andrzej roslanowski and saharon shelah- universal forcing notions and ideals

8/3/2019 Andrzej Roslanowski and Saharon Shelah- Universal Forcing Notions and Ideals

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UNIVERSAL FORCING NOTIONS AND IDEALS

ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

Abstract. The main result of this paper is a partial answer to [6, Problem5.5]: a finite iteration of Universal Meager forcing notions adds generic filtersfor many forcing notions determined by universality parameters. We also givesome results concerning cardinal characteristics of the ideals determined bythose universality parameters.

0. Introduction

One of the most striking differences between measure and category was discov-ered in Shelah [8] where it was proved that the Lebesgue measurability of 13 setsimplies 1 is inaccessible in L, while one can construct (in ZFC) a forcing notionP such that VP |= projective subsets of R have the Baire property. For thelatter result one builds a homogeneous ccc forcing notion adding a lot of Cohenreals. Homogeneity is obtained by multiple use of amalgamation (see [4] for a fullexplanation of how this works), the Cohen reals come from compositions with theUniversal Meager forcing notion UM or with the Hechler forcing notion D. Themain point of that construction was isolating a strong version of ccc, so calledsweetness, which is preserved in amalgamations. Later, Stern [10] introduced aweaker property, topological sweetness, which is also preserved in amalgamations.

Sweet (i.e., strong ccc) properties of forcing notions were further investigated in[6], where we introduced a new property called iterable sweetness (see [6, Definition4.2.1]) and we proved the following two results.

Theorem 0.1. (1) (See [6, Proposition 4.2.2]) IfP is a sweet ccc forcing notion(in the sense of [8, Definition 7.2]) in which any two compatible elementshave a least upper bound, thenP is iterably sweet.

(2) (See [6, Theorem 4.2.4]) IfP is a topologically sweet forcing notion (in thesens of Stern [10, Definition 1.2]) andQ

is aPname for an iterably sweet

forcing, then the compositionP Q

is topologically sweet.

In [6, 2.3] we introduced a scheme of building forcing notions from so calleduniversality parameters (see 1.2 later). We proved that typically they are sweet

(see [6, Proposition 4.2.5]) and in natural cases also iterably sweet. So the questionarose if the use of those forcing notions in iterations gives us something really new.Specifically, we asked:

Date: April 2004.1991 Mathematics Subject Classification. 03E40, 03E17.The first author thanks the Hebrew University of Jerusalem for support during his visit to

Jerusalem in Spring2003. He also thanks his wife, Malgorzata JankowiakRoslanowska for sup-porting him when he was working on this paper.

Both authors acknowledge support from the United States-Israel Binational Science Foundation(Grant no. 2002323). This is publication 845 of the second author.

1


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2 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

Problem 1 (See [6, Problem 5.5]). Is there a universality parameter p satisfyingthe requirements of [6, Proposition 4.2.5(3)] such that no finite iteration of the

Universal Meager forcing notion adds a Qtree

(p)generic real? Does the UniversalMeager forcing add generic reals for the forcing notions Qtree(p) defined from p asin 1.11, 1.7, 1.9 here?

Bad news is that Problem 1 has a partially negative answer: if the universalityparameter p satisfies some mild conditions (i.e., is regular, see 1.14), then finiteiteration ofUM will add a generic filter for the corresponding forcing notion, seeCorollary 2.2.

Good news is that we have more examples of iterably sweet forcings, and theywill be presented in a subsequent paper [7].

The structure of the present paper is as follows. In the first section we recall in asimplified form all the definitions and results we need from [6], and we define regularuniversality parameters. We also re-present the canonical examples we keep in mind

in this context. In the second section we prove our main result: a sequence Cohenreal dominating real Cohen real produces generic filters for forcing notionsQtree(p) determined by regular p (see Theorem 2.1). In the following section welook at the idealsIp for regular p and we prove a couple of inequalities concerningtheir cardinal characteristics.

Notation Our notation is rather standard and compatible with that of classicaltextbooks (like Jech [3] or Bartoszynski and Judah [1]). In forcing we keep theolder convention that a stronger condition is the larger one. Our main conventionsare listed below.

(1) For a forcing notion P, all Pnames for objects in the extension via P willbe denoted with a tilde below (e.g.,

, X

). The complete Boolean algebra

determined by P is denoted by RO(P).

(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) A tree is a family T of finite sequences such that for some root(T) T wehave

( T)(root(T) ) and root(T) T T.

For a tree T, the family of all branches through T is denoted by [T], andwe let

max(T)def= { T : there is no T such that }.

If is a node in the tree T then

succT() = { T : & lh() = lh() + 1} andT[] = { T : }.

(4) The Cantor space 2 and the Baire space are the spaces of all functionsfrom to 2, , respectively, equipped with the natural (Polish) topology.

(5) The quantifiers (n) and (n) are abbreviations for

(m )(n > m) and (m )(n > m),

respectively. For f, g we write f


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UNIVERSAL FORCING NOTIONS AND IDEALS 3

(6) R0 stands for the set of non-negative reals.

Basic convention: In this paper, H is a function from to \2 and X = i


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(2) We define a forcing notion Qtree(p):A condition in Qtree(p) is a pair p = (Np, Tp) such that Np < and Tp

is an infinite pnarrow Htree.The order on Qtree(p) is given by:(N0, T0) (N1, T1) if and only if

N0 N1, T0 T1, and T1

i


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()

max(S)

i A [ndn, nup)

i max(Yi)

.

If A = then we may omit it and write GgF

.

Proposition 1.8. LetFH be an increasing function such thatn <

(n + 1)2

in

H(i) < FH(n)

and let g , A []. If a function F : FT[H] R0 satisfiesS FT[H]

| max(S)| = 1 F(S) = 0

,

then (Gg,AF

, FH) is a suitable simplified universality parameter.

Example 1.9. Let g , A [].

(1) Let F0, F1 : FT[H] R0 be defined by

F0(S) = max |succS(s)| : s S\ max(S)

1 and F1(S) = | max(S)| 1

(for S FT[H]). Then both (Gg,AF0 , FH) and (Gg,AF1 , FH) are suitable simpli-fied universality parameters.

(2) Let F2 : FT[H] R0 be defined by F2({}) = 0 and

F2(S) =lev(S) 1 : max(S) 1

when lev(S) > 0. Then (Gg,AF2

, FH) is s suitable simplified universalityparameter.

(3) Suppose that (K, ) is a local tree creating pair for H (see [5, 1.3, Def.1.4.3]) such that

for each n < , i


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Proposition 1.12. (1) Let FH be as in 1.8. Then pcmzH = (GcmzH

, FH) is asuitable simplified universality parameter.

(2) An infinite Htree T is pcmzH narrow if and only if [T] is of measure zero

(with respect to the product measure on X).(3) Ipcmz

His the ideal of subsets of X generated by closed measure zero sets.

Definition 1.13. (1) A coordinate-wise permutation for H is a sequence =n : n < such that (for each n < ) n : H(n) H(n) is a bijection.We say that such is anncoordinate-wise permutationifi is the identityfor all i > n.

(2) A rational permutation for H is an ncoordinate-wise permutation for H(for some n < ). The set of all ncoordinate-wise permutations for H willbe called rpn

Hand the set of all rational permutation will be denoted by

rpH

(so rpH

=n

rpnH

).

(3) Let be a coordinate-wise permutation for H. We will treat as a bijectionfrom

n

i


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Proof. We will define nk and T

nnk+1

i


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(a) T IFT[H] is a pnarrow tree such that for every closed set A I0pcoded in V, there is n < with A {[[T]] : rp

nH

}, and

(b) c is a Cohen real over V

.Then, in V, there is a generic filter G

Qtree(p)

Vover V.

(2) Suppose thatV V V are universes of set theory, p V, c V

and d V are such that(a) c is a Cohen real over V, and(b) d is dominating over V.

Then, in V, there is a pnarrow tree T IFT[H] such that for everyclosed set A I0p coded in V, there is n < with A

{[[T]] : rpn

H}.

Proof. (1) The proof essentially follows the lines of that of this result for the caseof the Universal Meager forcing notion by Truss [11, Lemma 6.4]. So suppose thatT, c are as in the assumptions. Let n = nk : k < , T V be as given by 1.16for T (so they satisfy 1.16(ac)).

Consider the following forcing notion C = C(n, T):A condition in C is a finite Htree S such that lev(S) = nk + 1 for some k < .The order relation C on C is given by:S0 C S1 if and only if S0 S1 and S1

i


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if S+, T and lh() = lh() = nk0 + 1 and rpnk0H

are such that() = ,

then (S+

)[]

= [(T

)[]

].In V, take a maximal antichain A D ofQtree(p)

Vsuch that Np > nk0 for each

p A. It follows from 1.4 that then also A is a maximal antichain ofQtree(p)V

inV. Therefore some condition p = (Np, Tp) A is compatible with (nk0 + 1, S

+) Qtree(p)

V. Note that then (Np > nk0 and)

Tp

ink0

H(i) = S+

ink0

H(i) = max(S0)

and S+

i nk0 and

Tp

{[T] : rpnkH

} (remember the assumption 2.1(1)(a) on T). Let

S1def= (S+ Tp)

nnk+1i


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Claim 2.1.2. Let

Gdef= p Qtree(p)V : Tp Tc & Tp

iNp

H(i) = Tc iNp

H(i) V.

Then G is a generic filer inQtree(p)

Vover V.

Proof of the Claim. By 1.6(2), G is a directed subset ofQtree(p)

V. We need that

G D = for every open dense subset D V ofQtree(p)

V. So let D V

be an open dense subset ofQtree(p)

Vand let CD be as defined before 2.1.1. It

follows from 2.1.1 that Gc CD = , say S Gc CD. Then for some k < andT IFT[H] V we have

(nk, T) D, and

T

inkH(i) = max(S), and T

{[T] : prnk

H}.

Now, by (), we may conclude that T Tc getting (nk, T) G.

(2) Suppose that c, d and V V V are as in the assumptions. In V,consider the following forcing notion C:

A condition in C is a pair (n, S) such that

() n = ni : i k is a strictly increasing finite sequence (so k < ),() S FT[H] is a finite Htree such that lev(S) = nk + 1, and for < k:

() if0, 1 S, lh(0) = lh(1) = n +1, and rpnH

is such that (0) = 1,

then [S[0]] = S[1], and() if

T FT[H], lev(T) = n+1 + 1 and

for each 0 S, 1 T, lh(0) = lh(1) = n + 1 and rpnH such

that (0) = 1 we have T[1] [S[0]],

then there is n < such that n+1 < n F(n) < n+1 and (T, n+1, n) G.

The order relation C on C is essentially that of the end-extension:

(n0, S0) C (n1, S1) if and only if n0 n1, S0 S1 and S1

i 2).


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Let A be the set of all X =i


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Hence we may conclude that

i N (ki , ki+1)T in+1+1

H(i) {[Tc] : rpnH }and therefore T

{[T] : rpnN

H} (just look at the choice of T).

Corollary 2.2. Suppose that p is a regular universality parameter for H andP iseither the Hechler forcing (i.e., standard dominating real forcing) or the Univer-sal Meager forcing (i.e., the amoeba for category forcing). ThenQtree(p) can becompletely embedded into RO(P P

).

3. Ideals Ip

Let us recall that for an ideal I of subsets of the space X we define cardinal

coefficients of I as follows:the additivity of I is add(I) = min{|A| : A I &

A / I };the covering of I is cov(I) = min{|A| : A I &

A = X };

the cofinality of I is cof(I) = min{|A| : A I & (B I)(A A)(B A)};the uniformity of I is non(I) = min{|A| : A X & A / I }.The dominating and unbounded numbers are, respectively,

d = min{|F| : F & (g )(f F)(g f)}b = min{|F| : F & (g )(f F)(f g)}.

Below, M denotes the ideal of meager subsets ofX (or of any other Polish perfectspace).

For the rest of this section let us fixed a regular universality parameter p = (G, F)for H.

Corollary 3.1. add(M) add(Ip).

Proof. It should be clear how the proof of 2.1(2) should be rewritten to provideargument for

minb, cov(M)

add(Ip).

(Alternatively, see the proof of the dual version of this inequality in 3.2 below.) Bywell known results of Miller and Truss we have min

b, cov(M)

= add(M) (see

[1, Corollary 2.2.9]), so the corollary follows.

Corollary 3.2. cof(Ip) cof(M).

Proof. By a well known result of Fremlin we have cof(M) = maxd, non(M)

(see

[1, Theorem 2.2.11]). Thus it is enough to show that

cof(Ip) maxd, non(M)

.

Let C be the forcing notion defined at the beginning of the proof of 2.1(2). Let

Ydef=

(n, T) IFT[H] :

k <

(n(k+1), { T : lh() nk+1) C

be equipped with the natural Polish topology. Let = maxd, non(M)

and

choose sequences K : < and (n, T) : < so that

(i) K = {ki : i } [] (the enumeration is increasing),

(ii) (K [])( < )(i )(|K (ki , ki+1)| > 2),


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(iii) the set {(n, T) : < } is not meager (in Y).

For , < and N < let

AN,def=

X :

i N

[ki , ki+1)

(n+1+1)

{[T] : rp

n

H}

.

Then:

()1 Each AN, is a closed pnarrow subset of X.

[Why? See the proof of 2.1.3.]

()2 For each pnarrow tree T IFT[H], there is < such that the set

KTdef=

< : for all 0 T, 1 T

such that lh(0) = lh(1) = n + 1,

{[(T)[1]

in

+1

H(i)] : rpn

H& (1) = 0} T

is infinite.

[Why? By (iii) and an argument similar to the one in the proof of 2.1.4.]

()3 For each pnarrow tree T IFT[H] there are , < and N < such

that [T] AN,.

[Why? By ()2+(ii) and an argument as in the proof of 2.1.4.]Consequently, {AN, : , < & N < } is a cofinal family in I

0p

. Hence, by

1.17(2), {[A0,] : rpH} : , <

is a basis of Ip.

Proposition 3.3. add(Ip) b andd cof(Ip).

Proof. Recall that X = n n, Sn i


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Proof of the Claim. Assume that (n < )((B)(n) < f(n)). Then the set

K = n < : (k < )(f(2n) (B)(k) < (B)(k + 1) < f(2n + 2))is infinite. Now we may pick X such that for each n < we have:

(i) mf(2n) Sf(2n), and(ii) if n K, then for some k such that

f(2n) (B)(k) < (B)(k + 1) < f(2n + 2)

we have (B)(k + 1) / Tn+1.

It should be clear that the choice is possible; note that for n, k as in (ii) we have

f(2n) < lev(Sf(2n)) = mf(2n) < Mf(2n) M(B)(k).

The proposition follows from 3.3.1: if F is an unbounded family, then

{(f) : f F} / Ip, and if B Ip is a basis of Ip, then {(B) : B B} is adominating family in .

It was shown in [2] that the additivity of the ideal generated by closed measurezero sets (i.e., the one corresponding to pcmz

Hof 1.11) is add(M). We have a similar

result for another specific case of Ip:

Proposition 3.4. Suppose that H : \ 2 and g : \ 2 is such that

g(n) + 1 < H(n) for all n < . Let A [] and p = (Gg,AF2

, FH) (see 1.9(2)).Then add(Ip) = add(M).

Proof. Since p is a regular universality parameter (by 1.15), we know that add(M) add(Ip) b (by 3.1, 3.3). So for our assertion it is enough to show that add(Ip) cov(M).

Let us start with analyzing sets in Ip. Suppose that n, w are such that()0 n = nk : k < is a strictly increasing sequence of integers such that

A [nk, nk+1) = for each k < ,

()1 w = wi : i A, wi [H(i)]g(i) + 1 for each i A.

Put

Z(n, w)def=

i


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Proof of the Claim. Pick an increasing sequence ai : i < of members of A so

that f(i) < [H(ai)]g(ai) + 1 for all i < ). For each i fix a one-to-one mappingi : f(i) [H(ai)]g(ai) + 1 . Now, for < , k < and j A let

nk = ak and wk =

i

f(i)

if j = ai, i < ,g(j) + 1 if j / {ai : i < }.

Then n, w satisfy ()0 + ()1 above and thus Z(n, w) Ip (for all < ).

Since < add(Ip) we know that


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Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182-

0243, USA

E-mail address: [email protected]: http://www.unomaha.edu/aroslano

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew

University of Jerusalem, Jerusalem, 91904, Israel, and Department of Mathematics,

Rutgers University, New Brunswick, NJ 08854, USA

E-mail address: [email protected]: http://www.math.rutgers.edu/shelah

andrzej roslanowski and saharon shelah- universal forcing notions and ideals

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