] k "K and K+ are still cardinals and lK".

Since in V[G r K + 1], (K + 1) - (K+) , P(K + 1) adds no new K+ sequences to V[G r K + 1]. Hence v[G] I= "K &K+ are cardinals and ElK "-

Finally, suppose V l= K is superstrong. Let j: V -* M witness this (so Vj(K) c M). Let G c JP be generic. So V[G] = V[G1][H1][H2] where GI c P I K, H1 C 1P(K) r j(K) and H2 C P(j1(K)) come from G. Let H1 = {p E HIlp ends in a tail of 1's}.

Claim. G1 * H1 is generic for j(IP r K) over M.

Proof of Claim. Suppose D c j(P r K) iS dense and in M. Since M l= "j(K) is Mahlo" (it is superstrong), there is an inaccessible A < j(K) such that {p E P f Alp-(I, ...) E D} is dense in IP r A. (Note that (IP r A)M - IP r A.) So there is a p E G [ A such that p-(I, ... ) E D. Hence p(,...) E D n (G1 *H1). This completes the proof of the claim. o

Hence we can extend j to an elementary embedding]: V[G1] M[G1 ][H1 ] Now let E be the (K, j(K)) extender derived from ]. So we get the following commutative diagram:

V[G1I] ) M[G, ][RI

iE\ k

Ult(V[G1], E)

Note that iE(K) = j(K), cp(k) > j(K) and M[G1][H1] 'j(K) Ult(V[G1], E). Since V[G] adds no new K sequences (of ordinals) to V[G1], Ult(V[G], E) makes sense and is well founded. Note that V[G1] 1= K is strongly inaccessible and so V[G] -K+1 V[G1] and therefore Ult(V[G], E) j(K) Ult(V[GI], E)

(and iVj(G](K) = j(K) also). We now show that V[G] -j(K) M[G1][H1] and so V[G] I= K is superstrong. Clearly M[Gl][Hl]j(K) c V[G]. So suppose x E (V[G])j(K) . We may assume that x c A where A < j(K) is a cardinal (j(K) is a strong limit cardinal in V[G] since it is superstrong in M and therefore a limit of Mahlos in V). So x E V[G r A]]. Hence x E Vj(K)[G r A] c M[G1][H1]. El

This theorem shows that we cannot prove there is a successor cardinal K+

such that wJ1 -* K+ from a superstrong cardinal. Does a supercompact cardinal imply the existence of such a K+ ?

ACKNOWLEDGMENT

The author thanks Ernest Schimmerling and James Cummings for helpful discussions.

REFERENCES

[B] J. Baumgartner, Iterated forcing (A. R. D. Mathis, ed.), Surveys in Set Theory, Cambridge Univ. Press, Cambridge, 1983.

[F] M. Foreman, Potent axiomns, Trans. Amer. Math. Soc. 294 (1986), 1-28.

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GENERIC EMBEDDINGS AND THE FAILURE OF BOX 2871

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Notes in Math., vol. 669, Springer-Verlag, Berlin and New York, 1978, pp. 99-275. [LMS] J.-P. Levinski, M. Magidor, and S. Shelah, Changs conjecture for Rc,+1 , Israel J. Math. 69

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vol. 3, Springer-Verlag, Berlin, 1994. [S] E. Schimmerling, Combinatorial principles in the core model for one Woodin cardinal, Ph.D.

thesis, UCLA, 1992. [T] J. Tryba, No Jdnsson filters over R,,, J. Symbolic Logic 52 (1987), 5 1-53. [W88] H. Woodin, Supercompact cardinals, sets of reals and weakly homogeneous trees, Proc. Nat.

Acad. Sci. U.S.A. 85 (1988), 6587-6591.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NORTH TEXAS, DENTON, TEXAS 76203 Current address: Department of Mathematical Sciences, University of Nevada, Las Vegas,

Nevada 89154 E-mail address: dburke~nevada. edu

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