mirna dzamonja and saharon shelah- saturated filters at successors of singulars, weak reflection and...

8/3/2019 Mirna Dzamonja and Saharon Shelah- Saturated filters at successors of singulars, weak reflection and yet another

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Saturated filters at successors of singulars, weak reflection andyet another weak club principle

by 1

Mirna Dzamonja Saharon ShelahUniversity of Wisconsin-Madison Hebrew University of JerusalemDepartment of Mathematics Institute of MathematicsMadison, WI 53706 91904 Givat RamUSA [email protected] [email protected]

Abstract

Suppose that is the successor of a singular cardinal whose cofinality is an un-countable cardinal . We give a sufficient condition that the club filter of concentratingon the points of cofinality is not +-saturated.2 The condition is phrased in terms of anotion that we call weak reflection. We discuss various properties of weak reflection.

We introduce a weak version of the -principle, which we call , and show that if itholds on a stationary subset S of , then no normal filter on S is +-saturated. Under theabove assumptions, (S) is true for any stationary subset S of which does not containpoints of cofinality . For stationary sets S which concentrate on points of cofinality , weshow that (S) holds modulo an ideal obtained through the weak reflection.

1

Authors partially supported by the Basic Research Foundation Grant number 0327398 administered by theIsrael Academy of Sciences and Humanities. For easier future reference, note that this is publication [DjSh 545]

in Shelahs bibliography. The research for this paper was done in the period between January and August 1994.

The most sincere thanks to James Cummings for his numerous comments and corrections, and to Uri Abraham,

Menachem Magidor and other participants of the Logic Seminar in Jerusalem, Fall 1994, for their interest. The

first author also wishes to thank the Hebrew University and the Lady Davis Foundation for the Forchheimer

Postdoctoral Fellowship.2

added in proof: M. Gitik and S. Shelah have subsequently and by a different technique shown that the club

filter on such is never saturated.

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0. Introduction. Suppose that = + and is an infinite cardinal of cofinality .

We revisit the classical question of whether a normal filter on can be +-saturated. We

are in particular concerned with the case of uncountable and less than . We are mostly

interested in the club filter on .

While the richness of the literature on the subject provides us with a strong motivation

for a further study, it also prevents us from giving a complete history and bibliography

involved. We shall give a list of those references which are most directly connected or used

in our results, and for further reading we can suggest looking at the references mentioned

in the papers that we refer to.

It is well known that for no regular > 0 the club filter on can be -saturated,

but modulo the existence of huge or other large cardinals it is consistent that 1 carries

an 2-saturated normal filter (Kunen [Ku1]), or in fact that any uncountable regular

carries an +1-saturated normal filter (Foreman [Fo]). In these arguments the saturated

filter obtained is not the club filter. As a particular case of [Sh 212, 14] or [Sh 247, 6], if

= +, then the club filter D restricted to the elements of of a fixed cofinality = cf()

is not +-saturated. It is consistent that D1 is 2-saturated, as is shown in [FMS] and

also in [SvW].

If < and is a regular cardinal, we use S to denote the set of elements of

which have cofinality . Let be as above.

In the first section of the present paper, we give a sufficient condition that D S

is not +-saturated. Here 0 < = cf() < . The condition is a reflection property,

which we shall call weak reflection, as we show that it is weaker than some known reflection

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properties. We discuss the properties of weak reflection in more detail in 1. Of course,

the main part of the section is to show that the appropriate form of the weak reflection

indeed suffices for D S not to be

+-saturated, which is done in 1.13. In 1.15 the

argument is generalized to some other normal filters on , and [Sh 186,3] is revisited.

In the second section we introduce the combinatorial principle which has the

property that, if (S) holds, then no normal filter on S is +-saturated. Here S is a

stationary subset of . The is a weak form of. The -principle was first introduced

for 1 in Ostaszewski [Os], and later investigated in a more general setting in [Sh 98] and

elsewhere.

If is the successor of the singular cardinal whose cofinality is , then ( \ S)

is true just in ZF C. As a corollary of this we obtain an alternative proof of a part of the

result of [Sh 212, 14] or [Sh 247, 6]. On the other hand, (S) only holds modulo an

ideal defined through the weak reflection (in 1), so this gives a connection with the results

of the first section. For the case of being a strong limit, just becomes the already

known . Trying to apply to arguments which work for the corresponding version of

, we came up with a question we could not answer, so we pose it in 2.6.

Before we proceed to present our results, we shall introduce some notation and con-

ventions that will be used throughout the paper.

Notation 0.0(0) Suppose that and is a regular cardinal. Then

S 0.

S = { < : cf() = }.

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More generally, we use Sr for r {, } to describe

Sr = { < : 0 cf() & cf() r }.

(1) SING denotes the class of singular ordinals, that is, all ordinals with cf() < .

LIM denotes the class of all limit ordinals.

(2) For a cardinal with cf() > 0, we denote by D the club filter on . The ideal

of non-stationary subsets of is denoted by J.

(3) If C , then

acc(C) = { C : = sup(C )} and nacc(C) = C\ acc(C).

We now go on to the first section of the paper.

1. Saturated filters on the successor of a singular cardinal of uncountable

cofinality. Suppose that is a cardinal with the property

> cf() = > 0,

and that = +. We wish to discuss the saturation of D S and some other normal

filters on . For the readers convenience, we recall some relevant definitions and notational

conventions. The cardinals , and as above will be fixed throughout this section.

Notation 1.0(0) Suppose that D is a filter on the set A. The dual ideal I of D is

defined as

I = {a A : A \ a D}.

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A set a A is D-positive or D-stationary, if a / I. The family of all D-stationary sets is

denoted by D+. A subset ofA which is not D-stationary, is referred to as D-non-stationary.

(1) Suppose that is a cardinal. A filter D on a set A is -saturated iff there are no

sets which are all D-stationary, but no two of them have a D-stationary intersection. If we

denote the dual ideal ofD by I, then this is equivalent to saying that P(A)/I has -ccc.

For a cardinal , the filter D is -complete if it is closed under taking intersections of

< of its elements.

(2) A -complete filter D on a cardinal is normal, if for any sequence X( < ) of

elements of D, the diagonal intersection

0 is a regular cardinal and > is an ordinal. We

say that has the strong non-reflection property for , if there is a function h :

such that for every S, there is a club subset C of with h C strictly increasing. In

such a case we say that h witnesses the strong non-reflection of for .

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If does not have the strong non-reflection property for , then we say that has the

weak reflection for , or that is weakly reflective for .

If h : is a function, we define

ref(h)def= { S : h C is not strictly increasing for any club C of }.

We shall first make some general remarks about Definition 1.1.

Observation 1.2(1) If is weakly reflective for > 0, and > , then is weakly

reflective for .

(2) If > = cf() > 0, then

has the strong non-reflection property for iff there is an h : such that for

all S, there is a club C of with the property that h C is 1-1.

In fact, the two sides of the equivalence can be witnessed by the same function h.

(3) Suppose that is a given uncountable regular cardinal such that there is an

which weakly reflects at .

Then the minimal such , which we shall denote by (), is a regular cardinal > .

Consequently, weakly reflects at iff there is a regular cardinal such that

weakly reflects at .

(4) Suppose that is a regular cardinal, and = () or is an ordinal of cofinality

() such that S() is stationary in . (This makes sense, as by (3), if() is defined,

then it is a regular cardinal.)

Then not only that is weakly reflective for , but for every h : , the set

ref(h) is stationary.

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Proof. (1) Follows from the definition.

(2) If has the strong non-reflection for , the other side of the above equivalence is

obviously true.

In the other direction, suppose that h satisfies the conditions on the right hand side,

and fix a S. Let C be a club of on which h is 1-1. In particular note that ran(h) is

cofinal in and that otp(C) = . By induction on < , define

= Min

C :

(C\ )

< )

h() > h()

.

Then D = { : < } is a club of and h D is strictly increasing.

(3) Let = () be the minimal ordinal which weakly reflects at , so obviously

> . If is not a regular cardinal, we can find an increasing continuous sequence of

ordinals i : i < which is cofinal in , and such that < . We can also assume that

0 = 0 and 1 > . In addition, for every i < , if i is a successor, we can assume that i

is also a successor.

Then, for every i < , there is an hi+1 : i+1 which exemplifies that i+1 has

the strong non-reflection property for . There is also a g : which witnesses that

has the strong non-reflection property for . Define h : by:

h() = hi+1() if (i, i+1)g(i) if = i.

Since is weakly reflective for , there must be a S such that for no club C of , is

h C strictly increasing. We can distinguish two cases:

Case 1. (i, i+1] for some i < .

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Then there is a club C of on which hi+1 is strictly increasing. But then h (C \

(i + 1)) = hi+1 (C \ (i + 1)) is strictly increasing, and C \ (i + 1) is a club of . A

contradiction.

Case 2. = i for some limit i < .

Notice that D = {j : j < i} is a club of i. We know that there is a club C of i

on which g is increasing. Then setting E = C D, we conclude that h E is strictly

increasing. Therefore, a contradiction.

(4) We first assume that = (). By (2), () is necessarily a regular cardinal

> . Let h : () be given. Suppose that ref(h) is non-stationary in () and

fix a club E in () such that E ref(h) = . Without loss of generality, otp E = ().

Let us fix an increasing enumeration E = {i : i < ()}. Without loss of generality,

0 = 0 and 1 > and i+1 is a successor for every i. So, for each i, there is a function

hi : i+1 which witnesses that i+1 is strongly non-reflective at .

But now we can use h and hi for i < () to define a function which will contradict

that () weakly reflects at , similarly to the proof of (3).

The other case is that > (), so S() is stationary in . Let h : be given.

If S(), then let us fix a club C of such that otp(C) = (). Let C = {i :

i < ()} be an increasing enumeration. Then h C induces a function g : () ,

given by g(i) = h(i).

We wish to show that ref(h) is stationary in . So, let C be a club of . As we know

that S() is stationary in , we can find a S() which is an accumulation point of

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C. Then C C is a club of, so Edef= {i < () : i C} is a club in

(). Therefore,

there is an i E ref(g), by the first part of this proof. Then i ref(h) C.1.2.

Remark 1.2.a. One can ask the question of 1.2.3 in the opposite direction: suppose

that reflects at some , what can we say about the first such ? This is an independence

question, for more on this see [CDSh 571].

We find it convenient to introduce the following

Definition 1.3. For ordinals > > 0, where is a regular cardinal, we define

I[, ) = {A : there is a function h : such that for every A S,

there is a club C of with h C strictly increasing.}

Saying that has the strong non-reflection property for is equivalent to claiming that

I[, ), or S I[, ). In such a case, we say that I[, ) is trivial.

I(, ) is the statement:

There is a (, ) with I[, ) non-trivial.

Theorem 1.4. Suppose that > > 0 and is a regular cardinal.

Then I[, ) is a -complete ideal on .

Proof. I[, ) is obviously non-empty and downward closed.

Suppose that Ai (i < i < ) are sets from I[, ), and that hi : witnesses

that Ai I[, ) for i < i. Let A = i


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C increasingly as { : < }. Therefore, the sequence hi() : < is a strictly

increasing sequence in , and = sup


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Observation 1.6(0) Note that by a closed unbounded set of , we simply mean an

unbounded subset of. Similarly, any with cf() = 0 will have a club subset consisting

of an unbounded -sequence in . So, 1 trivially holds.

(1) If1 2 < , then(1,) = (2,). If < 1 2, then(,2) = (,1).

Theorem 1.7. If > is regular , then (,) implies that has the strong non-

reflection property for .

Proof.From [Sh -g VII 1.7], we recall the following Fact 1.7.a. For the readers

convenience, we also include the proof.

Fact 1.7.a. Assume that (,) holds.

Then, there is a partial (,)-sequence

C : (SING LIM) & < < & cf() < ,

such that for each for which C is defined,

(i) C is a club subset of .

(ii) otp(C) < & cf() < = otp(C) < .

(iii) acc(C) & C defined = C = C .

Proof of 1.7.a. We start with a sequence D : (SING LIM) & < <

which exemplifies (,).

We can without loss of generality assume that each D satisfies D = . For each

for which D is defined, we can define a 1-1 onto function f : D otp(D) by

f()def= otp(D ).

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Note that acc(D) = f D = f . Now we define C for < with cf() < by

induction on < .

If D is not defined, then C is not either.

If D is defined, and Dotp(D) is not defined, we set C = D. Note that otp(D) <

in this case.

Finally, suppose that both D and Dotp(D) are defined. Then > otp(D) > and

cf

(otp(D)

= cf() < . So Cotp(D) is already defined and we can define

Cdef

= { D : f() Cotp(D)}.

We can check that

Cdef= C : (SING LIM) & < < & cf() <

is as required. One thing to note is that if acc(C) and Dotp(D) is defined, then

Dotp(D) must be defined too.1.7.a.

Fixing a sequence C like in Fact 1.7.a, the following defines a function h : :

h() =

otp(C) if S


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REF is consistent modulo the existence of infinitely many supercompact cardinals

[Sh 351]. The consistency of Ref(,,) has also been extensively studied, starting with a

result of J. Baumgartner in [Ba] that CON(Ref(2, 1, 0)) follows from the existence of

a weakly compact cardinal.

(1) M. Magidor points out the following equivalent definition of the weak reflection,

from which it is easy to see that it is weaker than what is usually meant by reflection:

We can say that > weakly reflects at iff for any partition

S


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We now go on to present the last two facts before we proceed to the Main Theorem.

Fact 1.11. Suppose that cf() > 0 and f : is a function which is not

increasing on any club of . Then there is a stationary set S in such that S =

f() < Min(S). (Hence, there is also a stationary subset of on which f is a constant.)

Proof. We fix an increasing continuous sequence of ordinals : < cf() which is

cofinal in . Let Tdef= { : f() < }. So T cf().

We shall see that T is stationary in cf(). Let us first assume that it is true, and

define for T \ {0},

g()def= Min{ : f() < +1}.

Therefore, g is regressive, and we can find a stationary T1 T such that g T1 is constantly

equal to some .

Let Sdef= { : T1 \ (

+ 2)}. We can check that this S is as required.

It remains to be seen that T is stationary in cf(). Suppose not. Then we can find a

club C in cf() such that C T = . Let

Edef= { : LIM C& ( < )(f() < )}.

Then E is also a club of cf(). But then Ddef= { : E} is a club of and f D is

strictly increasing, contradicting our assumption.1.11.

Remark 1.12. As a remark on the side: we cannot improve the previous result to

conclude, from the assumptions given above, that there is a club C of such that f C is

constant. Namely, if we take a stationary costationary set S in , and define f on by

f() =

1 if S0 otherwise,

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then this f is neither increasing nor constant on any club of . In the following Fact

1.12.a we recall a more general result along the lines of Fact 1.11. This Fact is not used in

the proof of the Main Theorem 1.13, and a reader who is in a hurry may without loss of

continuity proceed directly to 1.13.

Fact 1.12.a. Suppose that is an ordinal with cf() > 0 and f is a function from

to the ordinals. Then, there is a stationary S such that

either f S is constant,

or f S is strictly increasing.

Proof. Let = cf() > 0, and : < a strictly increasing enumeration of a

club of . Let Edef= LIM. We define a partial function g : E as follows:

g() = if

(a) f() f()(b)

<

f() f() = f() f()

(c) < is minimal under (a) and (b).

Note that g() < for all Dom(g). Now we consider three cases:

Case 1. S0def= E\ Dom(g) is stationary in .

Then Sdef= { : S0} is stationary in . We claim that f S is strictly increasing

on S. Otherwise, there would be an S such that there is a 0 < with f() f(0),

which contradicts the fact that / Dom(g).

Case 2. Dom(g) is stationary in and S1def= { Dom(g) : f() = f(g())} is

stationary in .

Since g S1 is regressive, there is a stationary S2 S1 such that g S2 is constant.

Then Sdef= { : S2} is stationary in , and f S is constant.

Case 3. Dom(g) is stationary in , but S1 is not stationary.

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Then S3def= Dom(g) \ S1 = { Dom(g) : f() < f(g())} is stationary, and there

is a stationary S4 S3 such that g S4 is a constant. Let S = { : S4}, so S is a

stationary subset of . We claim that f S is strictly increasing.

Otherwise, there are 1 < 2 S4 such that f(2) f(1). On the other hand,

f(1) < f(g(1)) = f(g(2)), since both 1 and 2 are members of S4. This contradicts

the definition of g(2), since 1 < 2.1.12.a.

We now present our main result.

Main Theorem 1.13. Assume that = + and > cf() = > 0. In addition,

for some (, ), we know that has the weak reflection property for .

Then D S is not +-saturated.

Proof. By 1.2.3, without loss of generality, is a regular cardinal, so < .

Fact 1.13.a. There is a stationary subset S of S such that:

For some S+

S and S+

\ , there is a sequence

C = C : S+,

such that, for every S+,

1. C is a subset of \ and otp(C) .

2. is a limit ordinal = sup(C) = .

3. C = ( S+ and C = C ).

4. cf() = iff S.

5. For every club E in , there are stationarily many S, such that for all , :

( < & , C) = (, ] E = .

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Proof of 1.13.a. Since + < we can apply [Sh 420, 1.5], so there is a stationary

subset S1 of S \ {} with S1 I[]. By [Sh 420, 1.2] this means that there is a sequence

D : < such that

(a) D is a closed subset of .

(b) nacc(D) = D = D .

(c) For some club E of , for every S1 E,

= sup(D) & otp(D) = .

(d) nacc(D) is a set of successor ordinals.

Observation. We do not lose generality if we in addition require that for each ,

otp(D) .

[Why? Let E be the club guaranteed by (c). Since S1 is stationary, so is S1 E, so

we can define for

D =

D if S1 E

or >

S1 E& nacc(D)

. otherwise.

Then we can set S2def= S1 E, so the sequence D : < will satisfy (a)(d), with S1

replaced by S2, and otp(D) will hold for each .]

Continuation of the Proof of 1.13.a. Now, for any club F of , we let

C[F]def= C[F] : S

+[F]def= S1 E acc(F) (successors) \ ,

where

C[F]def=

nacc(D) \ : F

sup( D),

=

.

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We claim that C can be set to be equal to C(F) for some club F, with S = S[F]def=

E S1 acc(F) \ and S+ = S+[F]. We are using the ideas of [Sh 365,2].

It is easily checked that (1)(4) are satisfied for any club F. So, let us suppose that

(5) is not satisfied for any choice of F, and we shall obtain a contradiction.

By induction on < +, we define a club F of .

If = 0, we let F = LIM.

If is a limit ordinal, we let F =


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Notice that def= sup{ : D} <

+, since otp(D) = .

Since / G [F+1], by the definition of G [F+1], there are < C[F ] such

that

(, ] F+1 = .

On the other hand, C [F ] =

sup( D),

F = = sup( D)

sup( F) = = sup( F+1). But < and nacc(D), so sup( D) .

Therefore . Note now that must be a limit ordinal, since F+1 LIM.

But is a successor, by (d) in the definition of D. So, < F

+1 and

(, ] F+1 = , by the choice of F . A contradiction.1.13.a.

Continuation of the proof of 1.13. Let us fix S, S+ and Cdef= C : S

+ as in Fact

1.13.a. Denote by cl(C) the following

cl(C) = {C : C S+ C(C = C )}.

Now fix the following enumerations:

For S+, let

C = {(, ) : < otp(C)},

such that (, ) is increasing in . For each , let

(, )def= (, ).

Let = i .

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Claim 1.13.b. Given enumerations as above, we can for each < , find sets

ai (i < ) such that

(I) = i


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Now, fix a sequence E = {E : S}, where each E is a club subset of with

otp(E) = and consisting only of elements of cofinality < . We prove the following

claim, with the intention of applying it later on the functions h ( < +).

Claim 1.13.c. Suppose that h : is non-decreasing, and E is given as above.

Then , there is an i = i(h) < such that for stationarily many S , there is a

C = C() cl(C) with:

(i) C = sup(C).

(ii) C Eai (hence Ca

i Ea

i , by the choice of as).

(iii) = sup{ C : h() Eai and h(sup(C )) Ea

i }.

Proof of Claim 1.13.c. Recall the definition of S and Cs from Claim 1.13.a. In

particular, for all S S ,

C = {(, ) : < otp(C) = }.

Let us first fix a S and suppose that for all C,

h() < Min(C \ ( + 1)). (b)

We define a function f = f : by:

If ( \ S


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Remember that all S 0 and f (, ) : (, ) (, ), as (, ) > .

So Fact 1.11 applies and f is constant on some stationary subset of(, ). Let us denote

that constant value by i = i(). Let def= (, ).

Observation. The set Eai

is an unbounded (even stationary) subset of C

(, ).

[Why? We can check that edef= { < : (, ) E} is a club of , so s

def=

e : f() = i

e is stationary, and we conclude that {(, ) : s} (, )

is stationary in (, ). Note that for each e, by the choice of E, we have that

cf((, )) < .

Now we take any s. Since cf((, )) < , by the definition of f, we have that

a(,)i

C is unbounded in (, ). Then

Eai C (,)sa

(,)i

C,

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and since s is stationary in (, ), we derive the desired conclusion.]

Continuation of the Proof of 1.13.c. So, by (V) in the choice of as and (3) of Fact

1.13.a, we conclude that Eai

contains the entire C (, ). Hence, by (IV) of

1.13.b,

Eai {a

i

: C (, )}.

Now, there must be some j = j < , such that { S : i is well defined and = j}

is a stationary subset of .

If we let vary, then j is defined for every S

which satisfies (b). We show that

the set of such is stationary in S .

We now get to use 5. from 1.13.a. Namely,

E = { < : ( < ) (h() < )}

is a club in . Therefore,

T = Thdef= { S : < (, C = (, ] E = )}

is stationary. It is easily seen that every element of T satisfies (b), so the set of all S

for which j is defined, is stationary.

Now, for some i() < , the set { S : j = i()} is stationary. Therefore,

{ = (, ) : S & i() is defined and = i()}

is stationary in . For every such = (, ), we define C = C() by Cdef= C (, ).

We can easily check that this is a well posed definition and that i = i() is as

required.1.13.c.

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Continuation of the proof of 1.13. Now we apply the previous claim to h ( < +).

For every < +, we fix i() < as guaranteed by the claim. Then note that for some

i() < , the set

W = { < + : i() = i()}

is unbounded in +. Of course, we can in fact assume W to be stationary, but we only

need it to be unbounded.

We now define a new family of functions, based on the hs.

For every W and S

let h, be the function with domain a

def= Ea

i()

,

defined by

h,() = Min(a \ h()).

Observe that a and = sup(a). We have noted before that for < < , we can

fix an , such that

h [, , ) < h [, , ).

So, if W, then

[, , ) = h,() h,().

Also, if < W, then

h() < h() h,() = h,() = [h(), h()) = & [h(), h()) a = .

Using the above functions, we define the following sets A, for < W.

A,def= { S : for unboundedly a

, we have h, () < h,()}.

We now show that these sets witness that D S is not +-saturated.

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Note:

(A) If < W, then A, is a stationary subset of .

[Why? To see this, fix such < and suppose that E is a club in which misses

A, .

Suppose that > , and S E. Then / A, , so there is 0 < sup(a

) = ,

such that for all (a \ 0),

h, () = h,(),

or, equivalently

h(), h()

a = .

We can also assume 0 , , so

h(), h()

= . In particular, h() / a. We can

further assume that satisfies the conclusion of Claim 1.13.c, with h = h and i = i()

(since W). Let C be as there.

But then the conclusion of Claim 1.13.c tells us that we can find a in C which is

greater than 0 and such that h() a.]

(B) If1 2 2 1 W, then A2,2 \ A1,1 is bounded.

[Why? This follows easily by the remarks after the definition of h, .]

Now assume, for contradiction, that D S is

+-saturated. Then, by (B):

(C) For each W, we have A,/D : (, +

) W is eventually constant, say for

[ , +).

So, Adef= A, satisfies 1 < 2 W = A1 A2(mod D) (again by (B)). Hence,

again by the +-saturation of D S ,

(D) A/D : W is eventually constant, say for .

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Choose W \ for < +i() such that

(1) < (2) = (1) < (2),

which is possible since W is unbounded. By (a) after the definition of h, we can find an

< such that

(1) < (2) < +i()

< = h(1)() < h(1) () < h(2)().

By clauses (C) and (D), for all , W and < +i(),

A = A = A, = A,+1 = (mod D).

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So we get a contradiction. 1.13.

A similar argument can be applied to other normal filters D on , under certain con-

ditions. In addition we shall see that, under some assumptions on the cardinal arithmetic,

the fact that D is not +-saturated is strongly witnessed, by the existence of a on D.

That is, we obtain D(S). This notation is explained in the following

Definition 1.14. If D is a normal filter on , and S is D-stationary, then D(S)

means:

There is a sequence A : S such that each A and for every A []

,

{ S : A = A} D+.

The statement D(S) means:

There is a sequence P : S such that each P P() and |P| , and for

every A , there is a C D such that for all ,

S C = A P.

By a well known result of Kunen (see [Ku 2]),

D(S) = D(S).

It is easily seen that D(S) implies the existence of an almost disjoint family of D-

stationary subsets of , of size 2. Therefore, ifD(S) holds, D is not 2-saturated.

Looking back at the proof of Theorem 1.13, there are two important facts that we

were using. The first is that there is a (, ) such that has the weak reflection

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property for . The other important ingredient of the proof is Fact 1.13.c. Dealing with

filters other than D, to obtain the corresponding version of 1.13.c, we have to strengthen

our assumptions, Here, I[, ) is as in the Definition 1.3.

Theorem 1.15. Assume > = cf() > = cf() > 0, while = + and

S / I[, ). Suppose that D is a normal filter on such that S D.

If, for some S,

() For every A D,

S : C a club in

{otp(C ) : C A} = modI[, )

is a D-stationary set in ,

then:

(1) Claim 1.13.c holds for any C and ai as in the assumptions of the Claim 1.13.c, with

stationary replaced by D-stationary .

(2) D S is not +-saturated.

(3) If 2 = and [] = , then D(S) holds.

We remind the reader of the notation [] for the revised cardinal power, from [Sh

460].

Definition 1.16. Suppose that > are infinite cardinals and is regular. Then

[] = Min{|P| : P [] and A [] (A the union of< elements of P)}.

Proof of 1.15. (1) and (2) are easily adjusted from Theorem 1.13.

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(3) The conclusion is the same as that of [Sh 186,3], and the proof is the same. The

assumptions on the cardinal arithmetic are here the same as in [Sh 186,3], with from

there equal to our . The only difference is that the proof in [Sh 186,3] started from ,

but we can use Fact 1.13.a instead.1.15.

2. The

(S) principle. Suppose that S is a stationary subset of a regular

uncountable cardinal . The aim of this section is to formulate a combinatorial principle

which suffices to show that certain normal filters on cannot be +-saturated. The

principle,

(S), is a form of(S), where is a sequence of ordinals. Our interest in this

comes from two facts presented in 2.8. Firstly, if is a successor of a singular cardinal

of uncountable cofinality , then ( \ S) is always true. This can be used to obtain

an alternative proof of that part of the result from [Sh 212, 14]=[Sh 247, 6] which states

that no normal filter concentrating on \ S is +-saturated. Secondly, if I[, ) S

contains only nonstationary sets, (S) is true, so by 2.5. we can conclude that D S

is not +-saturated. The key to the proof of 2.8. is the combinatorial lemma 2.7.

We commence by recalling the definition of some versions of .

Definition 2.0. Let be an uncountable regular cardinal and S stationary.

Then:

(0) (S) means:

There is a sequence A = A : S LIM such that:

(i) For each S LIM, the set A is an unbounded subset of .

(ii) For all A [], the set

GA(S)[A]def= { S LIM : A A}

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is non-empty.

(1) (S) means:

There is a sequence P = P

: S LIM such that

(i) For each S LIM, we have |P| ||.

(ii) If B P, then B is an unbounded subset of .

(iii) For every A [], there is a club CA of such that

GP(S)[A]def= { S LIM : B P (B A)} CA S LIM.

Many authors use a different definition of, in which every unbounded set is required

to be guessed stationarily many times. The following well known fact shows that the two

definitions are equivalent. We also include some other easy observations about Definition

2.0.

Fact 2.1. Assume that and S are as in Definition 2.0.

(0) If A = A : S LIM is a (S) -sequence, then, for every A [], the set

GA(S)[A] is stationary.

(1) If (S) holds, then there is a (S)-sequence A : S LIM such that for

each S LIM, we have that otp(A) = cf().

(3) Suppose that D is a normal filter on and S D+ is such that P exemplifies that

(S) holds. Then, for every A [], the set GP(S)[A] is D-stationary.

Proof. (0) Otherwise, we could find an A [] and a club C in , such that for all

in C S LIM, the set A is not a subset ofA. Then set A = {Min(A \ ) : C},

so A []. But if A A, then, A is also a subset of A. On the other hand, since

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A is unbounded in (as A is), also C is unbounded in , therefore C. This is a

contradiction.

(1) Suppose that B : S LIM exemplifies (S) and define for each

S LIM, the set A to be any cofinal subset ofB with otp(A) = cf().

(3) We simply remind the reader of the following elementary

Observation. For every club C of, we have C D (so the set S C is D-stationary).

[Why? Suppose that C is a club of such that C / D, so S\ C D +. We define the

following function, for S\ C:

f()def= sup(C )

and we note that f is regressive on S \ C. Then we can find a T S \ C which is

D-stationary and such that f T is a constant. Then T must be bounded, which is a

contradiction.]

2.1.

Now we introduce the version of the principle that will mainly interest us. The

guessing requirement is weaker, while the order type of the sets entering each family in the

-sequence is controlled by a sequence of ordinals.

Definition 2.2. Let be a regular uncountable cardinal and S stationary. Let,

= : S be a sequence of ordinals. Then

(S) means:

There is a sequence P = P : S LIM such that:

(i) For each S LIM, the family P consists of || unbounded subsets of .

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(ii) For each S LIM and B P, we have otp(B) < .

(iii) For A [], there is a club CA of such that

GP

(S)[A] def= { :

B P

= sup(B A)

} CA S LIM.

If for each S, = cf() + 1, we omit in the above notation.

If for some we have that = for all S LIM, then we write (S) rather

than

(S).

We make some easy remarks on Definition 2.2.

Observation 2.3. Assume that , S and are as in Definition 2.2.

(0) If the set of all S LIM for which > cf() is non-stationary, then (S)

is false. Otherwise (S) =

(S).

(1) If cf()

{} otherwise.

Then |P| = |P|cf() and, by (0), P : S LIM exemplifies

(S).

(2) Like 2.1.3. 2.3.

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We can consider also -sequences whose failure to guess is always confined to a set in

a given ideal on . In this context, for example (S) will mean (S)/J.

Definition 2.4. Let and S be as above, while I is an ideal on . Then

(S)/I means:

There is a sequence A = A : S such that each A is cofinal subset of , and

this sequence has the following property. For every A [], the set

GA(S)/I[A]def= { : A A}

satisfies GA(S)/I[A] / I.

If D is the dual filter of the ideal I, then (S)/D means the same as (S)/I. We

extend this definition in the obvious way to the other mentioned versions of the principle.

Theorem 2.5. Suppose that = + and is a limit cardinal.

(1) Assume that

(S) holds for a stationary S

. Then no normal filter

Don S

is +-saturated.

(2) If S is stationary and D is a normal filter on S such that (S)/D holds,

then D is not +-saturated.

Proof. Let us fix a sequence P = P : S LIM which exemplifies (S).

Therefore, for each S LIM, we have a family P = {P,i : i < i } such that each

P,i is a cofinal subset of of order type otp(P,i) < .

We fix a 1-1 onto pairing function pr : \ such that for each , , we

have

Max{, } pr(, ) < (|| + ||)+.

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(Here, we use the convention that n+ = 1 for n .)

We shall also fix a club C of \ such that

, < & C = pr(, ) < .

Now, we choose sets b for < + and < , and the functions h : for <

+

as in the proof of 1.13, and with the same properties as there. For < +, we define the

unbounded subset X of by

Xdef

=

pr

, h()

: <

,

and partial functions g by

g()def= Min{i < : sup(X P,i) = }.

In fact, the domain of each g is exactly the set GP

(S)[X ], so Dom(g) is D-stationary

(by 2.3.2). For Dom(g) C, we can define

f()def= pr

g(), |P,g()|

+.

Notice that g is regressive on its domain, so if and f() is defined, then f() <

(as C). Therefore, there is a D-stationary set B such that f B is constantly equal

to pr(i , ) for some i and a regular cardinal < . Then there are i < and a regular

cardinal < such that the set

Wdef=

< + :

i ,

=

i,

is unbounded.

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Let us define for S acc(C) LIM with i < i the set

ddef=

:

pr(, ) P,i

:

pr(, ) P,i

.

By the definition of pr, we have d . In fact, since acc(C), the set d is an

unbounded subset of. Like in 1.13, we can define for W and for which d is defined,

a function h, on d given by

h, () =

Min

d \ (h() + 1)

if d \

h() + 1

=

otherwise.

Therefore

h, : d d {}.

For , W we define sets

A, = { S : d is defined and for unboundedly d we have h, () < h,()}.

Assume now that D is +-saturated, and let us make some simple observations about the

just defined sets:

(a) A,/D increase with and decrease with .

(b) Since D is +-saturated, for any fixed W, the sequence A,/D : < + is

eventually constant, let us say for [, +) W.

Similarly,

(c) Adef= A,/D : <

+ are eventually constant, say for W \ ().

We choose by induction on < ordinals such that

(i)

(), +

.

(ii) W.

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(iii) are strictly increasing with .

(iv) +1 > .

Therefore

(1) < (2) < = A(1),(2)/D = A(1),(1) /D = A()/D.

Let () < be such that for all

(), +

, the sequence h() : < is strictly

increasing. (Recall that for < < +, we know that h is eventually strictly less than

h.) We can as well assume that () > .

By the above and the fact that < , we can find a set E D such that

(1) < (2) < = A(1),(2) E = A(1),(1) E = A() E.

We can assume that Min(E) > (). Without loss of generality, we can also add that

E acc(C) and

< & E S LIM = h

,

since < .

Now we discuss the two possible cases, the first of which corresponds to the situation

in 1.13.

Case 1. For some (), we have W and A() B is D-stationary.

We choose a E B A(). In particular, d is defined. We consider h, : h(1).

On the other hand, h(1) < h(1) d, so h, (1) h(1). Therefore, h, (1) =

h,(1), which is a contradiction to 1 > 1.

(2) Follows from the proof of (1).2.5.

Like and by the same proof, the usual principle on can be used for guessing not

just unbounded subsets of, but any other structure which can be coded by the unbounded

subsets of . With the principle, this does not seem to be the case, or at least the

-like proof fails. In particular, we do not know if the following is true:

Question 2.6. Suppose that is a regular uncountable cardinal and S a stationary

subset of such that

(S) holds. Is it true that there is a sequence

F:

S

LIM

with the following properties:

(i) Each F consists of partial functions from to , each of which has an unbounded

subset of as its domain.

(ii) |F| .

(iii) For every function f : , there is a club Cf of with the property

Cf S LIM =

g F

sup{ Dom(g) : g() = f()} =

?

We note that a positive answer to 2.6. would quite simplify the proof of 2.5.

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Our next goal is to prove that ( \ S) is true for any which is the successor of

a singular cardinal of uncountable cofinality . The key is the following

Lemma 2.7. Assume that = +

and 0 < = cf() < . Also = i .

Then there is a sequence

ai : i < , <

such that for every < , the sets ai (i < ) are subsets of which are -increasing in i,

with |ai | i, and such that:

For any f : , if

Afi =

< : = sup{ ai : f() ai }

then

(A) A

f

i are -increasing in i.

(B) \ S i 0.)

So

(C

) If Sfdef= S \ i


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First we fix a sequence e : LIM such that e is a club of with otp(e) =

cf(). Then we define by induction on < sets ai for all i < , requiring that for all

limit

i cf() = eai a

i ,

in addition to the requirements that we already mentioned in the statement of the lemma.

Now we suppose that f : is given, and check claims (A)(D).

(A) This follows from the fact that for every < ,

i < j < = ai aj .

(B) Let

Edef=

: = sup{ < : f() < }

.

Then E is a club of . We shall show that

( \ S) acc(E) i


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(C) Suppose that cf() > 0 and Afi reflects on . In particular, A

fi e is

stationary in . Given an < , we can find a Afi e such that < . Then, there

is a ai such that < and f() ai , by the definition of A

fi . Since cf() i,

we have that ai ai , and we are done.

(C

) This follows immediately by (C).

(D) Define h : S 0.

Then,

( \ S) holds and

(S

)/I[, ) holds.

(2) If in (1) we in addition assume that is a strong limit, then,

( \ S ) holds and (S)/I[, ) holds.

Proof. Let us fix sets ai (i < ) for < as guaranteed by Lemma 2.7. We fix for

all a cofinal subset P of such that otp(P) = cf().

(1) We shall define for ,

P =

{ai : i < & sup(a

i ) = } {P} if ||

{} otherwise.

Now, certainly each P is a family of || subsets of .

Suppose that A [] is given, and let f be an increasing enumeration of A. In

particular, f() for all . The set

Cdef= { < : (f() < )} \

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is a club of. Let us take any C S=. By (B) of Lemma 2.7, there is an i < such

that Afi , where Afi is as defined in Lemma 2.7. Then = sup{ a

i : f() a

i },

so = sup(A ai ).

This proves ( \ S ). With the same definition of P

def= P : < , let us

start again from an A [], and f and C as above. Let Sf and N be as in Lemma

2.7, so Sf \ N I[, ). If C S \ Sf, then we argue as above, to conclude that

GP

(S)/I[,)[A].

(2) With the same notation as above, we define

Pdef=

{all cofinal sequences of included in ai : i < } {P} if || {} otherwise.

Then |P| 2|| + < .

We argue similarly on S , using the set Sf as above.2.8.

As a consequence, we obtain another proof of (a part of) a theorem from [Sh 212, 14]

and [Sh 247, 6], as well as some other statements.

Corollary 2.9. Suppose that = + and > cf() = > 0. Then:

(1) No normal filter on \ S is +-saturated.

(2) If D is a normal filter on S such that I[, ) S contains only non-stationary

sets, then D is not +-saturated.

(3) If (S) holds, then D S is not +-saturated.

Proof. (1) ( \ S) holds, by 2.8. By 2.5, D cannot be

+-saturated.

(2) Similar.

(3) See 2.10.2 below.2.9.

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Concluding Remarks 2.10. (0) Another useful version of the -principle on for

a stationary S is (S), which says that

There is a sequence P = P : S LIM such that P is a family of

unbounded subsets of with the property that for all unbounded subsets A of ,

GP(S)[A]def= { S : B P(B A )}

is non-empty.

As opposed to the situation with , it does not have to be true that (S) = (S).

Of course, still

(S) =

(S), so we do not consider

here.

(1) We can also consider versions of or for which the size of each P is deter-

mined by some cardinal , not necessarily equal to ||. Also, we can combine this idea

with the idea of , so also the order type of sets in P is controlled by some prescribed

sequence .

(2) If we now define

(S) in the obvious way, then it follows from the proof of

2.5. that for = + and singular, (S) is enough to guarantee that D S is not

+-saturated. Therefore, in particular, (S) suffices.

For the successor of a strong limit, most reasonable versions of coincide.

(3) After hearing our lecture at the Logic Seminar in Jerusalem, Fall 1994, M. Magidor

showed us an alternative proof of 2.5 using elementary embeddings and ultrapowers and

not requiring to be a limit cardinal.

(4) The assumptions of 1.13. and 2.5. seem similar, but we point out that there are

in fact different. The existence of a < which weakly reflects at is not the same as

the assumption that I[, ) S contains only bounded sets.

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REFERENCES

[Ba] J. E. Baumgartner, A new class of order types, Annals of Mathematical Logic (9)187-222, 1976.

[CDSh 571] J. Cummings, M, Dzamonja and S. Shelah, A consistency result on weak reflection,

Fundamenta Mathematicae (148) 91-100, 1995.[FMS] M. Foreman, M. Magidor and S. Shelah, Martins maximum, saturated ideals and

nonregular ultrafilters I, Annals of Mathematics, Second Series (27) 1-47, 1988.[Fo] M. Foreman, More saturated ideals, Cabal Seminar 79-81, Proceedings, Caltech-UCLA

Logic Seminar 1979-81, A.S. Kekris, D.A. Martin and Y.N. Moschovakis (eds.) , Lec-ture Notes in Mathematics (1019) 1-27, Springer-Verlag

[MkSh 367] A. H. Mekler and S. Shelah, The consistency strength of every stationary set reflects,Israel Journal of Mathematics, (67) 353-366, 1989.

[Ku 1] K. Kunen, Saturated ideals, Journal of Symbolic Logic, (43) 65-76, 1978.[Ku 2] K. Kunen, Set theory, an Introduction to Independence Proofs, North-Holland, Ams-

terdam 1980.[Os] A . J. Ostaszewski, On countably compact perfectly normal spaces, Journal of LondonMathematical Society, (14) 505-516, 1975.

[Sh -g] S. Shelah, Cardinal Arithmetic, Oxford University Press 1994.[Sh 88a] S. Shelah, Appendix: on stationary sets ( to Classification of nonelementary classes.

II. Abstract elementary classes), in Classification theory (Chicago, IL, 1985), Pro-ceedings of the USA-Israel Conference on Classification Theory, Chicago, December1985; J.T. Baldwin (ed.) , Lecture Notes in Mathematics, (1292) 483495, SpringerBerlin 1987.

[Sh 98] S. Shelah, Whitehead groups may not be free, even assuming CH, Israel Journal ofMathematics, (35) 257-285, 1980.

[Sh 108] S. Shelah, On Successors of Singular Cardinals, In Logic Colloquium 78, M. Boffa, D.van Dalen, K. McAloon (eds.) 357-380, North-Holland Publishing Company 1979.

[Sh 186] S. Shelah, Diamonds, Uniformization, Journal of Symbolic Logic (49) 1022-1033, 1984.[Sh 212] S. Shelah, The existence of coding sets, In Around classification theory of models,

Lecture Notes in Mathematics, (1182) 188-202, Springer, Berlin 1986.[Sh 247] S. Shelah, More on stationary coding, In Around classification theory of models, Lec-

ture Notes in Mathematics, (1182) 224-246, Springer, Berlin 1986.[Sh 351] S.Shelah, Reflecting stationary sets and successors of singular cardinals, Archive for

Mathematical Logic (31) 25-53, 1991.[Sh 355] S. Shelah, +1 has a Jonsson Algebra, in Cardinal Arithmetic, Chapter II, Oxford

University Press 1994.[Sh 365] S. Shelah, Jonsson Algebras in Inaccessible Cardinals, in Cardinal Arithmetic, Chapter

III, Oxford University Press 1994.[Sh 420] S. Shelah, Advances in Cardinal Arithmetic, In Finite and Infinite Combinatorics in

Sets and Logic, N. Sauer et al (eds.), pp. 355-383, Kluwer Academic Publishers, 1993.[Sh 460] S. Shelah, The Generalized Continuum Hypothesis revisited, Israel Journal of Math-

ematics, submitted.[StvW] J. R. Steel and R. van Wesep, Two Consequences of Determinacy Consistent with

Choice, Transactions of the AMS, (272)1, 67-85, 1982.

mirna dzamonja and saharon shelah- saturated filters at successors of singulars, weak reflection and...

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