Arthur W. Apter and Saharon Shelah- "Menas' Result is Best Possible"

8/3/2019 Arthur W. Apter and Saharon Shelah- "Menas' Result is Best Possible"

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Menas Result is Best Possible

by

Arthur W. Apter*

Department of Mathematics

Baruch College of CUNYNew York, New York 10010

and

Saharon Shelah**


The Hebrew University

Jerusalem, Israel

and


Rutgers University

New Brunswick, New Jersey 08904

December 11, 1995

Abstract: Generalizing some earlier techniques due to the second author, we show that

Menas theorem which states that the least cardinal which is a measurable limit of

supercompact or strongly compact cardinals is strongly compact but not 2 supercompact

is best possible. Using these same techniques, we also extend and give a new proof of a

theorem of Woodin and extend and give a new proof of an unpublished theorem due to

the first author.

*The research of the first author was partially supported by PSC-CUNY Grant 662341 and

a salary grant from Tel Aviv University. In addition, the first author wishes to thank the

Mathematics Departments of Hebrew University and Tel Aviv University for the hospitalityshown him during his sabbatical in Israel.

**Publication 496. The second author wishes to thank the Basic Research Fund of the

Israeli Academy of Sciences for partially supporting this research.

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0 Introduction and Preliminaries

It is well known that if is 2 supercompact, then is quite large in both size and

consistency strength. As an example of the former, if is 2 supercompact, then has a

normal measure concentrating on measurable cardinals. The key to the proof of this fact

and many other similar ones is the existence of an elementary embedding j : V M with

critical point so that M2

M. Thus, if 2 > +, one can ask whether must be large

in size if is merely supercompact for some < < 2.

A natural question of the above venue to ask is whether a cardinal can be both the

least measurable cardinal and supercompact for some < < 2 if 2 > +. Indeed,

the first author posed this very question to Woodin in the spring of 1983. In response,

using Radin forcing, Woodin (see [CW]) proved the following

Theorem. Suppose V |= ZFC + GCH + < are such that is + supercompact and

is regular. There is then a generic extension V[G] so that V[G] |= ZFC + 2 = +

is supercompact for all regular < + is the least measurable cardinal.

The purpose of this paper is to extend the techniques of [AS] and show that they can

be used to demonstrate that Menas result of [Me] which says that the least measurable

cardinal which is a limit of supercompact or strongly compact cardinals is strongly

compact but not 2 supercompact is best possible. Along the way, we generalize and

strengthen Woodins result above, and we also produce a model in which, on a proper

class, the notions of measurability, supercompactness, and strong compactness are all

the same. Specifically, we prove the following theorems.

Theorem 1. Suppose V |= ZFC + GCH + < are such that is < supercompact,

> + is a regular cardinal which is either inaccessible or is the successor of a cardinal

of cofinality > , and h : is a function so that for some elementary embeddingj : V M witnessing the < supercompactness of , j(h)() = . There is then a

cardinal and cofinality preserving generic extension V[G] |= ZFC + For every inaccessible

< and every cardinal [, h()), 2 = h() + For every cardinal [, ), 2 =

+ is < supercompact + is the least measurable cardinal.

Theorem 2. Let be a (class) function such that for any infinite cardinal, () > + is a

regular cardinal which is either inaccessible or is the successor of a cardinal of cofinality > ,

(0) = 0, and () is below the least inaccessible > if is singular. Suppose V |= ZFC

+ GCH + A is a proper class of cardinals so that for each A, h : is a functionand j : V M is an elementary embedding witnessing the < () supercompactness

of with j(h)() = () < for the least element of A > . There is then a

cardinal and cofinality preserving generic extension V[G] |= ZFC + 2 = () if A

and [, ()) is a cardinal + There is a proper class of measurable cardinals + [ is

measurable iff is < () strongly compact iff is < () supercompact] + No cardinal

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is () strongly compact.

Theorem 3. Suppose V |= ZFC + GCH + is the least supercompact limit of super-

compact cardinals + > + is a regular cardinal which is either inaccessible or is the

successor of a cardinal of cofinality > and h : is a function so that for some ele-

mentary embedding j : V M witnessing the < supercompactness of , j(h)() = .

There is then a generic extension V[G] |= ZFC + For every cardinal < which is an

inaccessible limit of supercompact cardinals and every cardinal [, h()), 2

= h() +For every cardinal [, ), 2 = + is < supercompact + < [ is strongly

compact iff is supercompact] + is the least measurable limit of strongly compact or

supercompact cardinals.

Let us take this opportunity to make several remarks concerning Theorems 1, 2, and

3. Note that we use a weaker supercompactness hypothesis in the proof of Theorem 1 than

Woodin does in the proof of his Theorem. Also, since Woodin uses Radin forcing in the

proof of his Theorem, cofinalities are not preserved in his generic extension (cardinals may

or may not be preserved in Woodins Theorem, depending upon the proof used), although

they are in our Theorem 1 when the appropriate forcing conditions are used. Further,

in Theorem 2, the model constructed will be so that on the proper class composed of all

cardinals possessing some non-trivial degree of strong compactness or supercompactness,

is strongly compact iff is supercompact, although there wont be any (fully) strongly

compact or (fully) supercompact cardinals in this model. (This is the generalized version

of the theorem that inspired the work of this paper and of [AS]. The original theorem was

initially proven using an iteration of Woodins version of Radin forcing used to prove his

above mentioned Theorem.) Finally, Theorem 3 illustrates the flexibility of our forcing

as compared to Radin forcing. Since iterating a Radin, Prikry, or Magidor [Ma1] forcing

(for changing the cofinality of to some uncountable < ) above a strongly compact

or supercompact cardinal destroys the strong compactness or supercompactness of ,

it is impossible to use any of these forcings in the proof of Theorem 3. Our forcing for

Theorem 3, however, has been designed so that, if is a supercompact cardinal which is

Laver [L] indestructible, then we can force above , destroy measurability, yet preserve the

supercompactness of .

The structure of this paper is as follows. Section 0 contains our introductory comments

and preliminary material concerning notation, terminology, etc. Section 1 defines and

discusses the basic properties of the forcing notion used in the iterations we employ toconstruct our models. Section 2 gives a proof of Theorem 1. Section 3 contains a proof of

Theorems 2 and 3. Section 4 concludes the paper by giving an alternate forcing that can

be used in the proofs of Theorems 1 and 2.

Before beginning the material of Section 1, we briefly mention some preliminary in-

formation. Essentially, our notation and terminology are standard, and when this is not

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the case, this will be clearly noted. For < ordinals, [, ], [, ), (, ], and (, ) are

as in standard interval notation. If f is the characteristic function of a set x , then

x = { < : f() = 1}. If < , f is a characteristic function having domain , and

f is a characteristic function having domain , we will when the context is clear abuse

notation somewhat and write f f, f = f, and f = f when we actually mean that the

sets defined by these functions satisfy these properties.

When forcing, q p will mean that q is stronger than p. For P a partial ordering, a

formula in the forcing language with respect to P, and p P, p will mean p decides .

For G V-generic over P, we will use both V[G] and VP to indicate the universe obtained

by forcing with P. If x V[G], then x will be a term in V for x. We may, from time

to time, confuse terms with the sets they denote and write x when we actually mean x,

especially when x is some variant of the generic set G, or x is in the ground model V.

If is a cardinal and P is a partial ordering, P is -closed if given a sequence p :

< of elements of P so that < < implies p p (an increasing chain of length

), then there is some p P (an upper bound to this chain) so that p p for all < .

P is < -closed if P is -closed for all cardinals < . P is (, )-distributive if for anysequence D : < of dense open subsets ofP, D =


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the supercompactness of are presumed to come from some fine, -complete, normal

ultrafilter U over P() = {x : |x| < }, and all elementary embeddings witnessing

the < supercompactness of for a limit cardinal are presumed to be generated by the

appropriate system U : < of ultrafilters over P() for [, ) a cardinal. Also,

where appropriate, all ultrapowers will be confused with their transitive isomorphs.

Finally, we remark that a good deal of the notions and techniques used in this paper

are quite similar to those used in [AS]. Since we desire this paper to be as comprehensible

as possible, regardless if readers have read [AS], many of the arguments of [AS] will be

repeated here in the appropriately modified form.

1 The Forcing Conditions

In this section, we describe and prove the basic properties of the forcing conditions we

shall use in our later iteration. Let < , > + be regular cardinals in our ground model

V, with inaccessible and either inaccessible or the successor of a cardinal of cofinality

> . We assume throughout this section that GCH holds for all cardinals in V,

and we define three notions of forcing. Our first notion of forcing P0, is just the standard

notion of forcing for adding a non-reflecting stationary set of ordinals of cofinality to .

Specifically, P0, = {p : For some < , p : {0, 1} is a characteristic function of

Sp, a subset of not stationary at its supremum nor having any initial segment which is

stationary at its supremum, so that Sp implies > and cof() = }, ordered by

q p iff q p and Sp = Sq sup(Sp), i.e., Sq is an end extension of Sp. It is well-known

that for G V-generic over P0, (see [Bu] or [KiM]), in V[G], since GCH holds in V for all

cardinals , a non-reflecting stationary set S = S[G] = {Sp : p G} of ordinals

of cofinality has been introduced, the bounded subsets of are the same as those in V,

and cardinals, cofinalities, and GCH at cardinals have been preserved. It is also

virtually immediate that P

0

, is -directed closed.Work now in V1 = V

P0,, letting S be a term always forced to denote the above set S.

P2,[S] is the standard notion of forcing for introducing a club set C which is disjoint to

S (and therefore makes S non-stationary). Specifically, P2,[S] = {p : For some successor

ordinal < , p : {0, 1} is a characteristic function of Cp, a club subset of , so that

Cp S = }, ordered by q p iff Cq is an end extension of Cp. It is again well-known (see

[MS]) that for H V1-generic over P2,[S], a club set C = C[H] = {Cp : p H} which

is disjoint to S has been introduced, the bounded subsets of are the same as those in V1,

and cardinals, cofinalities, and GCH for cardinals have been preserved.

More will be said about P0, and P2,[S] in Lemmas 4, 6, and 7. In the meantime, be-fore defining in V1 the partial ordering P

1,[S] which will be used to destroy measurability,

we first prove two preliminary lemmas.

Lemma 1. P0,

(S), i.e., V1 |= There is a sequence x : S so that for each

S, x is cofinal in , and for any A [], { S : x A} is stationary.

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Proof of Lemma 1: Since GCH holds in V for cardinals and V and V1 contain

the same bounded subsets of , we can let y : < V be a listing of all elements

x ([]

)V

= ([]

)V1

so that each x []

appears on this list times at ordinals of

cofinality , i.e., for any x [], = sup{ < : cof() = and y = x}. This then

allows us to define x : S by letting x be y for the least S ( + 1) so that

y and y is unbounded in . By genericity, each x is well-defined.

Let now p P0

,

be so that p A [] and K is club. We show that for some

r p and some < , r K S and x A. To do this, we inductively define an

increasing sequence p : < of elements of P0, and increasing sequences : <

and : < of ordinals < so that 0 0 1 1

( < ). We begin by letting p0 = p and 0 = 0 = 0. For = + 1 < a successor,

let p p and , max(, , sup(dom(p))) + 1 be so that p A

and K. For < a limit, let p =


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hypothesis on h. If2 = , then (x1 h(1))(x2 h(2)) = (x1 h(1))x2 =

by the definition of h(1). The sequence x h() : S is thus as desired.

If is a limit ordinal, then as S is non-stationary at its supremum nor has any

initial segment which is stationary at its supremum, we can let : < cof() be a

strictly increasing, continuous sequence having sup so that for all < cof(), S.

Thus, if S , then { : < } is bounded in , meaning we can find some

largest so that < . It is also the case that < +1. This allows us to define

h() = max({h+1(), }) for the just described. It is still the case that h() < .

And, if 1, 2 (, +1), then (x1 h(1)) (x2 h(2)) (x1 h+1(1))

(x2 h+1(2)) = by the definition of h+1 . If 1 ( , +1), 2 (, +1) with

< , then (x1 h(1)) (x2 h(2)) x1 (x2 ) 1 1 +1 = .

Thus, the sequence x h() : S is again as desired. This proves Lemma 2.

Lemma 2

At this point, we are in a position to define in V1 the partial ordering P1,[S] which will

be used to destroy measurability. P1,[S] is the set of all 5-tuples w,, r,Z, satisfying

the following properties.

1. w is so that |w| = .

2. < .

3. r = ri : i w is a sequence of functions from to {0, 1}, i.e., a sequence of subsets

of .

4. Z is a function so that:

a) dom(Z) {x : S} and range(Z) {0, 1}.

b) If z dom(Z), then for some y [w], y z and z y is bounded in the so that

z = x.

5. is a function so that:

a) dom() = dom(Z).

b) Ifz dom(), then (z) is a closed, bounded subset of such that if is inaccessible,

(z), and is the th element of z, then w, and for some w z,

r() = Z(z).

Note that the definitions of Z and imply |dom(Z)| = |dom()| .

The ordering on P1,[S] is given by w1, 1, r1, Z1, 1 w2, 2, r2, Z2, 2 iff the

following hold.

1. w1 w2.

2. 1

2

.3. Ifi w1, then r1i r

2i and |{i w

1 : r2i |(2 1) is not constantly 0}| < .

4. Z1 Z2.

5. dom(1) dom(2).

6. Ifz dom(1), then 1(z) is an initial segment of 2(z) and |{z dom(1) : 1(z) =

2(z)}| < .

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At this point, a few intuitive remarks are in order. If is measurable, then must

carry a normal measure. The forcing P1,[S] has specifically been designed to destroy

this fact. It has been designed, however, to destroy the measurability of as lightly as

possible, making little damage, assuming is < supercompact. Specifically, if is <

supercompact, then the non-reflecting stationary set S, having been added to , does not

kill the < supercompactness of by itself. The additional forcing P1,[S] is necessary

to do the job and has been designed so as not only to destroy the < supercompactness

of but to destroy the measurability of as well. The forcing P1,[S], however, has been

designed so that if necessary, we can resurrect the < supercompactness of by forcing

further with P2,[S].

Lemma 3. VP1,[S]

1 |= is not measurable.

Proof of Lemma 3: Assume to the contrary that VP1,[S]

1 |= is measurable. Let

p D is a normal measure over . We show that p can be extended to a condition q so

that q D is non-normal, an immediate contradiction.

We use a -system argument to establish this. First, for G1 V1-generic over P1,[S]and i < , let ri = {r

pi : p = w

p, p, rp, Zp, p G1[rpi r

p]}. An easy density

argument shows P1,

[S]ri : {0, 1} is a function whose domain is all of . Thus, we

can let ri = { < : ri() = } for {0, 1}.

For each i < , pick pi = wpi , pi, rpi , Zpi , pi p so that pi r

(i)i D for some

(i) {0, 1}. This is possible since P1,

[S]For each i < , ri0 r

i1 = and r

0i r

1i = .

Without loss of generality, by extending pi if necessary, since clause 4b) of the definition

of the forcing implies |dom(Zpi)| , we can assume that i wpi and z wpi for every

z dom(Zpi). Thus, since each wpi [] +, is either inaccessible or is the

successor of a cardinal of cofinality > , and GCH holds in V1 for cardinals , we can

find some A [] so that {wpi : i A} forms a -system, i.e., so that for i = j A,

wpi wpj is some constant value w which is an initial segment of both. (Note we can

assume that for i A, wi i = w, and for some fixed () {0, 1}, for every i A,

pi r()i D.) Also, by GCH in V1 for cardinals , |[P(w)]

| = |[+]

| = +.

Therefore, since |dom(Zpi)| for each i < and > +, we can assume in addition

that for all i A, dom(Zpi) P(w) = dom(pi) P(w) is some constant value Z. Hence,

since each Zpi is a function from a set of cardinality into {0, 1}, each pi is a function

from a set of cardinality into []

+

, GCH in V1 forcardinals allows us to assume that for i = j A, Zpi |Z = Zpj |Z and pi |Z = pj |Z.

Further, since each pi < , we can assume that pi is some constant 0 for i A.

Then, since any rpi = rpij : j wpi for i A is composed of a sequence of functions

from 0 to 2, 0 < , and |w| , GCH in V1 for cardinals again allows us to

assume that for i = j A, rpi |w = rpj |w. And, since i wpi , we know that we can also

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assume (by thinning A if necessary) that B = {sup(wpi) : i A} is so that i < j A

implies i sup(wpi) < min(wpj w) sup(wpj ). We know in addition by the choice of

X = x : S that for some S, x A. Let x = {i : < }.

We are now in a position to define the condition q referred to earlier. We proceed by

defining each of the five coordinates of q. First, let wq =


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any z dom(0), 0(z) contains no new element which is inaccessible, then p as just

defined is a condition. Therefore, we define a new partial ordering on P1,[S] by

p1 = wp1 , p1 , rp1 , Zp1 , p1 p2 = w

p2 , p2 , rp2, Zp2 , p2 iff p1 = p2 or p1 < p2and for z dom(p1), i f p1(z) = p2(z), then < max(p2(z)). If the sequence

pi = wpi , pi , rpi, Zpi , pi : i < is a directed sequence of conditions in P1,[S] with

respect to , then since sup( i of cofinality , the up-

per bound p as defined earlier exists. Further, if p1

= wp1 , p1 , rp1 , Zp1 , p1 p2

=

wp2 , p2 , rp2 , Zp2 , p2, for any z dom(p1) so that p1(z) = p2(z), we can define

a function having domain p2 so that |(dom(p2) dom(p1)) = p2 |(dom(p2)

dom(p1)) and such that for z dom(p1), (z) = p2(z) {z}, where z < is the

least cardinal above max(max(p2(z)), ). If q = wp2 , p2 , rp2 , Zp2 , , then q P1,[S]

is a valid condition so that p1 q and p2 q. This easily implies that G is (appropri-

ately) generic with respect to P1,[S], , an ordering that is +-directed closed, iff G

is (appropriately) generic with respect to P1,[S], , i.e., forcing with P1,[S],

and

P1,[S], are equivalent. This key observation will be critical in the proof of Theorem

3.Recall we mentioned prior to the proof of Lemma 3 that P1,[S] is designed so that

a further forcing with P2,[S] will resurrect the < supercompactness of , assuming

the correct iteration has been done. That this is so will be shown in the next section.

In the meantime, we give an idea of why this will happen by showing that the forcing

P0, (P1,[S] P

2,[S]) is rather nice. First, for 0 1 regular cardinals, let K(0, 1) =

{w,, r : w 1 is so that |w| = 0, < 0, and r = ri : i w is a sequence of

functions from to {0, 1}}, ordered by w1, 1, r1 w2, 2, r2 iff w1 w2, 1 2,

r1i r2i if i w

1, and |{i w1 : r2i |(2 1) is not constantly 0}| < 0. Given this

definition, we now have the following lemma.Lemma 4. P0, (P

1,[S] P

2,[S]) is equivalent to C() C(

+, ) K(, ).

Proof of Lemma 4: Let G be V-generic over P0,(P1,[S]P

2,[S]), with G

0,, G

1,, and

G2, the projections onto P0,, P

1,[S], and P

2,[S] respectively. Each G

i, is appropriately

generic. So, since P1,[S] P2,[S] is a product in V[G

0,], we can rewrite the forcing in

V[G0,] as P2,[S] P

1,[S] and rewrite V[G] as V[G

0,][G

2,][G

1,].

It is well-known (see [MS]) that the forcing P0, P2,[S] is equivalent to C(). That

this is so can be seen from the fact that P0, P2,[S] is non-trivial, has cardinality ,

and is such that D = {p,q P0

, P2

,[

S] : For some , dom(p) = dom(q) = + 1,p / S, and q C} is dense in P0, P

2,[S] and is < -closed. This easily

implies the desired equivalence. Thus, V and V[G0,][G2,] have the same cardinals and

cofinalities, and the proof of Lemma 4 will be complete once we show that in V[G0,][G2,],

P1,[S] is equivalent to C(+, ) K(, ).

To this end, working in V[G0,][G2,], let R = {Z : Z is a function from {x : S}

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into {0, 1} so that |dom(Z)| }, ordered by inclusion. Since |{x : S}| = and R is

-closed, it is clear R is equivalent to C(+, ). Further, the following facts are easy to see.

1. Ifp = wp, p, rp, Zp, p P1,[S], then Zp R.

2. If p1 = wp1 , p1 , rp1 , Zp1 , p1, p2 = wp2 , p2 , rp2, Zp2 , p2 are so that p1, p2

P1,[S] and p1 p2, then Zp1 Zp2 .

3. Ifp1 = wp1 , p1 , rp1 , Zp1 , p1 P1,[S] is so that Z

p1 Zp2 for some Zp2 R, then

there exists p2 P1,[S] with p1 p2, p2 = w

p2 , p2 , rp2 , Zp2 , p2.

From these three facts, it then easily follows that H = {Z R : p G1,[Z = Zp]}

is V[G0,][G2,]-generic over R. This means we can rewrite P

1,[S] in V[G

0,][G

2,] as

R (P1,[S]/R), which is isomorphic to C(+, ) (P1,[S]/R). We will thus be done if

we can show in V[G0,][G2,][H] (which has the same cardinals and cofinalities as V and

V[G0,][G2,]) that P

1,[S]/R is equivalent to K(, ).

Working now in V[G0,][G2,][H], we first note that as S is in V[G

0,][G

2,] a non-

stationary set all of whose initial segments are non-stationary, by Lemma 2, for the sequence

x : S, there must be a sequence y : S V[G0,][G

2,] V[G

0,][G

2,][H]

so that for every S, y x , x y is bounded in , and if 1

= 2

S, then

y1 y2 = . Given this fact, it is easy to observe that P1 = {w,, r, P1,[S]/R :

For every S, either y w or y w = } is dense in P1,[S]/R. To show this, given

w,, r, P1,[S]/R, r = ri : i w, let Yw = {y y : S : y w = }. As

|w| and y1 y2 = for 1 = 2 S, |Yw| . Hence, as |y| = < for y Yw,

|w| for w = w (Yw). This means w, , r, for r = ri : i w

defined by

ri = ri if i w and ri is the empty function if i w

w is a well-defined condition

extending w,, r, . Thus, P1 is dense in P1,[S]/R, so to analyze the forcing properties

of P1,[S]/R, it suffices to analyze the forcing properties of P1.

For S, let Q = {w,, r, P1

: w = y}, and let Q

= {w,, r, P1

:w

Sy}. Let Q be those elements of

S

Q Q of support so that for p =

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dom(pi) dom(pj ) = for i < j < (since if z dom(pi), z = x for some S,

meaning wpi = y by the definitions ofP1,[S], P

1,[S]/R, and Q), and dom(

p) = (since

for every S, wp y = , y x, and x y is bounded in ), conditions 3) and 4)

above on the definition ofQ show the function F(p) =

i


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V[G0,][G2,][H][I], i.e., since T is -directed closed, RT is isomorphic to K(, )C(),

i.e., Q, is isomorphic to K(, ). Since |{x : S}| = , conditions 3) and 4) on

the definition of Q ensure the ordering composed of those elements of

S

Q having

support ordered by Q |

S

Q is isomorphic to K(, ). Then, ifw,, r, Q, since

we have already observed dom() = , Q can easily be seen to be isomorphic to K(, ).

Putting all of this together yields Q ordered by Q is isomorphic to K(, ). Thus, since

Q, Q and Q, Q are forcing equivalent, this proves Lemma 4.Lemma 4

We remark that in what follows, it will frequently be the case that a partial ordering

P is forcing equivalent to a partial ordering P in the sense that a generic object for one

generates a generic object for the other. Under these circumstances, we will often abuse

terminology somewhat by saying that either P or P satisfies a certain chain condition, a

certain amount of closure, etc. when this is true of at least one of these partial orderings. We

will then as appropriate further compound the abuse by using this property interchangeably

for either partial ordering.

As the definition of K(0, 1) indicates, without the last coordinates Zp and p of a

condition p P1,[S] and the associated restrictions on the ordering, P1,[S] is essentially

K(, ). These last coordinates and change in the ordering are necessary to destroy the

measurability of when forcing with P1,[S]. Once the fact S is stationary has been

destroyed by forcing with P2,[S], Lemma 4 shows that these last two coordinates Zp and

p of a condition p P1,[S] can be factored out to produce the ordering C(+, ) K(, ).

K(, ), although somewhat similar in nature to C(, ) (e.g., K(, ) is -directed

closed), differs from C(, ) in a few very important aspects. In particular, as we shall see

presently, forcing with K(, ) will collapse +. An indication that this occurs is provided

by the next lemma.

Lemma 5. K(, ) satisfies ++-c.c. whenever 2 = +.

Proof of Lemma 5: Suppose w, , r : < ++ is a sequence of ++ many

incompatible elements ofK(, ). Since > +, each w [] , we can find some A ++,

|A| = ++ so that {w : A} forms a -system, i.e., {w : A} is so that for

1 = 2 A, w1 w2 is some constant value w. Since each < , let B A,

|B| = ++

be so that for 1 = 2 B, 1

= 2

= . Since for B, r

|w is a sequenceof functions from < into {0, 1}, the facts |w| and 2 = + together imply there

is C B, |C| = ++ so that for 1 = 2 C, r1 |w = r2 |w. It is thus the case that

w, , r : C is now a sequence of ++ many compatible elements of K(, ), a

contradiction. This proves Lemma 5.

Lemma 5

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It is clear from Lemmas 4 and 5 and the definition of K(, ) that since GCH holds in

V for cardinals , P0,(P1,[S]P

2,[S]), being equivalent to C()C(

+, )K(, ),

preserves cardinals and cofinalities and ++, has a dense subset which is -directed

closed, satisfies +-c.c., and is so that VP0,(P

1,[S]P

2,[S]) |= For every cardinal

[, ), 2 = . Our next lemma shows that the forcing P0, P1,[S] is also rather nice,

with the exception that it collapses +. By Lemma 4, this has as an immediate consequence

that the forcing K(, ) also collapses +.

Lemma 6. P0, P1,[S] preserves cardinals and cofinalities and

++, collapses+,

is < -strategically closed, satisfies +-c.c., and is so that VP0,P

1,[S] |= 2 = for all

cardinals [, ).

Proof of Lemma 6: Let G = G0,G1, be V-generic over P

0,P

1,[S], and let G

2, be

V[G]-generic over P2,[S]. Thus, G G2, = G is V-generic over P

0, (P

1,[S] P

2,[S]) =

P0, (P1,[S] P

2,[S]).

By Lemmas 4 and 5 and the remarks immediately following, since GCH holds in

V for cardinals , V[G] |= 2

= for all cardinals [, ) and has the samecardinals and cofinalities as V and ++. Hence, since V[G] V[G], forcing with

P0, P1,[S] over V preserves cardinals and cofinalities and

++ and is so that

VP0,P

1,[S] |= 2 = for all cardinals [, ).

We now show forcing with P0, P1,[S] over V collapses

+. Since forcing with P0,over V collapses no cardinals and preserves GCH for cardinals , we assume without

loss of generality our ground model is V[G0,] = V1.

Using the notation of Lemma 3, i.e., that for i < , ri is a term for {rpi : p =

wp, p, rp, Zp, p G1,[rpi r

p]}, we can now define a term for < by =

min({ : < + and is the order type of {i < : ri() = 1}). To see that is well-

defined, let p = wp, p, rp, Zp, p P1,[S] be a condition. Without loss of generality,

we can assume that p > . Further, since |wp| = , sup(wp +) < +, so we can let

< + be so that > sup(wp +). We can then define q p, q = wq, q, rq, Zq, q

by letting q = p, Zq = Zp, q = p, wq = wp [, + ), and rq by rq = rqi : i wq

where rqi is rpi if i w

p, and rqi is the function having domain q which is constantly 1

if i wq wp. Clearly, q is well-defined, and p q. Also, by the definition of q, since

< q, q i [, + )[ri() = 1]. This means q + , so is well-defined.

We will be done if we can show P

1

,[S]

: < is unbounded in +. Assume

now towards a contradiction that p = wp, p, rp, Zp, p is so that psup( : .

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To see that the claim is true, let s q, s = ws, s, rs, Zs, s be so that s > q.

By clause 3 in the definition of on P1,[S], s |{i < : ri() = 1}| < . This means

q > , thus proving our claim and showing that + is collapsed.

We next show the < -strategic closure of P0, P1,[S]. We first note that as (P

0,

P1,[S]) P2,[S] = P

0, (P

1,[S] P

2,[S]) has by Lemma 4 a dense subset which is < -

closed, the desired fact follows from the more general fact that if P Q is a partial ordering

with a dense subset R so that R is < -closed, then P is < -strategically closed. To show

this more general fact, let < be a cardinal. Suppose I and II play to build an increasing

chain of elements of P, with p : + 1 enumerating all plays by I and II through

an odd stage + 1 and q : < + 1 and is even or a limit ordinal enumerating a

set of auxiliary plays by II which have been chosen so that p, q : < + 1 and

is even or a limit ordinal enumerates an increasing chain of elements of the dense subset

R P Q. At stage + 2, II chooses p+2, q+2 so that p+2, q+2 R and so that

p+2, q+2 p+1, q; this makes sense, since inductively, p, q R P Q, so

as I has chosen p+1 p, p+1, q P Q. By the < -closure of R, at any limit

stage , II can choose p, q so that p, q is an upper bound to p, q : <

and is even or a limit ordinal. The preceding yields a winning strategy for II, so P is

< -strategically closed.

Finally, to show P0, P1,[S] satisfies

+-c.c., we simply note that this follows from

the general fact about iterated forcing (see [Ba]) that if P Q satisfies +-c.c., then P

satisfies +-c.c. (Here, P = P0, P1,[S] and Q = P

2,[S].) This proves Lemma 6.

Lemma 6

We remark that P0,

P1,[S] is ++-c.c.. Otherwise, if A = p : <

++

were a size ++ antichain of elements of P1,[S] in V[G0,], then (using the notation

of Lemma 4) since P1,[S] is isomorphic to C(

+

, ) (P1,[S]/R) and P

1,[S]/R has a

dense subset which is isomorphic to Q, Q, without loss of generality, A can be taken

as an antichain in Q, Q. Since Q Q, A must also be an antichain with respect

to Q, and as V[G0,], V[G

0,][G

2,], and V[G

0,][G

2,][H] all have the same cardinals,

A must be a size ++ antichain with respect to Q in V[G0,][G

2,][H]. The fact that

V[G0,][G2,][H] |= 2

= + and Q, Q is isomorphic to K(, ) then tells us that A is

isomorphic to a size ++ antichain with respect to K(, ) in V[G0,][G2,][H]. Lemma 5,

which says that K(, ) is ++-c.c. in any model in which 2 = +, now yields an immediate

contradiction.

We conclude this section with a lemma that will be used later in showing that it ispossible to extend certain elementary embeddings witnessing the appropriate degree of

supercompactness.

Lemma 7. For V1 = VP0,, the modelsV

P1,[S]P2,[S]

1 and VP1,[S]

1 contain the same <

sequences of elements of V1.

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Proof of Lemma 7: By Lemma 4, since P0, P2,[S] is equivalent to the forcing C()

and V VP0, VP

0,P

2,[S], the models V, VP

0,, and VP

0,P

2,[S] all contain the same

< sequences of elements of V. Thus, since a < sequence of elements of V1 = VP0,

can be represented by a V-term which is actually a function h : V for some < ,

it immediately follows that VP0, and VP

0,P

2,[S] contain the same < sequences of

elements of VP0,.

Let now f : V1 for < be so that f (VP0,P

2,[S])P

1,[S] = V

P1,[S]P2,[S]

1

,

and let g : V1, g VP0,P

2,[S] be a term for f. By the previous paragraph, g VP

0,.

Since Lemma 5 shows that P1,[S] is ++-c.c. in VP

0,P

2,[S] and ++ , for each < ,

the antichain A defined in VP0,P

2,[S] by {p P1,[S] : p decides a value for g()} is

so that VP0,P

2,[S] |= |A| < . Hence, by the preceding paragraph, since A is a

set of elements of VP0,, A V

P0, for each < . Therefore, again by the preceding

paragraph, the sequence A : < VP0,. This just means that the term g VP

0,

can be evaluated in VP1,[S]

1 , i.e., f VP1,[S]

1 . This proves Lemma 7.

Lemma 7

2 The Proof of Theorem 1We turn now to the proof of Theorem 1. Recall that we are assuming our ground

model V |= ZFC + GCH + is < supercompact + > + is regular and is either

inaccessible or is the successor of a cardinal of cofinality > + h : is so that

for some elementary embedding j : V M witnessing the < supercompactness of

, j(h)() = . By reflection, we can assume without loss of generality that for every

inaccessible < , h() > + and h() is regular. Given this, we are now in a position to

define the partial ordering P used in the proof of Theorem 1. We define a stage Easton

support iteration P = P, Q : < , and then define P = P+1 = P Q for a

certain partial ordering Q definable in VP . The definition is as follows:

1. P0 is trivial.

2. Assuming P has been defined for < , let be so that is the least cardinal


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all cardinals [, ), 2 = , and no cardinal < is measurable.

Proof of Lemma 8: We first note that Easton support iterations of-strategically closed

partial orderings are -strategically closed for any regular cardinal. The proof is via

induction. If R1 is -strategically closed and R1R2 is -strategically closed, then let

p R1 be so that p g is a strategy for player II ensuring that the game which produces

an increasing chain of elements of R2 of length can always be continued for . If

II begins by picking r0 = p0, q0 R1 R2 so that p0 p has been chosen according tothe strategy f for R1 and p0 q0 has been chosen according to g, and at even stages

+ 2 picks r+2 = p+2, q+2 so that p+2 has been chosen according to f and is so

that p+2 q+2 has been chosen according to g, then at limit stages , the chain

r0 = p0, q0 r1 = p1, q1 r = p, q ( < ) is so that II can find an

upper bound p for p : < using f. By construction, p q : < is so that at

limit and even stages, II has played according to g, so for some q, p q is an upper

bound to q : < , meaning the condition p, q is as desired. These methods,

together with the usual proof at limit stages (see [Ba], Theorem 2.5) that the Easton

support iteration of -closed partial orderings is -closed, yield that -strategic closure ispreserved at limit stages of any Easton support iteration of -strategically closed partial

orderings. In addition, the ideas of this paragraph will also show that Easton support

iterations of -strategically closed partial orderings are -strategically closed for any

regular cardinal.

Given this fact, it is now easy to prove by induction that VP |= For all inaccessible

< and all cardinals [, h()), 2 = h(), and no cardinal < is measurable.

Given < , we assume inductively that VP |= For all inaccessible < and all

cardinals [, h()), 2 = h(), and no cardinal < is measurable, where is

as in the definition of P. By Lemmas 3 and 6 and the definition of P, since inductively

PGCH holds for all cardinals , VPQ = VP+1 |= For all inaccessible

and all cardinals [, h()), 2 = h(), and no cardinal is measurable.

If now is a limit ordinal, then we know by induction that for all < ,

VP |= For all inaccessible < and all cardinals [, h()), 2 = h(), and no

cardinal < is measurable. If we write P = P R, then by the definition of P, the

proof of Lemma 4, Lemma 6, and the fact contained in the first paragraph of the proof of

this lemma, PR is forcing equivalent to a < -strategically closed partial ordering,

so V

PR

= V

P

|= For all inaccessible < and all cardinals [, h()), 2

= h(),and no cardinal < is measurable. If we let = , since < is arbitrary in the

preceding, it thus follows by the definition of for < that VP |= For all inaccessible

< and all cardinals [, h()), 2 = h(), and no cardinal < is measurable.

The proof of Lemma 8 will be complete once we show VPQ = VP is so that

VP |= For all inaccessible < and all cardinals [, h()), 2 = h(), for all

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cardinals [, ), 2 = , and no cardinal < is measurable. By the last paragraph,

P is inaccessible, and by the definition of P, |P| = , meaning PGCH holds

for all cardinals . Therefore, by Lemma 4 and the definition of Q, VP |= Q

is equivalent to C() C(+, ) K(, ), so VP |= For all inaccessible < and all

cardinals [, h()), 2 = h(), for all cardinals [, ), 2 = , and no cardinal <

is measurable. This proves Lemma 8.

Lemma 8

We now show that the intuitive motivation for the definition of P as set forth in the

paragraph immediately preceding the statement of Lemma 8 actually works.

Lemma 9. For G V-generic over P, V[G] |= is < supercompact.

Proof of Lemma 9: Let j : V M be an elementary embedding witnessing the <

supercompactness of so that j(h)() = . We will actually show that for G = G G

our V-generic object over P = PQ, the embedding j extends to k : V[GG] M[H]

for some H j(P). As j() : < M for every < , this will be enough to allow

for every < the definition of the ultrafilter x U iff j() : < k(x) to be given

in V[G G], thereby showing V[G] |= is < supercompact.

We construct H in stages. In M, as is the critical point of j, j(P Q) = P

R R R

, where R

will be a term for P

0, P

1,[S] (note that as M

, and j(h)() = , R is indeed as just stated), R

will be a term

for the rest of the portion of j(P) defined below j(), and R will be a term for j(Q).

This will allow us to define H as H H H

H

. Factoring G

as G

0, (G

1, G

2,),

we let H = G and H = G

0, G

1,. Thus, H

is the same as G

, except, since

M |= < j() and j(h)() = , we omit the generic object G2,

.

To construct H , we first note that the definition ofP ensures |P| = and, since is

necessarily Mahlo, P is -c.c. As V[G] and M[G] are both models of GCH for cardinals

, the definition of R in M[H] and the remark following Lemma 6 then ensure that

M[H] |= R is a < -strategically closed partial ordering followed by a

++-c.c. partial

ordering and ++ . Since M


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of the dense open subsets of R present in M[H H]. The -strategic closure of R

in both M[H H] and V[G H

] now allows us to meet all of these dense subsets as

follows. Work in V[G H]. Player I picks p D extending sup(q : < ) (initially,

q1 is the trivial condition), and player II responds by picking q p (so q D). By

the -strategic closure of R in V[G H], player II has a winning strategy for this

game, so q : < can be taken as an increasing sequence of conditions with q Dfor < . Clearly, H = {p R

: < [q p]} is our M[H H

]-generic object over

R which has been constructed in V[G H] V[G G

], so H

V[G G

].

By the above construction, in V[G G], the embedding j extends to an embedding

j : V[G] M[H H H

]. We will be done once we have constructed in V[G G

]

the appropriate generic object for R = P0j(),j() (P

1j(),j()[Sj()] P

2j(),j()[Sj()]) =

(P0j(),j() P

2j(),j()[Sj()]) P

1j(),j()[Sj()]. To do this, first rewrite G

as (G

0, G

2,)

G1,. By the nature of the forcings, G0, G

2, is V[G]-generic over a partial ordering

which is (< , )-distributive. Thus, by a general fact about transference of generics

via elementary embeddings (see [C], Section 1.2, Fact 2, pp. 5-6), since j : V[G]

M[H H H

] is so that every element of M[H H

H

] can be written j

(F)(a)with dom(F) having cardinality < , jG0, G

2, generates an M[H H

H

]-generic

set H4.

It remains to construct H5, our M[H H H

H

4]-generic object over

P1j(),j()[Sj()]. To do this, first recall that in M[H H

], as previously noted, R

is

-strategically closed. Since M[H H] has already been observed to be closed under

< sequences with respect to V[GH], and since any sequence of elements for < a

cardinal ofM[HHH

] can be represented, in M[HH

], by a term which is actually

a function f : M[H H], M[H H

H

] is closed under < sequences with

respect to V[GH], i.e., iff : M[HHH ] for < a cardinal, f V[GH],

then f M[H H H

].

Choose in V[G G] an enumeration p : < of G

1,. Adopting the notation

of Lemma 4 and working now in V[G G], first note that by Lemma 4, there is an

isomorphism between P1,[S] and R (P1,[S]/R). Again by Lemma 4, since P

1 is dense

in P1,[S]/R and Q, Q and P1 are isomorphic, there is an isomorphism g0 between a

dense subset of P1,[S] and R Q, Q. Therefore, as R is isomorphic to C(+, ) and

Q, Q is isomorphic to K(, ), we can let g1 : R Q, Q C(

+, ) K(, ) be an

isomorphism. It is then the case that g = g1 g0 is a bijection between a dense subset ofP1,[S] and C(

+, ) K(, ). This gives us a sequence I = g(p) : < of many

compatible elements of C(+, ) K(, ). By Lemma 7, V[G G0, G

1, G

2,] =

V[G G0, G

2, G

1,] = V[G G

] and V[G G

0, G

1,] = V[G H

] have the

same < sequences of elements of V[G G0,] and hence ofV[G H

]. Thus, any <

sequence of elements of M[H H H

] present in V[G G

] is actually an element

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of V[G H] (so M[H H

H

] is really closed under < sequences with respect to

V[G G].)

For (+, ), ifp C(+, ), let p| = {, , p : < }, and ifq K(, ),

q = w,, r, let q| = w ,, r|(w ). Call w the domain ofq. Since C(+,)There

are no new sequences of ordinals, we can assume without loss of generality that for

any condition p = p0, p1 C(+, ) K(, ), p1 is an actual condition and not just a

term for a condition. Thus, for (+, ) and p = p0, p1 C(+, ) K(, ), the

definitions p| = p0|, p1| and I| = {p| : p I} make sense. And, since GCH

holds in V[G G0, G

2,] for cardinals , it is clear V[G G

] |= |I|| + is a regular cardinal which

is either inaccessible or is the successor of a cardinal of cofinality > with () below

the least inaccessible > if is singular and (0) = 0 and that our ground model V is

such that V |= ZFC + GCH + A is a proper class of cardinals so that for each A,

h : is a function and j : V M is an elementary embedding witnessing the

< () supercompactness of with j(h)() = () < for the least element of A

> . Without loss of generality (by cutting off V if necessary), we assume that for each

A, sup({ A : < }) = < and isnt inaccessible if order type({ A : < })

is a limit ordinal.For each A, let P(, ()) be the version of the partial ordering P used in the

proof of Theorem 1 which ensures each inaccessible ((), ) isnt measurable yet

is < () supercompact, i.e., using the notation of Section 2, P(, ()) is the + 1 stage

Easton support iteration P, Q : defined as follows:

1. P0 is trivial.

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2. Assuming P has been defined for < , let be so that is the least cardinal

}

P(, ()) : support(p) is a set}, and T and T are both ordered

componentwise. For each A, the definition of the component partial orderings of Tand T ensures that T is ()

+-c.c. and T is ()+-strategically closed. This allows us

to conclude in the manner of [KiM] that VP |= ZFC.

Lemma 10. VP |= 2 = () if A and [, ()) + Every A is < ()

supercompact + [ is measurable iff is < () strongly compact iff is < ()supercompact].

Proof of Lemma 10: Using the notation above, for each A, write P = T T;

further, write T as T


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Keeping the preceding paragraph in mind, we take as our ground model VP = V1.

Working in V1, we let R = {p

A

P0,() : support(p) is a set}, ordered by com-

ponentwise extension. As before, we can write for each A R = R R, where

R =

{A:}

P0,() and R = {p

{A:>}

P0,() : support(p) is a set}. We can also

write R as R


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Lemma 12 (Menas [Me]). If is the least measurable limit of either strongly compact

or supercompact cardinals, then is strongly compact but isnt 2 supercompact.

Proof of Lemma 12: We assume without loss of generality that is the least measurable

limit of strongly compact cardinals. As readers will easily see, the proof given works equally

well if is the least measurable limit of supercompact cardinals.

Let : < enumerate in increasing order the strongly compact cardinals below

. Fix > an arbitrary cardinal. Let be any measure (normal or non-normal) over, and let : < be a sequence of fine, -complete measures over P(). The set

U given by X U iff X P() and { < : X| } , where for X P(),

< , X| = {p X : p P()}. It can easily be verified that U is a -additive, fine

measure over P(). Since > is arbitrary, is strongly compact.

Assume now that is 2 supercompact, and let k : V M be an elementary em-

bedding with critical point so that M2

M. By the fact that is the critical point

of k, if < is strongly compact, then M |= k() = is strongly compact. By the

fact M2

M, M |= is measurable. Thus, M |= is a measurable limit of strongly

compact cardinals, contradicting the fact that M |= k() > is the least measurablelimit of strongly compact cardinals. This proves Lemma 12.

Lemma 12

We return to the proof of Theorem 3. Recall that we are assuming our ground model

V |= ZFC + GCH + is the least supercompact limit of supercompact cardinals +

> + is a regular cardinal which is either inaccessible or is the successor of a cardinal

of cofinality > and h : is a function so that for some elementary embedding

j : V M witnessing the < supercompactness of , j(h)() = . As in the proof

of Theorem 1, we assume without loss of generality that for every inaccessible < ,

h() > + and h() is regular.

We also assume without loss of generality that h() = 0 if isnt inaccessible and that

if < is inaccessible and is the least supercompact cardinal > , then h() < . To

see that the conditions on h imply this last restriction, assume that h() on the set of

inaccessibles below . It must then be the case that M |= For some cardinal , is

supercompact. By the closure properties ofM, M |= is supercompact for all < .

It is a theorem of Magidor [Ma2] that if is < supercompact and is supercompact,

then is supercompact. It is thus the case that M |= is supercompact. Since V |=

is the least supercompact limit of supercompact cardinals and is the critical point of j,if V |= < is supercompact, M |= j() = is supercompact. Putting these last

two sentences together yields a contradiction to the fact that M |= j() > is the least

supercompact limit of supercompact cardinals.

We next show the following fact about Laver indestructibility [L] which will play a

critical role in the proof of Theorem 3.

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Lemma 13. If is a supercompact cardinal, then the definition of the partial ordering

which makes Laver indestructible under -directed closed forcings can be reworked so

that for any fixed < , there are no strongly compact cardinals in the interval (, ).

Proof of Lemma 13: Let f : V be a Laver function, i.e., f is so that for every

x V and every TC(x), there is a fine, -complete, normal ultrafilter U over P()

so that for j the elementary embedding generated by U, (jf)() = x. The Laver

partial ordering P

which makes Laver indestructible under -directed closed forcingsand destroys all strongly compact cardinals in the interval (, ) is as usual defined as

a stage Easton support iteration P, Q : < . As in the original definition, at

each stage < , an ordinal < is chosen, where at limit stages , = |TC(Q)| be a regular cardinal so

that PQ > for = max(|TC(Q)|, 2|[] , by Lemma 8, the closure properties of M[G H] with respect to

V[G H], and the definitions of P and k(P) (including the fact both P and k(P)

are Easton support iterations of partial orderings satisfying a certain degree of strategic

closure), the partial ordering T M[G H] so that P Q P0, T

= P Q T = Pk()(where T is a term for the appropriate partial ordering) is -strategically closed in both

M[G H] and V[G H]. As in Lemma 9, this means that if H is V[G H]-generic over

T, then M[G H H] is closed under sequences with respect to V[G H H], and the

embedding k extends in V[G H H] to k : V[G] M[G H H]. The definition of

and the fact k(Q) is -directed closed in both M[G H H] and V[G H H] allow

us to find in V[G H H] a master condition q extending each p kH. If H is

now a V[G H H

]-generic object over k

(Q) containing q, then k

extends further inV[G H H H] to k : V[G H] M[G H H H]. By the fact that T k(Q) is

-strategically closed in either V[G H] or M[G H] and the definition of , the ultrafilter

over (P())V[GH] definable via k in V[G H H H] is present in V[G H]. Since

was arbitrary, V[G H] |= is supercompact.

It remains to show that VP

|= No cardinal in the interval (, ) is strongly compact.

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To see this, choose an embedding k : V M so that (kf)() = Q, +, where Q is a

term with respect to P for the trivial partial ordering {}. By the definition of P, it

will then be the case that M |= P+1 = P Q P

0, = (P

)V Q P0, and the T so

that P+1 T = k(P) is such that k(P) contains a non-reflecting stationary set of

ordinals of cofinality . By reflection, { < : P+1 = P Q P

0, and the T so that

P+1 T = P is such that P contains a non-reflecting stationary set of ordinals

of cofinality } is unbounded in , where Q is a term with respect to P for the trivial

partial ordering {}. By a theorem of [SRK], if contains a non-reflecting stationary set of

ordinals of cofinality , then there are no strongly compact cardinals in the interval (, ).

Thus, the last two sentences immediately imply VP

|= No cardinal in the interval (, )

is strongly compact. This proves Lemma 13.

Lemma 13

We note that in the proof of Lemma 13, GCH is not assumed. If GCH were assumed,

then as in Lemma 9, we could have taken the generic object H as an element ofV[G H].

We can now define in the manner of Theorem 1 the partial ordering P used in the

proof of Theorem 3 by defining a stage Easton support iteration P = P, Q : <

and then defining P = P+1 = P Q for a certain partial ordering Q definable in VP .

We first let : < be the continuous, increasing enumeration of the set { < : is

a supercompact cardinal} { < : is a limit of supercompact cardinals}. We take as

an inductive hypothesis that the field ofP is { : < } and that if is supercompact,

then |P| < . The definition is then as follows:

1. P0 is trivial.

2. Assuming P has been defined for < , we consider the following three cases.

(1):

is a supercompact cardinal. By the inductive hypothesis, since |P

| <

,

the Levy-Solovay results [LS] show that VP |= is supercompact. It is thus

possible in VP to make Laver indestructible under -directed closed forcings.

We therefore let Q be a term for the partial ordering of Lemma 13 making Laver indestructible so that PQ is defined using partial orderings that are

at least (2)+-directed closed and add non-reflecting stationary sets of ordinals

of cofinality (2)+ for = max(sup({ : < }), h(sup({ : < }))), and

we define P+1 = P Q. (If = 0, then Q is a term for the Laver partial

ordering of Lemma 13 where = , and P+1 = P Q.) Since Q can be

chosen so that |P+1| = < +1, the inductive hypothesis is easily preserved.(2): is a regular limit of supercompact cardinals. Then P+1 = P Q, with

Q a term for P0,h()

P1,h()[Sh()], where Sh() is a term for the non-

reflecting stationary subset ofh() introduced by P0,h()

. Since +1 must be

supercompact, by the conditions on h, |P+1| < +1, so the inductive hypothesis

is once again preserved.

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(3): is a singular limit of supercompact cardinals. Then P+1 = P Q, where

Q is a term for the trivial partial ordering {}.

3. Q is a term for P0, (P

1,[S] P

2,[S]), where again, S is a term for the non-

reflecting stationary subset of introduced by P0,.

Note that if < is a limit ordinal, then since is the least supercompact limit of

supercompact cardinals, sup({ : < }) = < , where is the least supercompact

cardinal in the interval [, ). It is this fact that preserves the inductive hypothesis at limit

ordinals < and at successor stages + 1 when is a singular limit of supercompact

cardinals.

The intuitive motivation behind the above definition is much the same as in Theorem

1. Specifically, below at any inaccessible limit of supercompact cardinals, we must

force to ensure that becomes non-measurable and is so that 2 = h(). At , however,

we must force so as simultaneously to make 2 = while first destroying and then resur-

recting the < supercompactness of . The forcing will preserve the supercompactness

of every V-supercompact cardinal below and will ensure there are no measurable limits

of supercompacts below . In addition, the forcing will guarantee that the only strongly

compact cardinals below are those that were supercompact in V. Thus, will have

become the least measurable limit of supercompact and strongly compact cardinals in the

generic extension.

Lemma 14. VP |= If < is supercompact in V, then is supercompact.

Proof of Lemma 14: Let < be a V-supercompact cardinal. Write P = R R,

where R is the portion ofP whose field is all cardinals and R is a term for the rest of

P. By case 1 in clause 2 of the inductive definition of P, V1 = VR |= is supercompact

and is indestructible under -directed closed forcings.Assume now that VP = VR1 |= isnt supercompact, and let p = p : R

be so that over V1, p R isnt supercompact. By the remark after the proof of Lemma

3, case 1 in clause 2 of the inductive definition of P, and the fact each P0,h()

is -

directed closed for < , it inductively follows that if H is a V1-generic object over R so

that p H, then H must be V1-generic over a partial ordering T V1 so that p T and

so that V1 |= T is -directed closed. This means V1[H] |= is supercompact. This

contradicts that over V1, p R isnt supercompact. This proves Lemma 14.

Lemma 14

Lemma 15. VP |= No inaccessible < which is a limit ofV-supercompact cardinals is

measurable.

Proof of Lemma 15: If < is in VP an inaccessible limit ofV-supercompact cardinals,

then since VP VP, this fact must be true in VP as well. Hence, since is so that P

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is the direct limit of the system P, Q : < and V |= GCH, VP |= All cardinals

are the same as in V and GCH holds for all cardinals . Therefore, the same

arguments as in Lemmas 3 and 6 show that VP+1 |= isnt measurable and 2 = h()

if [, h()) is a cardinal. (The same argument as in Lemma 8 also tells us that

VP |= 2 = if [, ) is a cardinal.) It then follows by case 1 in clause 2 of the

inductive definition of P that VP |= isnt measurable and 2 = h() if [, h()) is a

cardinal. This proves Lemma 15.

Lemma 15

Lemma 16. VP |= For any < , is supercompact in V iff is supercompact iff is

strongly compact.

Proof of Lemma 16: Let < be strongly compact and not V-supercompact, and

let (, ) be the least V-supercompact cardinal > . Since Lemma 15 shows that

no inaccessible limit of V-supercompact cardinals is measurable, sup({ < : is a V-

supercompact cardinal}) = < , where < and is as in the definition of P.

(If < 0, then let = 1 and = 0.) Thus, (,

) and

= +1. By thedefinition of P, P+2 = P+1 Q+1, where Q+1 is so that P+1Q+1 destroys all

strongly compact cardinals in the interval (, +1) by adding non-reflecting stationary

sets of ordinals of cofinality (2+1)+ to unboundedly many in +1 cardinals, where as

before, +1 = max(sup({ : < + 1}), h(sup({ : < + 1}))) = max(, h()).

(If = 1, then Q+1 is so that P+1Q+1 destroys all strongly compact cardinals in

the interval (, +1) by adding non-reflecting stationary sets of ordinals of cofinality

to unboundedly many in +1 cardinals.) Again by the definition of P, for the T so that

P+2 T = P, P+2T is (2+2)

+= (2h(+1))

+-strategically closed, so PThere are

unboundedly many in +1 cardinals in the interval (, +1) containing non-reflectingstationary sets of ordinals of either cofinality (2+1)

+or . This means VP |= No

cardinal in the interval (, +1) is strongly compact, a contradiction. This, combined

with Lemma 14, proves Lemma 16.

Lemma 16

Lemma 17. VP |= is the least measurable limit of either strongly compact or super-

compact cardinals.

Proof of Lemma 17: By Lemma 16, if < is strongly compact, then must be

supercompact in both V and VP. By Lemma 15, there are no measurable limits of V-supercompact cardinals in VP below . This proves Lemma 17.

Lemma 17

Lemmas 13-17, together with the observation that the same arguments as in Lemma

9 yield VP |= is < supercompact, complete the proof of Theorem 3.

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Theorem 3

4 Concluding Remarks

In conclusion to this paper, we outline an alternate notion of forcing that can be used

to construct models witnessing Theorems 1 and 2 in which cardinals and cofinalities are

the same as in the ground model. The forcing we use is a slight variation of the forcing

used in [AS]. Specifically, as in Section 1, we let < be cardinals with inaccessible,

> + regular, and either inaccessible or the successor of a cardinal of cofinality > .

We also assume as in Section 1 that our ground model V is so that GCH holds in V for all

cardinals , and we fix < a regular cardinal. As before, we define three notions of

forcing. P0, is just the standard notion of forcing for adding a non-reflecting stationary

set of ordinals of cofinality to , i.e., P0, is defined as in Section 1, only replacing

in the definition with . If S is a term for the non-reflecting stationary set of ordinals of

cofinality introduced by P0,, then P2,[S] V1 = V

P0, is the standard notion of forcing

for introducing a club set C which is disjoint to S, i.e., P2,[S] essentially has the same

definition as in Section 1.

To define P

1

,[S] in V1, as in [AS] or Section 1, we first fix in V1 a (S) sequenceX = x : S. (Since each element of S has cofinality , either Lemma 1 of [AS] or

our Lemma 1 shows each x X can be assumed to be so that order type(x) = .) Then,

in analogy to the definition given in Section 1 of [AS], P1,[S] is defined as the set of all

4-tuples w,, r, Z satisfying the following properties.

1. w []


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measurability of , and be able to resurrect the < supercompactness of if necessary.

P1,[S] has been designed so as to allow us to do all of these things.

The proof that VP1,[S]

1 |= is non-measurable is as in Lemma 3 of [AS]. In

particular, the argument of Lemma 3 of [AS] will show that cant carry a -additive

uniform ultrafilter. We can then carry through the proof of Lemma 4 of [AS] to show

P0, (P1,[S] P

2,[S]) is equivalent to C() C(, ). The proofs of Lemma 5 of [AS] and

Lemma 6 of this paper will then show P

0

, P

1

,[

S] preserves cardinals and cofinalities,is +-c.c., and is so that VP

0,P

1,[S] |= 2 = for every cardinal [, ). Then, if

we assume is < supercompact with and h : as in Theorem 1 and define in

V an iteration P as in Section 2 of this paper, we can combine the arguments of Lemma

8 of [AS] and Lemma 8 of this paper to show V and VP have the same cardinals and

cofinalities and VP |= For all inaccessible < and all cardinals [, h()), 2 = h(),

for all cardinals [, ), 2 = , and no cardinal < is measurable. We can then

prove as in Lemma 9 of this paper that VP |= is < supercompact. The proof now of

Theorem 2 is as before, this time using the just described iteration as the building blocks

of the forcing. This will allow us to conclude that the model witnessing the conclusionsof Theorem 2 thereby constructed is so that it and the ground model contain the same

cardinals and cofinalities.

We finish by explaining our earlier remarks that it is impossible to use the just de-

scribed definitions of P0,, P1,[S], and P

2,[S] to give a proof of Theorem 3 of this paper.

This is since ifV |= is a supercompact limit of supercompact cardinals, < is super-

compact, and are both regular cardinals, and < < < , then forcing with either

P0, or P0, P

1,[S] will kill the strong compactness of , as S will be a non-reflecting

stationary set of ordinals of cofinality through in either VP0, or VP

0,P

1,[S]. (See

[SRK] or [KiM] for further details.) This type of forcing must of necessity occur if we usethe iteration described in the proof of Theorem 3.

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References

[AS] A. Apter, S. Shelah, On the Strong Equality between Supercompactness and Strong

Compactness, to appear in Transactions AMS.

[Ba] J. Baumgartner, Iterated Forcing, in: A. Mathias, ed., Surveys in Set Theory, Cam-

bridge University Press, Cambridge, England, 159.[Bu] J. Burgess, Forcing, in: J. Barwise, ed., Handbook of Mathematical Logic, North-

Holland, Amsterdam, 1977, 403452.

[C] J. Cummings, A Model in which GCH Holds at Successors but Fails at Limits,

Transactions AMS 329, 1992, 1-39.

[CW] J. Cummings, H. Woodin, Generalised Prikry Forcings, circulated manuscript of a

forthcoming book.

[J] T. Jech, Set Theory, Academic Press, New York, 1978.

[Ka] A. Kanamori, The Higher Infinite, Springer-Verlag, New York and Berlin, 1994.

[KaM] A. Kanamori, M. Magidor, The Evolution of Large Cardinal Axioms in Set Theory,in: Lecture Notes in Mathematics 669, Springer-Verlag, Berlin, 1978, 99275.

[KiM] Y. Kimchi, M. Magidor, The Independence between the Concepts of Compactness

and Supercompactness, circulated manuscript.

[L] R. Laver, Making the Supercompactness of Indestructible under-Directed Closed

Forcing, Israel J. Math. 29, 1978, 385388.

[LS] A. Levy, R. Solovay, Measurable Cardinals and the Continuum Hypothesis, Israel J.

Math. 5, 1967, 234248.

[Ma1] M. Magidor, Changing Cofinality of Cardinals, Fundamenta Mathematicae 99,

1978, 6171.[Ma2] M. Magidor, On the Role of Supercompact and Extendible Cardinals in Logic ,

Israel J. Math. 10, 1971, 147157.

[Me] T. Menas, On Strong Compactness and Supercompactness, Annals Math. Logic 7,

1975, 327359.

[MS] A. Mekler, S. Shelah, Does -Free Imply Strongly -Free?, in:

Proceedings of the Third Conference on Abelian Group Theory, Gordon and Breach,

Salzburg, 1987, 137148.

[SRK] R. Solovay, W. Reinhardt, A. Kanamori, Strong Axioms of Infinity and Elementary

Embeddings, Annals Math. Logic 13, 1978, 73116.

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Arthur W. Apter and Saharon Shelah- "Menas' Result is Best Possible"

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