pierre matet and saharon shelah- the nonstationary ideal on p-kappa(lambda) for lambda singular

8/3/2019 Pierre Matet and Saharon Shelah- The Nonstationary Ideal on P-kappa(lambda) for lambda singular

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THE NONSTATIONARY IDEAL ON P()

FOR SINGULAR

Pierre MATET and Saharon SHELAH

Abstract

We give a new characterization of the nonstationary ideal on P() inthe case when is a regular uncountable cardinal and a singular stronglimit cardinal of cofinality at least .

0 Introduction

Let be a regular uncountable cardinal and be a cardinal.

As [10] and [11] of which it is a continuation, this paper investigates idealson P() with some degree of normality. For , let N S, denotes the

least -normal ideal on P(). Thus N S, = the noncofinal ideal I, for

any < , and N S, = the nonstationary ideal N S,. N SS, denotes theleast seminormal ideal on P(). It is simple to see that N SS, = N S, in

case cf() < . If is regular, then by a result of Abe [1], N SS, = N S , is the restric-tion of a smaller ideal, i.e. whether N S, = J | A for some ideal J N S,and some A N S,. The question as stated has a positive answer (see [2])

with J = I,. By a result of Abe [1] we can also take J = N SS, in case

cf() < . We investigate the possibility of taking J =


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there is no A such that N S, = N SS, | A.

Let H, assert that cof(N S,) for every cardinal with < , wherecof(N S,) denotes the reduced cofinality of N S

,. Clearly, H, follows from

2 but, by results of [7], for every cardinal with < , cof(N S,) = + and hence cof(N S,) .

It is known ([16], [11]) that if cf() < , then H, holds just in case N S, =I, | A for some A. We will prove the following.

Proposition 0.1. Suppose cf() < and H, holds. Then (a)

N S, = N Scf()

,| A for some A, but (b) there is no B such that N S, =


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We establish the following.

Proposition 0.2. SupposeH, holds, and let > be such that (a) is eithera successor ordinal, or a limit ordinal of cofinality at least , and (b) , where

equals + 1 if cf() < , and cf() otherwise. Then cof u(, ). Goldring[7] and the second author proved that if is regular and > is Woodin, thenin VCol(,


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1 Ideals on P()

In this section we collect basic material concerning ideals on P().N S denotes the nonstationary ideal on .For a set A and a cardinal , let P(A) = {a A :| a |< }.Given four cardinals , , and , we define cov(,,,) as follows. If there isX P() with the property that for any a P(), we may find Q P(X)with a Q, we let cov(,,,) = the least cardinality of any such X.Otherwise we let cov(,,,) = 0.We let cov(,,,) = u(, ) in case = and = 2.

LEMMA 1.1. ([15], pp. 85-86)) Let , , and be four cardinals such that and 2. Then the following hold :

(i) If > , then cov(,,,) .(ii) cov(,,,) = cov(,,, ).

(iii) cov(+, , , ) = + cov(,,,)

(iv) If > and cf() < = cf(), then cov(,,,) =

and cf() , then cov(,,,) = and f : P() P(), we let

f(J) = {X P() : f1(X) J}.

MJ denotes the collection of all maximal antichains in the partially ordered set(J+, ), i.e. of all Q J+ such that (i) A B J for any distinct A, B Q,and (ii) for every C J+, there is A Q with A C J+.

For a cardinal , J is -saturated if | Q |< for every Q MJ.cof(J) denotes the least cardinality of any X J such that J =AX

P(A).

cof(J) denotes the least size of any Y J with the property that for everyA J, there is y P(Y) with A y.non(J) denotes the least cardinality of any A J+.Note that cof(J) non(J) non(I,) = u(, ).

4


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The following is well-known (see e.g. [10] and [11]).

LEMMA 1.2.

(i)


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(iii) ([11]) cof(N S,) cof(N S,) for every cardinal with < .

(iv) ([11]) Suppose cf() . Then cof(N S,) > .

The concept of []


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(iii) A {a C,g : a } = for some g : P.3() P3().

LEMMA 1.9. ([11]) Let and be two cardinals such that 2 .Then the following hold :

(i) Let J be a []


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The following is a straightforward generalization of a result of Foreman [4] :

PROPOSITION 1.13. Let and be two cardinals such that and

. Then every[]


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LEMMA 2.1. Let and be two cardinals such that 2 .

Then cof(N S[]


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PROPOSITION 2.3. Let be a cardinal with . Then cof(N S,) =

d

,.

Proof. By Lemmas 2.1 and 2.2.

COROLLARY 2.4. Let and be two cardinals such that < .

Then cof(N S,) cof(N S,).

3 N SS,

An ideal J on P() is seminormal if it is -normal for every < .N SS, denotes the smallest seminormal ideal on P().

LEMMA 3.1.

(i) (Folklore) Suppose cf() < . Then N S, = N SS,.

(ii) ([1]) Suppose cf() < . Then N S, = N SS, | A for some A.

PROPOSITION 3.2. Suppose cf() < . Then cof(N SS,) > .

Proof. By Lemmas 1.5 (iv) and 3.1.

We will see that cof(N SS,) > needs not hold in case is regular. Notethat if is regular, then by Lemma 1.5 (iv), cof(N S,) > .

LEMMA 3.3. ([1]) Suppose is regular. Then N SS, =


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some A

N S[]


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by Lemmas 1.4 (ii) and 1.10. Hence cof


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Proof. By Proposition 3.6 and Lemma 4.2.

In particular, if cf() < and H, holds, then cof(N S,) = .

Note that if cof

N S[]


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In particular ifH, holds and cf() < , then cof


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cof(I, | A) for some A

N S[]


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It follows that B

N S[]


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cof(N S,) u(, ) u(, ) cof(N S,)

by Lemma 1.11 (ii), so cof(N S,) = u(, ) = u(, ). Hence by Lemma 1.5(iv), SSH does not hold.

PROPOSITION 5.6. Let and be two cardinals such that 2 and

< . Suppose that cof(N S,)


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such that {b gy

C,g : b } H. Let


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N S[]


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cof

N S[]


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Suppose is a limit cardinal and is a cardinal with . If either

cf() < or cf() > , then by Lemmas 1.10 and 4.1

cof(N S,)

, and 0 in case < . For < , select a family G of functions from P3( ) to P3() so that

| G | cof(N S,

) and for any H (N S,

)

, there is y P(G) \ {}

with

b gy

C,g : b

H. Let


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Let us show that A tz

Dt

C,f . Thus let a A tz

Dt

and v

P3(a ). There must be a such that v . Then a x

C,g ,

so v f(v) for some a . Now a y

C,g , and therefore

f(v) a.

7 Nowhere precipitousness of N S,

Throughout this section it is assumed that cf() < . Let bea cardinal with cf() < . We will show that under certain conditions,

N S, is nowhere precipitous. Our proof will follow that of Theorem 2.1 in [13],except that we do not appeal to pcf theory.Set = cf(). We assume that c(, ) < in case > . Let < be aregular cardinal such that > if = , and > c(, ) otherwise.Select a continuous, increasing sequence < : < > of cardinals so that . Let E be the set of all limit ordinals < with

cf() < . We define D as follows. If = , we set D = E. Otherwise we pickD in N S, so that | D |= c(, ). For d D, put (d) = (d ). Note that(d) E. Moreover (d) = d in case = .

Let B be the set of all a P() such that (i) 0 a, (ii) + 1 a for every

a , (iii) a , (iv) a { : a}, and (v) a Din case > . Then clearly, B

N S,

. For d D, define Wd by letting

Wd = {a B : (a ) = d} if = , and Wd = {a B : a = d} otherwise.Note that (a (d)) = (d) for every a Wd.

LEMMA 7.1. Suppose there is T P() such that (a) |T P(a)| < foranya P(), and (b) u(, ) |T| for every cardinal with < . Thenfor every R (N S,)

+

{d D : |{a (d) : a R Wd} u(, (d))}

lies in N S+ if = , and in N S+, otherwise.

Proof. For , select Z I+, with |Z| |T|. Then clearly there is Q Twith |


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Claim. Let S

N S,+

. Then there is d D such that

k(d)({a (d) : a S Wd}) I+,(d)

.

Proof of the claim. Assume otherwise. For d D, select yd P((d)) so

that yd \ k(d)(a (d)) = for every a S Wd. Set y =dD

yd. Note that

y P(). For , pick z Z so that y z. Now let H be the set

of all a P() such that i(z)

a

P(a ) for every a . Since

H (N S,), we can find a S B H. Set d = (a ) if = , and

d = a otherwise. Then a Wd and

yd y (d) =

a

(y )

a

z =

a

j(i(z)) k(d)(a (d)).

This contradiction completes the proof of the claim.

It is now easy to show that the conclusion of the lemma holds: Fix R (N S,)

+, and A such that A N S if = , and A N S, otherwise.

Set Y =

dDA

Wd. Since Y (N S,), there must be some d D such that

k(d)

{a (d) : a (R Y) Wd}

I+,(d) .

Then clearly, d A and |{a (d) : a R Wd}| u(, (d)).

Concerning condition (a) in Lemma 7.1 let us observe the following :

LEMMA 7.2. Let T P() be such that | T P(a) | u(, ) for anya P(). Then | T | u(, ).

Proof. Suppose otherwise. Select A I+, with | A |= u(, ). Defineg : T A so that t g(t) for all t T. Then g must be constant onsome D T with | D |=| A |+. Contradiction.

Consider for instance the following situation : In V, GCH holds, is a strongcardinal with < < , and a cardinal greater than . Then by a result

of Gitik and Magidor [6], there is a cardinal preserving, +-cc forcing notion Psuch that in VP, (a) no new bounded subsets of are added, (b) changes itscofinality to , and (c) 2 . Now working in VP, let T = P1(). Then clearly|T P(a)| 2|a| for any a P(). Moreover for any two uncountablecardinals and with cf() = < ,

u(, ) = 2


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Hence by Lemma 7.1, for any R (N S,)+,

d D :| {a (d) : a R Wd} | u(, (d))

lies in N S+ if = , and in N S

+, otherwise. Note that for any cardinal with

, cof

N S,

= u(, ) since cof(N S,) ( in case .

PROPOSITION 7.3. Suppose there is T P() and a cardinal with < such that (a) |T P(a)| < for anya P(), and (b) u(, ) |T|for every cardinal with < < . Then N S, is nowhere precipitous.

Proof. By Lemma 1.12 it suffices to show that II has a winning strategy in thegame G(N S,). We can assume without loss of generality that 0 > . Forg : P3() P3() and < , define g : P3() P3() by g(e) = g(e).

Claim 1. Let g : P3() P3(). Thend D :

a Wd : a (d) C

,(d)g(d)

C,g

lies in (N S | E) if = , and in N S, otherwise.

Proof of Claim 1. We prove the claim in the case when > , and leave theproof in the case when = to the reader. Define h : P3() by h(e) =the least < such that g(e) . Let Q be the set of all d D such thath(e) d for every e P3(d). Then clearly Q N S

,. Now fix d Q and

a Wd such that a (d) C,(d)g(d) . Let e P3(a ). Then h(e) d, so

g(e) (d). It follows that g(e) a, since (d) g(e) a. Thus a C,g .

This completes the proof of Claim 1.

Claim 2. Let X (N S,)+ and Y B. Suppose that

{a Y Wd : a (d) C,(d)k } =

whenever d D and k : P3() P3((d)) are such that

|{a (d) : a A Wdanda (d) C,(d)k }| =

(d).

Then Y (N S,)+.

Proof of Claim 2. Fix g : P3() P3(). By Lemma 7.1 and Claim 1, theremust be d D such that

| {a (d) : a

X C,g

Wd} |= (d)and

a Wd : a (d) C,(d)g(d)

C,g .

Then a Y Wd : a (d) C

,(d)g(d)

=

24


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since a (d) C,(d)g(d) for every a X C

,g Wd. Hence Y C

,g = .

This completes the proof of the claim.

For d D, consider the following two-person game Gd consisting of moves,with player I making the first move : I and II alternately pick subsets ofWd, thusbuilding a sequence < Xn : n < > subject to the following two conditions :

(1) X0 X1 . . . , and (2) {a X2n+1 : a (d) C,(d)k } = for every

k : P3() P3((d)) such that

|{a (d) : a X2n and a (d) C,(d)k }| =

(d).

II wins the game if and only ifn


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(i) Suppose < for every cardinal < . Then N S, is nowhere precip-

itous.(ii) Suppose > , and c(,) < for every cardinal < . Then N S, is

nowhere precipitous.

(iii) Suppose H, holds, and < for every cardinal < . Then N S, isnowhere precipitous.

Proof. By Proposition 7.3 and Corollary 6.4 (i).

Question. Suppose H, holds and < for every cardinal < . Does itthen follow that is a strong limit cardinal ?

We conclude the section with the following remark. Suppose there exist T and as in the statement of Proposition 7.3. Then either cof(N S,) = u(, ), or


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[9] P. MATET, Weak saturation of ideals on P(); Mathematical Logic Quar-

terly, to appear.[10] P. MATET, C. PEAN and S. SHELAH, Cofinality of normal ideals on

P()I ; preprint.

[11] P. MATET, C. PEAN and S. SHELAH, Cofinality of normal ideals onP()II; Israel Journal of Mathematics 150 (2005), 253-283.

[12] P. MATET, A. ROSLANOWKI and S. SHELAH, Cofinality of the non-stationary ideal; Transactions of the American Mathematical Society 357(2005), 4813-4837.

[13] Y. MATSUBARA and S. SHELAH, Nowhere precipitousness of the non-stationary ideal over P; Journal of Mathematical Logic 2 (2002), 81-89.

[14] Y. MATSUBARA and M. SHIOYA, Nowhere precipitousness of some ide-als; Journal of Symbolic Logic 63 (1998), 1003-1006.

[15] S. SHELAH, Cardinal Arithmetic; Oxford Logic Guides vol 29 Oxford Uni-versity Press, Oxford, 1994.

[16] S. SHELAH, On the existence of large subsets of []

pierre matet and saharon shelah- the nonstationary ideal on p-kappa(lambda) for lambda singular

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