saharon shelah- the combinatorics of reasonable ultrafilters

8/3/2019 Saharon Shelah- The Combinatorics of Reasonable Ultrafilters

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THE COMBINATORICS OF REASONABLE

ULTRAFILTERS

SAHARON SHELAH

Abstract. We are interested in generalizing part of the theory of ul-trafilters on to larger cardinals. Here we set the scene for furtherinvestigations introducing properties of ultrafilters in strong sense dualto being normal.

0. Introduction

Questions concerning ultrafilters on have occurred to be very stimulat-ing for research in several subareas of Set Theory and Topology. We hopethat this success story could be repeated for ultrafilters on uncountable reg-ular cardinals , particularly if is strongly inaccessible. Our aim in thepresent paper is to introduce new properties of ultrafilters and argue thatthese properties could play the stimulating role that was once played byPpoints on .

In a long run, we plan to find generalizations of the following results:

(a) Consistently, some ultrafilters on are generated by < 20 manysets.

(b) P-points are preserved by some forcing notions (see, e.g., [14, V],[7]).

(c) Consistently, there is no Ppoint.(d) For a function f : and ultrafilter D on , let

D/fdef= {A : f1(A) D};

it is an ultrafilter on (of course, we are interested in the cases

when D and D/f are uniform, which in this case is the same asnon-principal). By Blass and Shelah [1], consistently for any twonon-principal ultrafilters D1, D2 on there are finite-to-one non-decreasing functions f1, f2 : such that D1/f1 = D2/f2.

Date: September 2005.1991 Mathematics Subject Classification. Primary 03E05; Secondary: 03E20.The author acknowledges support from the United States-Israel Binational Science

Foundation (Grant no. 2002323). Publication 830.1


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2 SAHARON SHELAH

(f) For a significant family of forcing notions built according to thescheme of creatures of [7] we may consider an appropriate filter,

i.e., if p : < 1 is -increasing it may define an ultrafilter (see[7, 5,6]) which is not necessarily generated by 1-sets, so we mayask on this.

There are many works on normal ultrafilters; their parallel on are Ram-sey ultrafilters. Now, every Ramsey ultrafilter on is a P-point but thereare P-points of very different characters, e.g., P-point with no Ramsey ul-trafilter below. Gitik [4] has investigated generalizations of P-points fornormal ultrafilters. But this paper goes in a different direction (which up torecently I have not considered to be fruitful) and we restrict our attention toultrafilters which are very non-normal the weakly reasonable ultrafilters.

What is a weakly reasonable ultrafilter on ? It is a uniform ultrafilter ona regular cardinal which does not contain some club of and such thatthis property is preserved if we divide it by a non-decreasing f : with unbounded range (see Definition 1.4 below).

We also want that our ultrafilters generalize Ppoints on and in thesecond section we introduce reasonable and very reasonable ultrafilters. Theproperty defining Ppoints is that countable families of sets from the ultra-filter have pseudo-intersections in the ultrafilter. We modify this propertyso that we involve some description of how the considered ultrafilter is gen-erated, and we postulate that the generating systems are suitably directed.This is a replacement for the existence of pseudo-intersections and it is theessence of Definition 2.5(4,5). The third section shows that the number ofgenerating systems (of our type) for somewhat reasonable ultrafilters cannotbe too small. We conclude the paper with a section listing open problemsand describing further research.

Notation: Our notation is rather standard and compatible with that ofclassical textbooks (like Jech [5]). In forcing we keep the older conventionthat a stronger condition is the larger one. (However, in the present paperwe use forcing notions only for combinatorial constructions and almost everymention of forcing just means that we a dealing with a transitive reflexive

relation P = (P, P).)(1) Ordinal numbers will be denoted be the lower case initial letters of

the Greek alphabet ( , , , . . .) and also by i, j (with possible sub-and superscripts).

(2) Cardinal numbers will be called ,,; will be always assumed tobe a regular uncountable cardinal (we may forget to mention it).

(3) D,U will denote filters on , G, G, G will be subsets of specificpartial orders used to generate filters on .


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THE COMBI NATORICS OF REASONABLE ULTRAFILTERS 3

(4) A bar above a letter denotes that the object considered is a sequence;usually X will be Xi : i < , where is the length lh(X) of X.

Sometimes our sequences will be indexed by a set of ordinals, sayS , and then X will typically be X : S.

Definition 0.1. A dominating family in is a family F such that

(g )(f F)( < )( > )(g() < f()).

The dominating number d is defined as

d = min

F : F is a dominating family in

.

A clubdominating family in is a family F such that

g f F{ < : g() f()} is non-stationary in .The cl()dominating number dcl() is defined as

dcl() = min

F : F is a cl()dominating family in

.

On d, dcl() see, e.g., in Cummings and Shelah [2].

Acknowledgment: I thank Tomek Bartoszynski and Andrzej Roslanowskifor stimulating discussions.

1. Weakly reasonable ultrafilters

In Definition 1.4(1) we formulate the main property of ultrafilters on which is of interest to us: being a weakly reasonable ultrafilter. In thespectrum of all ultrafilters, weakly reasonable ultrafilters are at the oppositeend to the one occupied by normal ultrafilters. We show that there exist(in ZFC) weakly reasonable ultrafilters (see 1.10) and we also give someproperties of such ultrafilters.

Definition 1.1. For a cardinal ,

(a) ulf() is the set of all ultrafilters on ,(b) uuf() is the family of all uniform ultrafilters on ,(c) ifD is an ultrafilter on and f , then

D/fdef= {A : f1(A) D}.

Let us note that in the literature D/f is also denoted by f(D) or f(D)and it is called the image or the projection of the ultrafilter D by f. Weuse the quotient notation and terminology because we will deal mostly withD/C, where:

Definition 1.2. Assume D is an ultrafilter on .


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4 SAHARON SHELAH

(1) IfE is an equivalence relation on , then fE is defined by

fE() = otp

{ < : = min(/E) < min(/E)}

,

and D/E is D/fE. (Here, /E stands for the Eequivalence classof .)

(2) For a club C of let EC be the following equivalence relation on :

EC iff ( C)( < < )

(so EC is the equivalence relation determined by the partition of into intervals [, ) for consecutive members < of C {0}). LetD/C be D/EC.

(3) F is the family of all non-decreasing unbounded functions from to .

Observation 1.3. Assume that is a regular cardinal, D ulf().

(1) If f : , then D/f ulf().(2) If f F and D is uniform, then also D/f is a uniform ultrafilter

on .(3) If C is a club of and : < is the increasing enumeration of

C {0}, then for a set A ,

A D/C if and only if

[, +1) : A

D.

Definition 1.4. Let D be a uniform ultrafilter on .

(1) We say that D is weakly reasonable if for every f F there is aclub C of such that

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

(2) We define a game D between two players, Odd and Even, as follows.A play ofD lasts steps and during a play an increasing continuoussequence = i : i < is constructed. The terms of arechosen successively by the two players so that Even chooses the ifor even i (including limit stages i where she has no free choice) and

Odd chooses i for odd i.Even wins the play if and only if

{[2i+1, 2i+2) : i < } D.

Observation 1.5. LetD uuf(). Then the following conditions are equiv-alent:

(A) D is weakly reasonable,


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(B) for every increasing continuous sequence : < there is aclub C of such that

[, +1) : C

/ D,

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

Proposition 1.6. Assume D uuf().

(1) If is strongly inaccessible and Odd has a winning strategy inD,then D is not weakly reasonable.

(2) If D is not weakly reasonable, then Odd has a winning strategy inthe game D.

(3) In part (1) instead is strongly inaccessible, it suffices to assume

.

Proof. (1) Suppose towards contradiction that is strongly inaccessible,Odd has a winning strategy st in the game D but D is weakly reasonable.By induction on < choose an increasing continuous sequence N : < of elementary submodels of H(++) so that for each :

(a) N (H(++), ,


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6 SAHARON SHELAH

model N2i+1 is closed under forming sequences of length 2i + 1 (by (b)),we conclude that j : j 2i N2i+1. Since st N0 N2i+1, clearly

2i+1 N2i+1 and therefore 2i+1 < 2i+1. Hence{[2i, 2i+1) : i < }

{[2i , 2i+1) : i < }

{[, +1) : C} / D.

But st is a winning strategy for Odd, so he wins the play and{[2i+1, 2i+2) : i < } / D,

a contradiction.

(2) Suppose that D uuf() is not weakly reasonable. Then we may findf F such that for every club C of we have

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

Let st be a strategy of Odd in D which instructs him to play as follows.For an odd ordinal i = i0 + 1 < , in the i-th move of a play, ifj : j i0has been played so far, then Odd plays i = i0 + f(i0) + 1.

We claim that st is a winning strategy for Odd (in D). To this endsuppose that j : j < is a result of a play ofD in which Odd usesthe strategy st. Let C = {i : i < is limit } it is a club of , so by thechoice of f we have

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

Since

{[, + f()) : C}

{[2i, 2i+1) : i < } we may nowconclude that Odd indeed wins the play.

Remark 1.7. Let us note that some assumptions on in 1.6(1) are needed.This will be shown in the subsequent paper Roslanowski and Shelah [11].

Lemma 1.8. Suppose that is a regular uncountable cardinal, D uuf()is a weakly reasonable ultrafilter and i : i < is an increasing contin-uous sequence of ordinals below . Then there is an increasing continuoussequence : < consisting of limit ordinals and such that

{[2+1, 2+2) : < } D.Proof. It follows from 1.5 that we may find a club C of such that allmembers of C are limit ordinals and

[, +1) : C

/ D. LetC+ = C { + 1 : C} (clearly it is a club of ) and let : < be the increasing enumeration of C+. Note that C = { : < is even }and, for an even ordinal < , +1 = + 1. Hence

[ , +1) : < is even

=

[ , +1) : < is even

=[, +1) : C

/ D


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Consequently,

[ , +1) : < is odd

D.

Theorem 1.9. If is a regular uncountable cardinal and D uuf() isweakly reasonable, then D is a regular ultrafilter.

Proof. Using Lemma 1.8 we may choose by induction on < a sequence : < so that

(a) = i : i < is an increasing continuous sequence of non-successor ordinals below , 0 = 0,

(b) the set Adef=

[2i+1, 2i+2) : i < } belongs to D,

(c) if < , i < , then i {j : j < is a limit ordinal or zero }.

For < let f : A be such that

[2i+1, 2i+2) f() = 2i+1.

Note that

() if < < , A A, then f() < f().

[Why? Let f() = 2i+1 (so [

2i+1,

2i+2)). It follows from (c) that

f() {j : j < is a limit ordinal or zero } and hence (since also

f() ) we may conclude that f() < f(). ]For < , let w = { < : A}. It follows from () that (for every

< ) the sequence f() : w is strictly decreasing, so necessarilyeach w is finite. Since A D for each < (by (b)), we have shown the

regularity of D. Theorem 1.10. Let > 0 be a regular cardinal. Then there is a uniformweakly reasonable ultrafilter D on .

Proof. Let {f : < d} be a dominating family and for < d let Cbe a club of such that members of C are limit ordinals and

( C)( < )(f() < ).

Let ,i : i < be the increasing enumeration of C.By induction on we will choose sets E, A so that for each < d:

(a) A

is an unbounded subset of and E

C

is a club of,(b) A

[,, ,+1) : E} = ,

(c) if n < , 0 < .. . < n1 < , then A i


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8 SAHARON SHELAH

() if ,i < ,i+1, then g() = min{ > ,i+1 : [,i+1, ) A =

}.

The family {g

: >} is a subset of of cardinality || + 0 < d, so

it cannot be a dominating family. Therefore we may pick a function h

such that

( >)( < )(g() < h()).

Put

E = { < : = , is a limit ordinal and ( < )(h() < )} andA =

[,+1, ,) : < are successive members of E}.

It should be clear that E, A satisfy demands (a), (b).Let us argue that also condition (c) holds true. Let > and we shall

prove that A A is unbounded in . By the choice of h, the set B ={ < : g

() < h()} is of cardinality . Let us fix for a moment B

and let i < be such that ,i < ,i+1. Let sup(E ,i+1) = = ,and min(E \ ,+1) = = ,. Then , are successive members of Eand

,i < ,i+1 < .

Hence (by the definition of E and by B) we get

[,i+1, g()) [,i+1, h()) [,+1, ,) A.

It follows from () that [,i+1, g()) A = , and consequently A A\

= . Since B = we may now easily conclude that A A = ,showing that A, E are as required.

After the construction is carried out (and we have the sequence E, A : < d) we may find a uniform ultrafilter D on such that {A : < d} D (remember the demand in (c)). We claim that D is weakly reasonable.To this end suppose that C is a club of and : < is theincreasing enumeration of C. By the choice of f, C (for < d) we mayfind < d and j0 < such that

(i j0)( [,i, ,i+1) C > 2).

LetC = { E C\j0 : = , = is a limit ordinal }

(it is a club of ). Since for C we have that , = < +1 < ,+1we may easily conclude from (b) that

[, +1) : C

/ D,

completing the proof (remember 1.5).


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2. More reasonable ultrafilters

In this section we propose a property of ultrafilters stronger than beingweakly reasonable (see Definition 2.5(5)). We believe that the notion ofveryreasonable ultrafilters is the right re-interpretation of being a Ppoint in thesetting of very non-normal ultrafilters on an uncountable regular cardinal. We start with describing a forcing notion Q1 which motivated our choiceof generating systems of 2.5.

Like before, is always assumed to be an uncountable regular cardinal.

Definition 2.1. We define a forcing notion Q1 as follows.A condition in Q1 is a tuple p = (

p, Cp, Zp : Cp, dp : C

p)such that

(i) p

< , Cp

a club of consisting of limit ordinals only, and for Cp:(ii) Zp =

, min

Cp \ ( + 1)

and

(iii) dp P(Zp ) is a proper ultrafilter on Z

p .

The order Q1

= ofQ1 is given byp Q1

q if and only if

(a) p q, Cp p Cq Cp and(b) if < are successive members of Cq (so Zq = [, )), then

A dq Cp [, )A Zp dp.Remark 2.2. The forcing notion Q1 can be represented according to theframework of [10, B.5].

Proposition 2.3. (1) Q1 is a partial order, Q1 = 2

2


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10 SAHARON SHELAH

It is easy to check that p = (,C, Z : C, d : C) belongs to Q1and that it is a condition stronger than all pi (for i < ).

(4) Let p Q1 and A . Just for simplicity we may assume that p Cp

(as we may always increase p). Put

Ydef= { Cp : A Zp d

p}

and let us consider two cases.Case 1: Y is unbounded in .

Then we may choose an increasing continuous sequence i : i < Cp

such that 0 = p and

i <

[i, i+1) Y =

. Put

= p, C = {i : i < } (Cp p),

if Cp

p

, then Z = Z

p

and d = d

p

, if = i, i < , then Z = [i, i+1) and

d =

B Z : B Zp

min(Y\) dp

min(Y\)

.

It is straightforward to verify that q = (,C, Z : C, d : C) Q1 is a condition stronger than p and it is also clear that ( C\ p)(A Z d).

Case 2: Y is bounded in .Then the set \ Y is unbounded, so we may apply the construction of qfrom Case 1 replacing Y by its complement \Y. It should be clear that thecondition q which we will get then satisfies ( Cq \ p)(A Zq / d

q).

Remark 2.4. The following discussion presents our motivations for the def-initions and concepts presented later in this section.

Suppose that G Q1 is a generic filter over V. In V[G] we defineC =

{Cpp : p G} and for Cwe let d = dp for some (equivalently:

all) p G such that < p and Cp (, p) = . Then C is a club of and (for C) d is an ultrafilter on [, min(C\ ( + 1))). Let

D =

A P()V :

<

>

A [, min(C\ ( + 1))) d

.

It follows from 2.3(4) that D is an ultrafilter on the Boolean algebra P()V.Let D

be a Q1name for the D defined as above. Note that if p Q1,

A and ( < )( Cp \ )(A Zp dp), then p Q1 A D.

Plainly, the family {A : p Q1

A D

} is a uniform filter on , and,

of course, for a generic filter G Q1 over V,

D

G =

{A : p Q1

A D

} : p G

.

Definition 2.5. (1) We define a forcing notion Q0 as follows.A condition in Q0 is a tuple p = (C

p, Zp : Cp, dp : C

p)


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THE COMBINATORICS OF REASONABLE ULTRAFILTERS 11

such that (0, Cp, Zp : Cp, dp : C

p) Q1;The order Q0

= ofQ0 is inherited from Q

1 in a natural way.

(2) We define a relation Q0

= on Q0 as follows:p q if and only if for some < we have

(Cp \ , Zp : Cp \ , dp : C

p \ ) Q0

(Cq \ , Zq : Cq \ , dq : C

q \ ).

(3) For a condition q Q0 we let

fil(q)def=

A : ( < )( Cq \ )(A Zq dq)

,

and for a set G Q0 we let fil(G)

def=

{fil(p) : p G}. We alsodefine a binary relation 0 on Q0 by

p 0 q if and only if fil(p) fil(q).

(4) We say that an ultrafilter D on is reasonable if it is weakly rea-sonable (see 1.4(1)) and there is a directed (with respect to 0) setG Q0 such that D = fil(G

). The family G may be called thegenerating system for D.

(5) An ultrafilter D on is said to be very reasonable if it is weaklyreasonable and there is a (


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12 SAHARON SHELAH

(b) there is < such that

C

q

\

A dq

Cp

A Zp

dp

,(c) there is < such that

if Cq \ , 0 = sup

Cp ( + 1)

, 1 = min

Cp \ min(Cq \( + 1))

,

then there is an ultrafilter e on [0, 1) Cp such that

dq =

A Zq : A e

{dp : [0, 1) Cp}

.

Proof. (a) (b) Assume towards contradiction that p 0 q, but (b) fails.Then we may pick a sequence , A : < such that for each < ,

(i) Cq, A dq ,(ii) if < < , Cpmin

Cq\(+1)

, then min

Cp\(+1)

< ,

(iii)

Cp

A Zp / d

p

.

It follows from (ii) that for every Cp there is at most one < such thatZp Z

q

= . Also if Cp and Zp Zq

dp, then (Zq

\ A) Zp d

p.

Put A =


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such that

dq =

B Zq : B

e{dp : [0, 1) Cp}

.

Note that 0 and hence A Zp d

p for all [0, 1) C

p. Conse-quently

A [0, 1) e

{dp : [0, 1) Cp}

and therefore also

A Zq =

A [0, 1)

Zq dq.

Now we easily conclude that A fil(q).

Definition 2.10. Let p Q0. Suppose that X [Cp] and C Cp is a

club of such that

if < are successive elements of C,then |[, ) X| = 1.

(In this situation we say that p is restrictable to X, C.) We define therestriction of p to X, C as an element q = pX, C Q0 such thatCq = C, and if < are successive elements of C, x [, ) X, thenZq = [, ) and d

q = {A Z

q : A Z

px d

px}.

Proposition 2.11. (1) Assume that G Q0 is 0directed and 0

downward closed, p G, X [Cp] and C Cp is a club of such that p is restrictable to X, C. If

xX

Zpx fil(G), then

pX, C G.(2) If G Q0 is

0directed and G , then G has a 0upperbound. (Hence, in particular, fil(G) is not an ultrafilter.)

Proof. (1) Suppose that G, p , X , C are as in the assumptions and

xX

Zpx

fil(G). Since G is 0directed (and p G) we may pick r G such that

p 0 r and

xXZpx fil(r). We are going to show that q

def= pX, C 0 r

(which will imply that q G as G is downward closed).Since

xX

Zpx fil(r), there is < such that

Cr \

xX

Zpx Zr d

r

and Cr \

A dr

Cp

A Zp d

p


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14 SAHARON SHELAH

(remember 2.9(b)). Now suppose that Cr \ and A dr. Then

xX

Zpx A dr so there is C

p such that xX

Zpx A Zp d

p. In

particular,

xX

Zpx Zp A = , so necessarily X. Let 0 < 1 be the

successive elements of C such that 0 < 1. Since

Zp A =xX

Zpx Zp A d

p,

we also have A Zq0 dq0

. Thus we have shown that

if Cr \ and A dr,then there is 0 Cq such that A Z

q0

dq0.

Consequently, q 0 r (remember 2.9).

(2) Let p : < list (with possible repetitions) all members of G. For < let C =

< : = sup( Cp)

(it is a club of), and for , <

let ({, }) < be such that

if p 0 p, then Cp \ ({, })

A d

p

Cp

A Z

p d

p )

(remember 2.9). Let

C =

< : is limit and {p : < } is 0directed

(again, it is a club of ). Finally, let

C =

C


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For basic information on the ideal of meager subsets of and its coveringnumber we refer the reader e.g. to Matet, Roslanowski and Shelah [6, 4].

Here we state only the definitions we will need.

Definition 2.13. (1) The space is endowed with the topology ob-tained by taking as basic open sets and Os for s >, whereOs = {f

: s f}.(2) The (


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16 SAHARON SHELAH

Proof of the Claim. We will consider two cases.

Case 1: For every p0, . . . , pn G, n < , there is p Q such thatA fil(p) and p0 0 p , . . . , pn 0 p.

Note that the assumption of the present case is equivalent to

() for every p0, . . . , pn G, n < , and < there are < <

and an ultrafilter d N on [, ) such that ()p0,...,pn(, , d) holdstrue and < and A [, ) d.

We let

TA =

T :

< lh()

Z, d

() = Z, d A Z d

.

Clearly, TA is a branching subtree ofTand TA is isomorphic to>. Now,

for p0, . . . , pn G, n < , and < let

IA (p0, . . . , pn)def=

lim(TA) :

>

, , d

()p0,...,pn(, , d) & () = [, ), d

.

It should be clear that IA (p0, . . . , pn) is an open dense subset of lim(TA)(remember ()). Therefore (as G < cov(M,)) we know that

IA (p0, . . . , pn) : n < & p0, . . . , pn G & <

=

and we may choose from the set on the left-hand side above. Let p Q

be such that = p. Since lim(TA) we know that A fil(p). Also, forevery p0, . . . , pn G we have


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Proof of the Claim. Let p0, p1 G. Note that

() for every p2, . . . , pn G, 2 n < , and < there are < < and an ultrafilter d N on [, ) such that ()p0,p1,p2,...,pn(, , d)holds true and < .

We let Tp0,p1 be the set T :

< lh()

, , d

() = [, ), d ()p0,p1(, , d)

and we note that Tp0,p1 is a branching subtree of T isomorphic to >.For p2, . . . , pn G, 2 n < , and < we let

Ip0,p1 (p2, . . . , pn)def=

lim(Tp0,p1) :

>

, , d

()p2,...,pn(, , d) & () = [, ), d

.

Then Ip0,p1 (p2, . . . , pn) is an open dense subset of lim(T

p0,p1

) (remember()). Since G < cov(M,), we may choose p Q such that

p

Ip0,p1 (p2, . . . , pn) : 2 n < & p2, . . . , pn G & <

= .

Like in the proof of 2.14.1 we argue that G {p} is linked. Since p lim(Tp0,p1) we easily see that p is 0stronger than both p0 and p1.

Claim 2.14.3. If G Q is a linked family, G < cov(M,), is a limit ordinal and a sequence p : < G

is 0increasing, thenthere is p Q such that

(a) G {p} is linked, and

(b) ( < )(p 0 p).

Proof of the Claim. First let us consider the case when < . Suppose thata sequence p = p : < G is 0increasing and let

Tp =

T :

,

, d

()p0,...,pn

(,

, d) & () = [,

), d

.Then each Ip(p

0, . . . , p

n) is an open dense subset of lim(Tp). [Why? Let

Tp. We may assume that for each < < and Cp \ lh() and

A dp there is C

p such that A Zp dp . We also may demand that

0def= sup

< :

< lh()

, d

() = [, ), d


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18 SAHARON SHELAH

(a) : < is an increasing continuous sequence of ordinals below,

(b) d N is an ultrafilter on [, +1), +1


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Like in the proof of 2.14.1 we argue that G {p} is linked. Put

C

=

< : = is limit &

, d

p

() = [,

), d

and note that C is a club of. Note that if C and p() = [, ), d,then +1 <

and [, +1) / d. Consequently,[+1, ) : < are successive members of C

fil(p).

To prove the theorem we construct inductively a sequence q : < 2

of elements of Q such that

for each < 2 the family {q : < } is linked,

for each A there is < 2

such that either A fil(q) or \ A fil(q),

for each < < 2 there is < 2 such that q 0 q and q

0 q, if and p : < is a 0increasing sequence of elements of

{q : < 2}, then there is < 2 such that q is a 0upper boundto all ps,

if a sequence : < is increasing continuous, then for some < 2 and a club C of we have

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

fil(q).

The construction is a straightforward application of a suitable bookkeepingdevice and Claims 2.14.12.14.4. After it is carried out put G = {q :

saharon shelah- the combinatorics of reasonable ultrafilters

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