andrzej roslanowski and saharon shelah- reasonably complete forcing notions

8/3/2019 Andrzej Roslanowski and Saharon Shelah- Reasonably Complete Forcing Notions

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REASONABLY COMPLETE FORCING NOTIONS

ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

Abstract. We introduce more properties of forcing notions which imply thattheir support iterations are proper, where is an inaccessible cardinal.This paper is a direct continuation of Roslanowski and Shelah [5, A.2]. As anapplication of our iteration result we show that it is consistent that dominatingnumbers associated with two normal filters on are distinct.

0. Introduction

There are serious ZFC obstacles to easy generalizations of properness to the caseof iterations with uncountable supports (see, e.g., Shelah [8, Appendix 3.6(2)]).This paper belongs to the series of works aiming at localizing good propernessconditions for such iterations and including Shelah [9], [10], Roslanowski and She-lah [4], [5] and Eisworth [2]. Our results continue Roslanowski and Shelah [5, A.2],but no familiarity with the previous paper is assumed and the current work is fullyself-contained.

In Section 2 we introduce 3 boundingtype properties (A, B, C) and we essen-tially show that the first two are almost preserved in support iterations (Theo-rems 2.5, 2.8). Almost as the limit of the iteration occurs to have a somewhatweaker property, but equally applicable. In the following section we show that rea-sonably Abounding forcing notions are exactly the ones introduced in [5, A.2],thus showing that Theorem 2.8 improves [5, Thm A.2.4]. In the fourth sectionof the paper, we give an example of an interesting reasonably Bbounding forcingnotion and we use it to show that it is consistent that dominating numbers associ-ated with two normal filters on are distinct (Corollary 4.13). Finally, in the lastsection we present two forcing notions that are not yet covered by existing iterationtheorems. We hope that the further development of the theory will include alsothem.

Like in [5], we assume here that our cardinal is inaccessible. We do not knowat the moment if any parallel work can be done for a successor cardinal, thoughsome progress will be presented in a subsequent paper [6].

Acknowledgment: We would like to thank the anonymous referee for very valu-able comments and corrections.

Notation: Our notation is rather standard and compatible with that of classicaltextbooks (like Jech [3]). In forcing we keep the older convention that a strongercondition is the larger one.

Date: September 2006.1991 Mathematics Subject Classification. Primary: 03E40 Secondary: 03E35, 03E17.Both authors acknowledge support from the United States-Israel Binational Science Foundation

(Grant no. 2002323). This is publication 860 of the second author.

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2 ANDRZEJ ROSLANOWSKI AND SAHARON SHELAH

(1) Ordinal numbers will be denoted by the lower case initial letters of theGreek alphabet ( , , , . . .) and also by i, j (with possible sub- and su-

perscripts).Cardinal numbers will be called ,,; will be always assumed to be

inaccessible (we may forget to mention it).By we will denote a sufficiently large regular cardinal; H() is the

family of all sets hereditarily of size less than . Moreover, we fix a wellordering


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REASONABLY COMPLETE FORCING NOTIONS 3

(2) We say that the forcing notion P is strategically (


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pt Prk(t) for t T, and if s, t T, s t, then ps = ptrk(s).

(3) Let p0

, p1

be standard trees of conditions in Q, pi

= pit : t T. We write

p0 p1 whenever for each t T we have p0t p1t .

Note that our standard trees and trees of conditions are a special case of (w, ) trees introduced in [5, Def. A.1.7] (for = 1). Our notation preserves the redun-dant 1 to keep the compatibility with the established terminology. For the samereason we use (t) instead of t().

Proposition 1.5. Assume that Q = Pi,Q

i : i < is a support iteration suchthat for all i < we have

Pi Q

i is strategically (


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REASONABLY COMPLETE FORCING NOTIONS 5

(b) if 0 < 1 , then T0 = {t0 : t T1} and r

0t0

q1t 0 for

t T1 ;

(c) if < , t T and rk(t) = (so rk(t) = ), then

qtpt[, ), r

t

pt[, )

: <

qt , rt

: <

is a partial play of0 (P , pt) in which Complete uses her winning strategyst( , pt);

(d) = t : t T, rk(t) =

;

(e) if < , t T and rk(t) = (so rk(t) = ), then pt qt P and

qt P =

t ;

(f) if < , t T and rk(t) = , then

< : t {,

T+1} =

t .

We let T0 = {} and we choose q0 P0 and

0 so that p q

0 and q

0 P0

0 = 0. Then we let r0 be the answer given by st(0, p) in 0 (P0 , p) to q0.Now suppose that we have defined T, q, r and for < .

If is a limit ordinal then the demands (a) and (b) uniquely define the standardtree T. Note that |T| < as is inaccessible; remember also clause (f). It followsfrom the choice of st(, r) (see clause 1.3(iii)) and demand (c) at previous stagesthat

() if t T, rk(t) = (so rk(t) = ), then the sequenceqt

pt[, ), rt

pt[, )

: <

is a partial play of0 (P , pt) in which Complete uses her winning strategyst( , pt).

For t T we define a condition qt P as follows:

Dom(qt) =


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if t T \ T0, then qt = r0t0

pt[0 , ).

Then q = qt : t T is a tree of conditions, r0t qt for t T0 . It follows from

1.5(1) that we may choose a tree of conditions q = qt : t T such that q q

and

if < , t T and rk(t) = , then the condition qt decides

and,

say, qt

= t .

Next, like in the limit case, r = rt : t T is obtained by applying the strategiesst(rk(t), pt) suitably. Easily, T, q, r and satisfy the demands (a)(f).

After the inductive construction is carried out look at T , q

and :

andrzej roslanowski and saharon shelah- reasonably complete forcing notions

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