heike mildenberger, saharon shelah and boaz tsaban- covering the baire space by families which are...

8/3/2019 Heike Mildenberger, Saharon Shelah and Boaz Tsaban- Covering the Baire Space by Families which are Not Finitely

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arXiv:math/0

407487v4

[math.L

O]31Oct2010

COVERING THE BAIRE SPACE BY FAMILIESWHICH ARE NOT FINITELY DOMINATING

HEIKE MILDENBERGER, SAHARON SHELAH, AND BOAZ TSABAN

Abstract. It is consistent (relative to ZFC) that each union ofmax{b, g} many families in the Baire space which are notfinitely dominating is not dominating. In particular, it is con-sistent that for each nonprincipal ultrafilter U, the cofinality of thereduced ultrapower /U is greater than max{b, g}. The model isconstructed by oracle chain condition forcing, to which we give aself-contained introduction.

1. Introduction

The undefined terminology used in this paper is as in [9, 2]. Afamily Y is finitely dominating if for each g there exist kand f1, . . . , f k Y such that g(n) max{f1(n), . . . , f k(n)} for all butfinitely many n. The additivity number for classes Y Z P()with

Y Z is

add(Y,Z) = min{|F| : F Y and F Z}.Let D (respectively, Dfin) be the collection of all subsets of whichare not dominating (respectively, finitely dominating). Define

cov(Dfin) = min{|F| : F Dfin and

F= }.

It is easy to see that add(Dfin,D) = cov(Dfin), so we will use this shorternotation.

In [8] it is pointed out that

max{b, g} cov(Dfin),

1991 Mathematics Subject Classification. 03E15, 03E17, 03E35, 03D65.Key words and phrases. Finitely dominating families, groupwise density number

g, unbounding number b, cofinality of ultrapowers.The authors were partially supported by: The Austrian Fonds zur wis-

senschaftlichen Forderung, grant no. 16334, and the University of Helsinki (firstauthor), the Edmund Landau Center for Research in Mathematical Analysis andRelated Areas, sponsored by the Minerva Foundation, Germany (first and thirdauthor), the United States-Israel Binational Science Foundation Grant no. 2002323(second author), and the Golda Meir Fund (third author). This is the secondauthors publication 847.

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2 HEIKE MILDENBERGER, SAHARON SHELAH, AND BOAZ TSABAN

the inequality b cov(Dfin) being immediate from the definitions,and the inequality g cov(Dfin) having been implicitly proved in [5,

Theorem 2.2]. (For the readers convenience, we give a short prooffor this in Corollary 2.3). In [8] it is shown that in all standardforcing extensions (e.g., those appearing in [2, 11]), equality holds. Itis conjectured in [8] that this equality is not provable. We prove thisconjecture. In fact, we prove a stronger result: Let M denote the idealof meager sets of real numbers.

Theorem 1.1. It is consistent (relative to ZFC) that 1 = non(M) =g < cov(Dfin) = cov(M) = c = 2.

The statement of Theorem 1.1 determines the values of almost allstandard cardinal characteristics of the continuum in the model wit-

nessing it: IfN is the ideal of null sets of real numbers, then by provableinequalities (see [9, 2]), we have that p, t, h, add(N), add(M), b,s, cov(N),and non(M) are all equal to 1, and cov(M), non(N), r, d, u, i, cof(M),and cof(N) are all equal to 2 in this model.

In [8] it is shown that for each nonprincipal ultrafilter U on ,cov(Dfin) cof(

/U).

Corollary 1.2. It is consistent (relative to ZFC) that for each non-principal ultrafilter U on , max{b, g} < cof(/U).

This corollary partially extends the closely related Theorems 3.1 and3.2 of [7], which are proved using the same machinery: Oracle chaincondition forcing.

2. Making cov(Dfin) and cov(M) large

From now on, by ultrafilter we always mean a nonprincipal ultra-filter on . We will use the following convenient characterization.For functions f, g and an ultrafilter U we write f U g for{n : f(n) g(n)} U.

Lemma 2.1 ([8]). For each cardinal number , the following are equiv-alent:

(1) < cov(Dfin);(2) For each -sequence (U, g) : < with each U an ultra-

filter and each g there exists g such that for each < , g U g.

We first show how this characterization easily implies an assertionmade in the introduction.

Definition 2.2. For A [], define the function A+ by A+(n) =min{k A : n < k} for all n.


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COVERING AND FINITE DOMINANCE 3

Corollary 2.3 ([5]). g cov(Dfin).

Proof. We use Lemma 2.1. Assume that < g, and (U, g), < ,are given with each U an ultrafilter and each g . We must showthat there exists g such that for each < , g U g. We willuse the following morphism.

Lemma 2.4. For each f and each ultrafilter U,

GU,f = {A [] : f U A

+}

is groupwise dense.

Proof. Clearly, GU,f is closed under taking almost subsets. Assumethat {[an, an+1) : n } is an interval partition of . By merging

consecutive intervals we may assume that for each n, and each k [an, an+1), f(k) an+2.Since U is an ultrafilter, there exists {0, 1, 2} such that

A =n

[a3n+, a3n++1) U

Take A = A+2 mod 3. For each k A, let n be such that k [a3n+, a3n++1). Then f(k) a3n++2 = A+(k). Thus A GU,f.

Thus, we can take A


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Proof. Assume that G is a Q-generic filter over V. Let g =

2[G],where 2 denotes the projection on the second coordinate. Clearly, g

is a partial function from to . By density arguments, we have thatg is as required. To see this, consider first the sets

Dm = {(n,h,F) Q : m n}

for m . Each Dm is dense in Q: Assume that (n,h,F) Q. Ifm n then [n, m) = ; therefore (n,h,F) (n,h,F {}) Dm.Otherwise, define h : m by h(k) = h(k) for k < n, and h(k) =max{f(k) : F} for k [n, m). Then (m, h, F) is a memberof Dm, extending (n,h,F). The density of the sets Dm implies thatdom(g) = . Moreover, for each < the set

E = {(n,h,F) Q : F}

is dense in Q (for each condition (n,h,F), (n,h,F {}) is a strongercondition which belongs to E). Now fix < and choose an el-ement (n0, h0, F0) G E. For each n A \ n0 choose an ele-ment (n1, h1, F1) G Dn+1, and a common extension (n2, h2, F2) of(n0, h0, F0) and (n1, h1, F1). As F0 and n [n0, n2) A, we havethat g(n) g(n). Since this holds for each n n0, we have thatA {n : g(n) g(n)}.

Consequently, doing an iteration of forcing notions with the aboveforcing used cofinally often, with = 1 and an appropriate book-keeping will increase cov(Dfin). We will be more precise in the proof ofTheorem 2.9.

Observe that the sets A played no special role and in fact we couldtake A = for each (in this case we obtain a dominating real).However, this freedom to choose A will play a crucial role in thesequel, where we would like to make sure that b (or non(M)) and gremain small while we increase cov(Dfin).

We now make some easy observations concerning our planned forc-ing. We will construct our model by a finite support iteration P,Q : < 2 of c.c.c. forcing notions Q which add reals for cofinally many < 2. Consequently, VP satisfies c 2, where P = P2 =


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imply cov(M) 2 in the final model (briefly: Each meager set in thefinal model is contained in an F, thus Borel, meager set. Each Borel

set is coded by a real, and every real appears at a stage < 2, soCohen reals added later will not belong to the Borel meager set whichis the interpretation of this code, and since this property is absolute,they will not belong to the interpretation in the final model. Since 2is regular, the codes for 1 many Borel meager sets all appear at anintermediate stage, so their union does not contain Cohen reals addedlater).

Corollary 2.7. In the final model, cov(M) = c = 2 holds.

Now we show how to impose some more constraints on our iterationP,Q : < 2 so that in V

P2 , cov(Dfin) = 2. Our exposition

follows closely the treatment of names given in [4].

Choice 2.8. We fix a 2(S21 )-sequence S : S

21 in the ground

model. The idea is that stationarily often S will guess a function

(1) f : (1 2) 1 ([2]0)0 .

(So for each < 2 of cofinality 1, S : (1 ) 1 ([]0)0 .)We identify 2 with the partial order P2 we are about to build.

Then [2]0 contains all of the maximal antichains. Thus ([2]0)0

contains a name for each subset of (which corresponds to an elementof). Now any sequence

(U, g) : < 1

in the extension has a ground model function f : (1 2) 1 ([2]0)0, such that f() is a name for g and f(, ) is a name foran enumeration of the elements of U.

For each f as in Equation (1),

{ S21 : S = f }

is stationary in 2. We will inductively define an 2-stage finite supportiteration and an injection function F : P 2 for < 2 such thatthe range of each F is an initial segment of 2 which includes , and

for < < 2, F F.For < 2 we will denote by name(S) the sequence of 1 sets ofreals U and of1 reals g of the form

({n

{n} F1 (S(, )(n)) : < },n

{n} F1 (S()(n))) :

< 1.


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At stage S21 in the construction, ifP name(S) is a sequenceof 1 ultrafilters and 1 functions, then we can take P-names A,

< 1, such that P A (U) , which means P A is in thefirst component of name(S).

Theorem 2.9. Let V |= 2(S21 ) and let P2 be any forcing as in

Choice 2.8. Then VP2 |= cov(Dfin) = 2.

Proof. If P2 (U, g) : < 1 is a sequence of functions andultrafilters, then at club many stages the restriction of the namesto is also forced to be a sequence of ultrafilters in VP . For a proofof this (even in the countable support proper scenario) see [1]. Butthe restriction of the name to is guessed by name(S) for stationarilymany s in this club. So at such a stage the forcing Q adds a

function h such that g U h for all < 1 and this shows that thesequence was not a witness for cov(Dfin) = 1.

3. Interlude: Oracle chain condition forcing

Usually, the major difficulty in forcing inequalities between combi-natorial cardinal characteristics of the continuum is to make sure thatthose which are required to be smaller (non(M) and g in our case)indeed remain small in the generic extension. In this section we de-scribe one such method, which is suitable for our purposes: Oraclechain condition forcing [6, Chapter IV] (see also [3, 4]).

Oracle chain condition forcing is a method for forcing with 2-stage

finite support iteration, in such a way that some prescribed intersec-tions of1 many (descriptively nice) sets which are empty in an inter-mediate model remain empty in the final model.

Definition 3.1. An oracle (or 1-oracle) is a sequence M = M : limit < 1 of countable transitive models of a sufficiently large finiteportion of ZFC (henceforth denoted ZFC), such that for each , Mis countable in M, and for each A 1, the set

TrapM(A) = { < 1 : is a limit ordinal, and A M}

is a stationary subset of 1.

Clearly, implies the existence of an oracle. The sets TrapM(A)generate a filter TrapM, which is normal and proper. Moreover, for each

A, B 1, there exists C 1 such that TrapM(C) = TrapM(A) TrapM(B).

Notation 3.2. Assume that P Q are forcing notions, and N is aset. Then P


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Lemma 3.3.

(1)


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This shows that for all (large enough) limit ordinals < 1, 1[]


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Lemma 3.10 ([6, IV:3.23.3]). Assume that M is an oracle.

(1) For a finite support iteration P,Q

: < , if eachP sat-

isfies the M-chain condition, then so does P =


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(so at the end, Q =


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Theng .

Proof. By a result of Blass [2], g cf(c), so we can assume that cf(c). We now define sets and then show that they are groupwisedense and that their intersection is empty.

Let n : < c list all strictly increasing sequences of natural num-bers, each sequence appearing cofinally often. By induction on < cwe choose , < c and C [] as follows.

If there is some < such that for each < with = we

have [ni , ni+1) C for all but finitely many i, then we take as the

minimal such . By the assumption (1), we can choose to be theminimal < c such that = for all < and there are infinitelymany i such that [ni , n

i+1) Y. In this case we set C = {[n

i , n

i+1) :

i , [n

i , n

i+1) Y}. Otherwise we set = and C = .For each < , define

G = {B [] : ( < c) and B

C}.

We show that each G is groupwise dense. Clearly, it is closed underalmost subsets. Let an increasing sequence n be given. Then for each < , there is by our construction some () < c such that () = and [ni, ni+1) C() for infinitely many i. As cf(c), () =sup{() : < } < c. By the choice of n : < c there is some ((), c) such that n = n. So , and

{[ni , n

i+1) : [n

i , n

i+1)

Y} = C G.

To see that

{G : < } = , assume that B is infinite and foreach , B G. Then for each < , there is < c such that = and B C Y . Since is regular, we can thin out and assume

that if 1 < 2, then 1 = 2 . Thus we have that for 1 < 2,1 = 2, and hence 1 = 2 . Consequently, |{ : < }| = .But { : < } { < c : B

Y}, contradicting the assumption(2).

As we already stated in the previous sections, we shall use a finitesupport iteration P,Q, : < 2 of c.c.c. forcing notions, and chooseconstant or increasing oracles M, such that P has the M

-chain con-

dition for each . We start with a ground model satisfying 1 and

2(S21 ). Let S : S

21 be a 2(S

21 )-sequence.

There are three possibilities for Q. Ifcf() = 0 or if is a successor,then Q is the Cohen forcing.

Ifcf() = 1 and P name(S) is a sequence of ultrafilters U andof functions g, < 1, then we choose A, < 1 as in Lemma 4.1but with additional provisos and force with Q = Q(A, g : < 1).


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For the premise of this sentence we shortly say: S guesses (U, g) : < 1. Otherwise, we set Q = {0}.

Definition 5.2. For 2 we consider the class K of-approximations

(P,Q

, M, W1, W2) : <

with the following properties:

(a) P,Q

: < is a finite support iteration of partial orders suchthat for each < , |P| 1.

(b) M : < is a constant sequence of oracles such that for all, P satisfies the M

-chain condition and for + 1 < , P Qsatisfies the (M+1)-c.c. (as in Lemma 3.10(2)). The constant

value of the oracle sequence is some oracle

M as in Lemma 3.9,keeping cov(M) = 1.

(c) W1, W2 2 \ S21 , W1 and W2 are disjoint and if is a limit of

cofinality 1, then W1 , W2 are both cofinal in .

(d) If (W1 W2) then Q

is the Cohen forcing adding the realr

2.

(e) If S21 and S guesses (U(), g()) : < 1, then there issome strictly increasing enumeration () : < 1 of a cofinalpart of W2 , and for every < 1 there is () {0, 1} such

that Y()

()

:= r1

()

({()}) U, and Q = Q(Y()

()

, g() :

< 1).2

(f) For all , P (A []) { W1 : A Y1

} is at

most countable.3 Here, for = limit, P is the direct limit ofP : < , and for = = + 1, P = P Q

.

With the help of several lemmas we will prove the following.

Theorem 5.3. If V |= 1 and 2(S21 ), then for each 2, K is

not empty.

2

The (), < 1, chosen here do not have to be coherent when regardingdifferent s and we index them with because we need it. Strictly speaking the() is a function ()(). And also strictly speaking we should index by aswell, but we are suppressing this because we are anyway only working with endextensions when increasing .

3Here it is W1. We use the Cohens in W2 to build the forcings of type Q =

Q(Y()

(), g() : < 1) and the Cohens Y1 , W1, to build the Y s as in

Lemma 5.1.


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Let V fulfill the premises and let P2 be the direct limit of the firstcomponents of an 2-approximation. If G is a P2-generic filter and

Y1

[G2] = Y for W1, then we have in the final model a sequence

Y : < c as in Lemma 5.1 with = 1.

Corollary 5.4. VP2 |= cov(M) = g = 1 < cov(Dfin) = 2.

We prove Theorem 5.3 by induction on and we shall work withend extensions. For some s, one has to work to show item (e). Wewill do this in our first lemma. For all s but maybe the successorsteps of points not in S21 , one has to work to show that item (f) can bepreserved in the induction. This will be done in the last three lemmas.

Lemma 5.5. Consider a successor = + 1, S21 . Given any

1-oracle (

M+1

)

, the sequence () : < 1 can be chosen as in(e) so that the forcings given in item (e) have the (M+1)-c.c.

Proof. This is a variation of Lemma 4.1. We suppress some of the s.We choose : < 1 enumerating W2 so that, given the oracle(M+1) = N : < 1, the Cohen real r is generic over N. Forthis it suffices that the countable model N VP , which means that just has to be sufficiently large. Let the ak be chosen as in the proofof Lemma 4.1. Then there are infinitely many k such that

r1 ({}) [a2k+1, a2k+) = ,

and as in the proof of Lemma 4.1 this suffices.

Choice 5.6. We start with M as described. By Lemma 3.10, all theP, 2, have the M-chain condition as soon as we can arrange thatall the Q have the (M)-chain condition in V

P . The Cohen forcinghas the M-chain condition for any M. The Q in the steps S21 canbe chosen by the previous lemma so that they have the (M)-c.c.

Lemma 5.7. If S21 , Q is chosen as in Lemma 5.5, andP satisfies(f) of Definition 5.2, thenP+1 has the property stated in item (f).

Proof. Suppose that p P+1 A [] and |{ W1 : A

Y

}| =

1, and w.l.o.g. p P+1 A []

and { W1 : A

Y

} isincreasingly enumerated by { : < 1} = W1(A).

We take for n a maximal antichain {pn,i : i } above pdeciding the statements n A

with truth value tn,i. Let Cn,i = {

: pn,i() = 1}. For Cn,i S21 with Q = {0}, let pn,i() =

(mn,i(), hn,i(), Fn,i()). Let Fn,i() = {() : Fn,i()}. Weassume that all these are objects not just names. For Cn,i \ S

21 let


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pn,i() = hn,i(), mn,i() = |hn,i()| and set the other two componentsfor simplicity zero. Set mn,i = max{mn,i() : Cn,i}. Set

C = (mn,i(), hn,i(), Fn,i(), Fn,i(), g() mn,i : Fn,i()) :

Cn,i : n, i .

For each 1, let p p, p P+1 A [s , ) Y

and pshall decide the value of 2 and s . For < 1 we set C ={ : p() = 1}. If C S21 , then p() = (m(), h(), F()).If C \ S

21 , then p() = h(), () = |h()| and F() = . For

all , C, let Let F() = {() : F()} W2.Set

R(m) = (m(), h(), F(), F

(), g() m : F()): C.

These are finite arrays of finite sets.

Now we thin out: First we assume that for some k for all < 1,|C| = k, s k. We apply the delta system lemma to C, 1, geta root C. We assume that C, as this is the difficult case. We applythe delta lemma for each C to the F(), 1, and get a rootF(), and to F(), 1, and get a root F

(). We further assumethat for each in the delta system and for all C, all F() \ F()are above max(C(F(

)) (C\ {})) and same for the primed ones.

We thin out further and assume that there are (m(), h(), F()) suchthat for all < 1, for all C, m() = m(), h() = h() m(),and for the C \ C, the increasingly enumerated s in C = {

i :

i < k}, are isomorphic to the lexicographically first i : i < k, i.e.,m(

i ) = m(i), h(

i ) = h(i)

m(i), and we use a delta system

argument on the F(i ) giving a root F(i) and again impose on the

parts F(i ) \ F(i), that they have to lie above

i


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(


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Then p(()) = 1 and hence () C W2. But then becauseof the indiscernibility over m = m mmax (which is a component

of C), () C and hence it is in the root C. So p forced by ourthinning out same fact about Y

()()

mmax. Hence, for all F for

all m [mn,i, m) Y()()

, (h(m) g()(m)). So, taking 1 a) and 1

b) together, p||pn,i.

Second case: m mn,i. Then h hn,i, and p||pn,i means that for

all F Fn,i for all m [m, mn,i) Y()

(), (hn,i(m) g()(m)).

This latter statement does hold also for F instead of F and m in-stead of m, beause m = m and (F, g() mn,i : F) and(F, g() mn,i : F) are part of R(mmax) and R(mmax) and

hence indiscernible over hn,i for arguments m Y

()

() , as for thesems, that are forced to be in a Cohen part, () C and hence by ourthinning out we have mmax m. Also h hn,i, and hence p||pn,i.

So the claim is proved and with it also Lemma 5.7.

Lemma 5.9. (1) Ifcf() = 1 andQ

and M are as in the previouslemma and if P,Q

, M, W1, W2) : < K, then

P,Q

, M, W1, W2) : < P,Q

, M K+1.

(2) Ifcf() = 0 and if P,Q

, M, W1, W2) : < K, then

P

,Q

, M, W1, W

2) : < P

,C, M K

+1.

(3) If cf() = 0 and if P,Q

, M, W1, W2) : < K for

each < , then P,Q

, M , W1, W2) : < K.

(4) If cf() = 1 or = 2, and if P,Q

, M, W1, W2) : <

K for each < , then P,Q

, M, W1, W2) : < K.

Proof. (1) This was proved in Lemma 5.7.(2) If A is an almost subset of uncountably many Ys, then there

is some 0 < that there are uncountably many such below 0.A is possibly a name using the last, new forcing. But this is just

Cohen forcing. So there is some finite part of a Cohen condition forcingthat A

is in uncountably many Ys. But then also the forcing P

already contains a name for some infinite B almost contained inthe intersection of uncountably many Ys with < 0. So P does notfulfill property (f) and hence the induction hypothesis is not fulfilled.

(3) First we use the pigeonhole principle for the Ys as in the previousitem. Then we use the following


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Lemma 5.10. Assume

(a) Pn : n is a-increasing sequence of c.c.c. forcing notions

with unionP,

(b) Y is a set ofP0-names of infinite subsets of ,

(c) for n we have Pn = cf() > |{Y

Y : B

Y

}|,whenever B

is aPn- name of an infinite subset of .

Then condition (c) holds forP too.

Proof. Since P is a c.c.c. forcing notion, also in VP we have is a regularcardinal.

If the desired conclusion fails, then we can find a P-name B

of aninfinite subset of and a sequence (p, Y

, m) : < such that

() m ,

() Y Y without repetitions,

() p P, p P B

\ m Y.

Since cf() > 0, for some n(), m() the set S =df { < :p Pn(), m = m()} has cardinality . We identify it with .

Now for every large enough S we have

p P = |{ S : p GPn()

}|.

Why? Else for an end segment of < there is q p such that forall but < many S, q p G

Pn()

. That means that for an endsegments of < , w.l.o.g., for all , Perp := { S : q q}contains an end segment of S. Then we take the diagonal intersectionD of all these end segments of S. Since is regular, D contains aclub in . But then {q : D} is an antichain in Pn() of size .Contradiction.

Let Gn() be a subset ofPn() generic over V, and let S := { S : p Gn()}. We choose Gn(), such that |S| = . We let B

={Y

\ m() : S}. Then in V[Gn()], B is an infinite subset

of included in members of Y, contradicting the assumption. SoLemma 5.10 is proved.

(4) IfP adds some A, then this already comes earlier, say in VP, < , because A and because of the c.c.c. If A Y is forced,then < . This contradicts the induction hypothesis for P. Thiscompletes the proof of Lemma 5.9.

The lemmas together give that there is an 2-approximation, andthe proof of Theorem 5.3 is completed.


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With some extra care our proof can be modified to yield the following(cf. [7, 4]).

Theorem 5.11. It is consistent (relative to ZFC) that all of the fol-lowing assertions hold:

(1) Each unbounded set of contains an unbounded subset of size1,

(2) Each nonmeager subset of contains a nonmeager subset ofsize 1,

(3) g = 1; and(4) cov(Dfin) = cov(M) = c = 2.

Proof. This time we work with a version of K with increasing oracles,

which means that the

M

-chain condition implies

M

-chain conditionfor > and that P P[,) has the M

+1-c.c., though the initialsegment need not yet fulfill it, and the name for this new oracle may notyet have an evaluation in an initial segment P, < . The new parts ofthe oracles take care of the unbounded and the nonmeager families thatappear later in the iteration and that are frozen by the next step if theirintersection with VP is guessed by the diamond sequence and happensto be unbounded or nonmeager at the current stage : The conservationof the unboundedness and nonmeagerness of the intersection is writteninto all the oracles from onwards.

References

[1] A. R. Blass and S. Shelah, There may be simple P1- and P2-points, and theRudin-Keisler ordering may be downward directed, Annals of Pure and AppliedLogic 33 (1987), 213243.

[2] A. R. Blass, Combinatorial cardinal characteristics of the continuum, in:Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor,eds.), Kluwer Academic Publishers, Dordrecht, to appear.

[3] M. Burke, Liftings for Lebesgue measure, Israel Mathematical Conference Pro-ceedings 6 (1993), 119150.

[4] M. Burke and A. W. Miller, Models in which every nonmeager set is nonmeagerin a nowhere dense Cantor set, Canadian Journal of Mathematics 57 (2005),11391154. http://arxiv.org/abs/math.LO/0311443

[5] H. Mildenberger, Groupwise dense families, Archive for Mathematical Logic40 (2001), 93112.

[6] S. Shelah, Proper and Improper Forcing (second edition), Springer, 1998.[7] S. Shelah and J. Steprans, Maximal Chains in and Ultrapowers of the

Integers, Archive for Mathematical Logic 32 (1993), 305319.[8] S. Shelah and B. Tsaban, Critical cardinalities and additivity properties of

combinatorial notions of smallness, Journal of Applied Analysis 9 (2003), 149162.


19/19


[9] J. Vaughan, Small uncountable cardinals and topology, in: Open Problemsin Topology (eds. J. van Mill and G. M. Reed), North-Holland, Amsterdam:1990, 195218.

Universitat Wien, Institut fur Formale Logik, Wahringer Str. 25,

1090 Vienna, Austria

E-mail address: [email protected]

Einstein Institute of Mathematics, The Hebrew University of Jerusalem,

Givat Ram, 91904 Jerusalem, Israel, and Mathematics Department,

Rutgers University, 110 Frelinghuysen Road, NJ 08854-8019, USA


Einstein Institute of Mathematics, The Hebrew University of Jerusalem,

Givat Ram, 91904 Jerusalem, Israel


URL: http://www.cs.biu.ac.il/~tsaban

heike mildenberger, saharon shelah and boaz tsaban- covering the baire space by families which are...

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