andrzej roslanowski and saharon shelah- generating ultrafilters in a reasonable way

8/3/2019 Andrzej Roslanowski and Saharon Shelah- Generating Ultrafilters in a Reasonable Way

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GENERATING ULTRAFILTERS IN A REASONABLE WAY

ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

Abstract. We continue investigations of reasonable ultrafilters on uncount-able cardinals defined in Shelah [8]. We introduce a general scheme of generat-ing a filter on from filters on smaller sets and we investigate the combinatoricsof objects obtained this way.

0. Introduction

Reasonable ultrafilters were introduced in Shelah [8] in order to suggest a lineof research that would in some sense repeat the beautiful theory created aroundthe notion of Ppoints on . The definition of reasonable ultrafilters involves twoconditions. The first one, so called the weak reasonability of an ultrafilter, is a wayto guarantee that we are not entering the realm of large cardinals: the consideredultrafilter is required to be very non-normal. Since this property will be also of someinterest in the present paper, let us recall the following definition and observation.

Definition 0.1 (Shelah [8, Def. 1.4]). (1) We say that a uniform ultrafilter Don is weakly reasonable if for every non-decreasing unbounded functionf there is a club C of such that

{[, + f()) : C} / D.

(2) Let D be an ultrafilter on , C be a club and let : < be theincreasing enumeration of C {0}. We define

D/C =

A :A

[, +1) D

.

(It is an ultrafilter on .) D/C will be called the quotient of D by C.

Observation 0.2 (Shelah [8, Obs. 1.5]). LetD be a uniform ultrafilter on a regularuncountable cardinal . Then the following conditions are equivalent:

(A) D is weakly reasonable,(B) for every increasing continuous sequence : < there is a club

C

of such that [, +1) : C

/ D,

Date: September 2007.1991 Mathematics Subject Classification. Primary 03E05; Secondary: 03E20.The first author would like to thank the Hebrew University of Jerusalem and the Lady Davis

Fellowship Trust for awarding him with Schonbrunn Visiting Professorship under which thisresearch was carried out.Both authors acknowledge support from the United States-Israel Binational Science Foundation(Grant no. 2002323). This is publication 889 of the second author.

1


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2 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

(C) for every club C of the quotient D/C does not extend the filter generatedby clubs of .

The second part of the definition of reasonable ultrafilters is directly related togeneralizing Ppoints to the context of weakly reasonable ultrafilters on an un-countable cardinal . To carry out this process we have to be somewhat creative inre-interpreting the property that any countable family of members of the ultrafilterhas a pseudo-intersection in the ultrafilter. An interesting way of doing this is toimpose some demands on how the ultrafilter on can be obtained from sequencesof objects on smaller cardinals (an approach motivated by Roslanowski and Shelah[3, 5, 6].) For instance we may consider sequences r = ( , d) : < such that : < is an increasing continuous sequence of ordinals below and d is anultrafilter on the interval [, +1). For each such sequence r we look at the familyof subsets of which are eventually large in every interval [, +1), that is weconsider the set fil(r) = A : ( < )( > )(A [ , +1) d). (The setfil(r) is a filter on .) There is a natural quasi-order on sequences r as above: we saythat r s if and only if fil(r) fil(s). Now the demand generalizing Ppointnessmay be phrased for an ultrafilter D on as follows: there is a (


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GENERATING ULTRAFILTERS IN A REASONABLE WAY 3

is inaccessible for that result: Theorem 4.8 shows that consistently there is a veryreasonable ultrafilter D on 1 such that Odd has a winning strategy in D .

Notation: Our notation is rather standard and compatible with that of classicaltextbooks (like Jech [1]).

(1) Ordinal numbers will be denoted be the lower case initial letters of theGreek alphabet ( , , , . . .) and also by i, j (with possible sub- and su-perscripts). Cardinal numbers will be called ,, (with possible sub- andsuperscripts). is always assumed to be an uncountable regularcardinal.

(2) For two sequences , we write whenever is a proper initial segmentof , and when either or = . The length of a sequence isdenoted by lh().

(3) We will use letters D , E , F and d (with possible indexes) to denote filterson various sets. Typically, D will be a filter on (possibly an ultrafilter),

while E, F will stand for filters on smaller sets. Also, in most cases d willbe an ultrafilter on a set of size less than .For a filter F of subsets of a set A, the family of all Fpositive subsets ofA is called F+. (So B F+ if and only if B A and B C = for allC F.)

(4) In forcing we keep the older convention that a stronger condition is thelarger one. For a forcing notion P, P stands for the canonical Pname forthe generic filter in P. With this one exception, all Pnames for objects inthe extension via P will be denoted with a tilde below (e.g.,

, X

).

1. Generating a filter from systems of local filters

Here we present the general scheme of generating a filter on a regular uncount-able cardinal by using smaller filters. Our approach is slightly different from theone in [8, 2] and/or [4, 1], but the difference is notational only (see 1.3 below).

Definition 1.1. (1) A system of local filters on is a family F such that all members of F are triples (,Z,F) such that Z , |Z| < ,

= min(Z) and F is a proper filter on Z, the set

< :

Z, F

(,Z,F) F

is unbounded in .

If above for every (,Z,F) F, the set Z is infinite and F is a non-principalultrafilter on Z, then we say that F is a system of local non-principalultrafilters.

(2) More generally, if is a property of filters, then a system of local filterson is a system of local filters F such that for every (,Z,F) F, the

filter F has the property . The full system of local filters is the familyof all triples (,Z,F) such that < , Z \ , |Z| < and F isa proper filter on Z with the property (assuming that it forms a systemof local filters). The full system of local non-principal ultrafilters on isdenoted by Fult or just F

ult (if is understood).

The next definition introduces the filters generated by some families of localfilters. As we have said in the introduction, our motivations have roots in forcingswith norms and this suggested us to use sometimes a forcing-like notation (e,g, Q)similar to that of [3]. It is also worth noticing that some families of generators may


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be used as forcing notions - for instance (Q0, ) is the forcing used in the end of

[4, Sec. 1].

Definition 1.2. Let F be a system of local filters on .

(1) We let Q(F) be the family of all sets r F such that <

|{(,Z,F) r : = }| <

and |r| = .

For r Q(F) we define

fil(r) =

A :

<

(,Z,F) r

A Z F

,

and we define a binary relation =F on Q(F) by

r1 F r2 if and only if (r1, r2 Q

(F) and) fil(r1) fil(r2).

(2) We say that an r Q(F) is strongly disjoint if and only if

<

|{(,Z,F) r : = }| < 2

, and

(1, Z1, F1), (2, Z2, F2) r1 < 2 Z1 2.We let Q

0(F) be the collection of all strongly disjoint elements ofQ

(F).

(3) We write Q,Q0 for Q

(F

ult),Q0(Fult), respectively (where, remember,

Fult is the full system of local non-principal ultrafilters).(4) For a set H Q(F) we let fil(H) =

fil(r) : r H}.

Remark 1.3. (1) Note that ifr Q0 then there is r Q0 such that fil(r

) =fil(r) and for some club C of we have

(, Z) :

d

(,Z,d) r

=

(, [, )) : C & = min

C\ ( + 1)

.

Thus Q0 is essentially the same as the one defined in [8, Def. 2.5].(2) If H Q(F) is

directed, then D = fil(H) is a filter on extendingthe filter of co-bounded sets. We may say the that the filter D is generatedby H or that H is the generating system for D.

Definition 1.4. Suppose that

(a) X is a non-empty set and F is a filter on X,(b) Fx is a filter on a set Zx (for x X).

We letF

xX

Fx =

A

xX

Zx : {x X : Zx A Fx} F

.

(Clearly,F

xX

Fx is a filter on

xX

Zx.) IfX is a linearly ordered set (e.g. it is a set

of ordinals) with no maximal element and F is the filter of all co-bounded subsets

of X, then we will write xX Fx instead ofF

xX Fx.Proposition 1.5 (Cf. [8, Prop. 2.9]). (1) LetF be a system of local filters on

and p, q Q(F). Then p q if and only if there is < such that

(,Z,F) q

A F+

>

(, Z, F) p

A Z (F)+

.

(2) Let p, q Q. Then the following are equivalent:(a) p q,(b) there is < such that

(,Z,d) q

A d

>

(, Z, d) p

A Z d

,


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(c) there is < such that if (,Z,d) q, , and X =

(, Z, d)

p : Z Z = , then X = and there is an ultrafilter e on X suchthat

d =

A Z : A e

{d : (, Z)((, Z, d) X)}

.

The quasi-orders (Q, ) and (Q0,

) are (


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Suppose rk(T) > 0. Let = { < : T} (so 0 < is acardinal). For < we put

T = {


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()2

(,Z,F) p

C

< Z

.

By the choice of we know that {[, +1) : C} fil(H), so necessarily{[, +1) : C}

fil(p)

+. Thus we may pick (,Z,F) p and C

such that Z [, +1) F+ (remember ()2), contradicting ()1.

Claim 1.9.2. (1) Ifp q, p, q H andC C are clubs of, then|S(q, C)\S(p,C)| < .

(2) If A , then there are p H and a club C such that either S(p,C) A or S(p,C) \ A.

Proof of the Claim. (1) Pick < so that(,Z,F) q

A F+

> ((, Z, F) p)(A Z (F)+)

(remember 1.5) and let < be such that < and

(,Z,F) q

Z

. Suppose that S(q, C

) \

. Then C

C and there is(,Z,F) q such that [, +1) Z F+. Since , we also have >

and hence there is (, Z, F) p such that [ , +1) Z Z (F)+. Hence we

may conclude that S(p,C).

(2) Assume A . Let A =

[, +1) : A

. Since fil(H) is an ultrafilter,then either A or \ A belongs to it. Suppose A fil(p) for some p H. Pick aclub C such that

() if (,Z,F) p and (sup(Z) + 1) C = , then A Z F.

Suppose S(p,C), so C and for some (,Z,F) p we have [, +1) Z F+. It follows from () that A Z F and therefore A. Thus S(p,C) A.

If \ A fil(H), then we proceed in an analogous manner.

Let

D =

A : |S(p,C) \ A| < for some p H and a club C

.

It follows from 1.9.1 that all members of D are stationary and since H is directedwe may use 1.9.2(1) to argue that D is a filter on . By 1.9.2(2) we see that Dis an ultrafilter on (so it also contains all clubs as its members are stationary).Since H is (


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2. Systems of local ultrafilters

In this section we are interested in the full system Fult of local ultrafilters on and Q,Q0. The first question that one may ask is whether weakly reasonableultrafilters on generated by some H Q0(F) can be obtained by the use ofQ

0.

It occurs that it does matter which system of local filters we are using.

Definition 2.1. A filter F on a set Z is called an unultra filter, if for every A F+

there is B A such that both B F+ and A \ B F+. The full system of localunultra filters on will be denoted by Funu. (Thus Funu consists of all triples(,Z,F) such that = Z , |Z| < , = min(Z) and F is an unultra filter onZ.)

Observation 2.2. (1) If F is an unultra filter on Z, A F+, then F + Adef=

{B Z : B (Z\ A) F} is an unultra filter.(2) Suppose that is a limit ordinal, {Z : < } is a family of pairwise

disjoint non-empty sets, F is a filter on Z (for < ). Then


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and otherwise p+1 =

(,Z,F) p : Z\ Y F

.(iv) p+2 p+1 and for some club C of , for every C we have

Z whenever (,Z,F) p+2, <

, and

+1 < min

:

Z, F

(,Z,F) p+2

.

(v) p+3 =

(,Z,F+ A) : (,Z,F) p+2

where for every (,Z,F) p+2the set A F+ is such that

( , Y , d) r

A Y / d

.

(vi) p+3+n = p+3 for all n < .(vii) For every r Q0 and <

+, if (,Z,F) p and A F+, then there isA A such that

A F+ and

( , Y , d) r

A Y / d

.

Conditions (o)(vi) fully describe how the construction is carried out and 1.5+2.2imply that p : < + Q0(F

unu) is increasing. However, we have to arguethat the demand in (vii) is satisfied, as it is crucial for the possibility of satisfying

the demand in (v). Let r Q0. By induction on < we show that for every(,Z,F) p we have

()(,Z,F) if A F+, then there is A A such that A F+ and

( , Y , d)

r

A Y / d

.

(For a set A as above we will say that it works for F and r.)

Step < .Note that for each limit ordinal < there is at most one ( , Y , d) r such thatY [, + ) is infinite. Assume A [, + ) is infinite. Considering any twodisjoint infinite sets A, A A we easily see that one of them must work for thefilter of co-finite subsets of [, + ) and r.

Step = + n + 1, < + is limit, n < .

If (,Z,F) p+n, A F+, A A and A A works for F and r, then also Aworks for F + A and r.

Step = < + is limit.Suppose that (,Z,F) p . If cf() = , then (,Z,F) p for some

< (see(ii)) so the inductive hypothesis applies directly. So assume that cf () < . ThenZ =

i


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Case C: For each < there is ( , Y , d) r such that < < sup(Y) < .Let A F+. Then the set I = {i < cf() : A Zi (Fi)+} is unbounded in

cf() and using the assumptions of the current case we may choose an increasingsequence ij : j < cf() I such that for every ( , Y , d) r there is at most onej < cf() such that Zij Y = . For each j < cf() pick A

ij

(Fij )+ included in

A Zij which works for Fij and r, and then put A =

j


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Theorem 2.10. Assume that = (,Z,d) / D

.

It follows from the choice of N thatif p0, . . . , pn Q

(F

ultN ) and (,Z,d) (p0) . . . (pn),

then there are Z, d such that (, Z, d) (p0) . . . (pn) FultN .

Consequently, we may repeat arguments of the previous case replacing in clause

(ii) D FultN by F

ultN \ D. Then we obtain q Q

0(F

ultN ) Q

0 such that H0 {q}is linked and q D = .

Claim 2.10.2. Assume that H0 Q(F

ultN ) Q

is linked, |H0| < cov(M

,) and

a sequence : < is increasing continuously. Then there are p Q0(FultN )

and a club C of such that

(a) H0 {p} is linked, and(b)

[+1, ) : < are successive members of C

fil(p).

Proof of the Claim. This is essentially [8, Claim 2.14.4].


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Now we employ a bookkeeping device to construct inductively a sequence q : < 2 Q0(F

ultN ) such that

for each < 2 the family {q : < } is linked, if D FultN , then for some < 2

we have q D or q D = , if : < is increasing continuous, then for some < 2 and a

club C of we have that[+1, ) : < are successive members of C

fil(q).

Since |FultN | = , so there are no problems with carrying out the construction.It should be clear that at the end the family {q : < 2

} is linked, big and itgenerates a weakly reasonable ultrafilter.

Note that we may modify the construction in the proof of Theorem 2.10 sothat the resulting H is directed. Namely, by an argument similar to the one inClaim 2.10.1 we may show, that if H0 Q

(F

ultN ) is linked, |H0| < cov(M

,)

and p0, p1 H0, then there is q Q0(FultN ) such that q (p0) (p1) andH0 {q} is linked. With this claim in hands we may modify the inductive choiceof q : < 2 so that at the end {q : < 2} is directed. However, we do notknow how to guarantee the opposite, that the family {q : < 2} is not directedor even better, that for no directed H Q0 do we have fil(H) = fil({q : < 2

}).Thus the following question remains open.

Problem 2.11. Does H Q0 is linked and big imply that H is directed?

3. Systems of local pararegular filters

In this section we are interested in filters associated with the full system Fpr oflocal pararegular filters on and we show their relation to numbers of generators

(in standard sense) of some filters on .Definition 3.1. Suppose that Z is an infinite set, = min(Z). A pararegular filter on Z is a filter F on Z such that for some system Au : u []


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(i) p = pi : i < + Q satisfies ( < )(i < +)(qi p

i ),

(ii) : + + is regressive, i.e., (i < +)((i) < 1 + i);

Player II answers choosing a sequence q

= qi : i <

+

Q suchthat (i < +)(pi q

i ).

If at some stage of the game Player I does not have any legal move, then heloses. If the game lasted steps, Player I wins a play p, q, : < if there is a club C of + such that for each distinct members i, j of Csatisfying cf(i) = cf(j) = and ( < )((i) = (j)), the set {qi :

andrzej roslanowski and saharon shelah- generating ultrafilters in a reasonable way

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