andrzej roslanowski and saharon shelah- simple forcing notions and forcing axioms

8/3/2019 Andrzej Roslanowski and Saharon Shelah- Simple forcing notions and Forcing Axioms

1/23

508

re

vision:1996-06-22

modified:1996-06-22

Simple forcing notions and Forcing Axioms

Andrzej Roslanowski

Institute of Mathematics

The Hebrew University of Jerusalem

91904 Jerusalem, Israel

and

Mathematical Institute of Wroclaw University

50384 Wroclaw, Poland

Saharon Shelah

Institute of Mathematics

The Hebrew University of Jerusalem

91904 Jerusalem, Israel

and

Department of Mathematics

Rutgers University

New Brunswick, NJ 08854, USA

October 6, 2003

The research supported by KBN (Polish Committee of Scientific Research) grant1065/P3/93/04

The research partially supported by Basic Research Foundation of the Israel Academyof Sciences and Humanities. Publication 508

0


2/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 1

0 IntroductionIn the present paper we are interested in simple forcing notions and ForcingAxioms. A starting point for our investigations was the article [JR1] in whichseveral problems were posed. We answer some of those problems here.

In the first section we deal with the problem of adding Cohen reals by simpleforcing notions. Here we interpret simple as of small size. We try to establishas weak as possible versions of Martin Axiom sufficient to conclude that someforcing notions of size less than the continuum add a Cohen real. For examplewe show that MA(-centered) is enough to cause that every small -linkedforcing notion adds a Cohen real (see 1.2) and MA(Cohen) implies that everysmall forcing notion adding an unbounded real adds a Cohen real (see 1.6). Anew almost -bounding -centered forcing notion Q appears naturally here.

This forcing notion is responsible for adding unbounded reals in this sense, thatMA(Q) implies that every small forcing notion adding a new real adds anunbounded real (see 1.13).

In the second section we are interested in AntiMartin Axioms for simpleforcing notions. Here we interpret simple as nicely definable. Our aim is to showthe consistency of AMA for as large as possible class of ccc forcing notionswith large continuum. It has been known that AMA(ccc) implies CH, but ithas been (rightly) expected that restrictions to regular (simple) forcing notionsmight help. This is known under large cardinals assumptions and here we tryto eliminate them. We show that it is consistent that the continuum is large(with no real restrictions) and AMA(projective ccc) holds true (see 2.5).

Lastly, in the third section we study the influence ofMA on 13absolutenessfor some forcing notions. We show that MA

1

(P) implies 13

(P)absoluteness(see 3.2).Notation: Our notation is rather standard and essentially compatible with thatof [Je] and [BaJu]. However, in forcing considerations we keep the conventionthat a stronger condition is the greater one.For a forcing notion P and a cardinal let MA(P) be the following statement:

If A P are maximal antichains inP (for < ), p Pthen there exists a filter G P such that p G and G A = forall < .

For a class K of forcing notions the sentence MA(K) means (P K)MA(P);MA is the sentence MA(ccc).For a forcing notion P, the canonical Pname for the generic filter on P will becalled P. The incompatibility relation on P is denoted by P (so P meanscompatible).c stands for the cardinality of the continuum. For a tree T 2< , [T] is theset of all branches through T.The family of all sets hereditarily of cardinality < (for a regular cardinal )is denoted by H().


3/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 2

1 Adding a Cohen realIn this section we obtain several results of the form a (weak) version of MAimplies that small forcing notions (of some type) add Cohen reals. As a con-sequence we answer Problem 5.3 of [JR1] (see 1.9, 1.10 below).

Proposition 1.1 Suppose P is a forcing notion and h is a function such that

1. dom(h) P, rng(h) 2< ,

2. if p1, p2 dom(h), p1Pp2 then either h(p1) h(p2) or h(p2) h(p1),

3. if q P then there is 0 2< such that

( 2< , 0 )(p dom(h))(p

P

q & h(p)).

ThenP adds a Cohen real.

Proof Though this is immediate, we present the proof fully for readersconvenience. Let h : dom(h) 2< be the function given by the assumptions.Define a P-name c by

P c =

{h(p) : p dom(h) P}.

First note that, by the properties of h, for every filter G P the set {h(p) : p dom(h) G} is a chain in (2< , ). Hence

P c 2

.

But really c is a name for a member of 2: suppose not. Then we have q P,m such that

q P c 2m.

Applying the third property of h we get 0 2< as there. Let 2< ,

0 be such that lh() > m. We find p dom(h) such that pPq and

h(p). Thus p P h(p) c, a contradiction.To show that

P c is a Cohen real over V

suppose that we have a closed nowhere dense set A 2 and a condition q Psuch that

q P c A.

Take 0 2< given by condition (3) (for q). Since A is nowhere dense we may

choose 2< such that 0 and [] A = . By the choice of 0, thereis a condition p dom(h) such that pPq and h(p

) (so p P c / A), acontradiction.


4/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 3

Theorem 1.2 Assume MA

(-centered). IfP

is a -linked atomless forcingnotion of size thenP adds a Cohen real.

Proof We may assume that the partial order (P, ) is separative, i.e.

if p, q P, p P qthen there is r P such that q P r and rPp.

Of course we may assume that P is a partial order on a subset of 2. We aregoing to show that (under our assumptions) there exists a function h as in theassumptions of proposition 1.1. Since P is -linked there are sets Dn P suchthat

n

Dn = P and any two members of Dn are compatible in P (i.e. each

Dn is linked). Let N be a countable elementary submodel of (H((7)+), ,


5/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 4

Claim 1.2.2R

is -centered.

Proof of the claim: Note that if r1, r2 R, h = hr1 = hr2 then h, wr1 wr2is a common upper bound of r1, r2.

Claim 1.2.3 Suppose p P N, q P, r0 R and m . Then the followingsets are dense inR:

1. I0pdef= {r R : (q wr)[pPq (p

dom(hr))(p P p & pPq)]},

2. I1qdef= {r R : q wr},

3. I2r0,mdef= {r R : rRr0 or for every q wr0 and p dom(hr0) such

that pPq & (p dom(hr0))([p P p & p = p] pPq)

and for every 2m such that hr0(p) there is

p dom(hr) with p P p, pPq and hr(p)}.

Proof of the claim: 1) Assume p P N, r0 R. Let ql : l < l be anenumeration of {q wr0 : qPp}. Choose conditions pl (for l < l

) such that

1. pl P N

2. for each p dom(hr0) either p P pl or pPpl,

3. p P pl,

4. pl : l < l are pairwise incompatible,

5. plPql.

For this we need the assumption that P is atomless and linked. First takep+l P such that p, q

l P p+l (for l < l

). Next we choose p++l P, p+l P p

++l

such that the clauses (2)(4) are satisfied (remember that P is atomless anddom(hr0) is finite). Let nl be such that p

++l Dnl . As N is an elementary

submodel of (H(+7 ), , N. Then we have a condition p q which forces r and thusp P lh() K.Suppose now that q P, g is an increasing function and N0 . TakeN1 > N0 such that (n N1)(qn / A) and a condition pqg(N1) such that

q P pqg(N1) and pqg(N1) P qg(N1) r (remember that q [Tq]). Nownote that

pqg(N1) P K [N1, g(N1)) = .

Hence we easily conclude that

P the increasing enumeration of K is an unbounded real over V

finishing the proof.

Remark 1.14 The forcing notion Q makes the ground model reals meagerin a soft way: it does not add a dominating real (see 1.12). However itadds an unbounded real (just look at {n : A 2n = }, for A as in

1.12(2)). Consequently it adds a Cohen real (by [Sh:480]; note that Q is aBorel ccc forcing notion). Hence we may put together 1.6 and 1.13 and we getthe following corollary.

Corollary 1.15 Assume MA(Q). Then every ccc forcing notion of size adding new reals adds a Cohen real.

2 Anti-Martin Axiom

In this section we are interested in axioms which are considered as strong nega-tions of Martin Axiom. They originated in Millers problem if it is consistent

with CH that for any ccc forcing notion of the size c there exists an 1-Lusinsequence of filters (cf [MP]). The question was answered negatively by Todorce-vic (cf [To]). However under some restrictions (on forcing notions and/or densesets under consideration) suitable axioms can be consistent with CH. Theseaxioms were considered by van Douwen and Fleissner, who were interested inthe axiom for projective ccc forcing notions, but they needed a weakly compact


15/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 14

cardinal for getting the consistency (cf [DF]). Cichon preferred to omit the largecardinal assumption and restricted himself to 12 ccc forcing notions and stillhe was able to obtain interesting consequences (see [Ci]). Here we show how toomit the large cardinal assumption in getting AntiMartin Axiom for projectiveccc forcing notions. This answers Problem 6.6(2) of [JR1].

Definition 2.1 For a forcing notion P and a cardinal let AMA(P) be the following sentence:

there exists a sequence Gi : i < of filters on P such that for everymaximal antichain A P for some i0 < we have

(i i0)(Gi A = ).

For a class K of forcing notions the axiom AMA(K) is for each P K,AMA(P) holds true.

Definition 2.2 1. For two models N, M and an integer n, M n+1 Nmeans:

for every n formula (x, y) and every sequence m M,if N |= x(x, m) then M |= x(x, m).

(Thus M N if and only if (n > 0)(M n N).)

2. IfP0,P1 are ccc forcing notions, n > 0 then P0


16/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 15

2. Suppose < +, cf() 1

andPi

Cn

(for i < ) are such that

i < j < Pi


17/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 16

4. (pi, i, i) : i < lists all triples (p, , ) such that = (x, y) is a

n-formula, is a (canonical) P-name for a finite sequence (of a suitablelength) of members of H(1), p P,

5. ifi is limit, cf(i) = then Pi Cn is such that


18/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 17

Let pm

: m < P

. For k < we try to define aP

-name k

for a real in

by(m Ai,j,k)(pm P k(i) = j).

If this definition is correct then we ask if these reals encode (in the canonicalway) elements of H(1) (which we identify with the names k themselves). Ifyes then we ask if

p0 P (H(1), ) |= x(x, 0, . . . , 1).

If the answer is positive then we fix a P-name for (a real encoding) a memberof H(1) such that

p0 P (H(1), ) |= ( , 0, . . . , 1).

This name can be represented similarly as names k (for k < ) so we have asequence qm : m < P encoding it. Finally we want F,m to be such thatif the above procedure for pm : m < works then F,m(pm : m < ) = qm.Now take all the functions F0k , F

,m; it is easy to check that they work.

Lastly note that the case n = follows immediately from the lemma forn < . (For 3. construct an increasing sequence Pi : i < 1 such that P


19/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 18

4. if i

is aPi

-name and

Pi i(x, i), i(x0, x1, i) defines in (H(1), ) a ccc partial order Qi

then Pi Qi


20/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 19

Remark 2.6 In 2.4 and 2.5 we used H(1

) as we were mainly interested inAMA1 and projective ccc forcing notions. But we may replace it by H() forany uncountable regular cardinal such that


21/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 20

J1 = {p P : either p P / T(r)or ( Ord)(p P T(r) & (r n, ) = )}.

Clearly these are dense subsets ofP. By MA1(P) we find a filter G on P suchthat G J0n = for n and G J

1 = for 1


22/23

508

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 21

Proof For each 2< choose a maximal antichain p,l

: l inQand a set I such that for each l :

l I p,l Q r() = 0 and l / I p,l Q r() = 1.

Now note that for each 2 we have

A[r] (p Q)(n )(l In)(pn,lQp).

Proposition 3.4 For every A 2 there exist a -centered forcing notionQA

and a QAname r (for a function from 2< to 2) such that A = A[r] and

Q = A + 0.

Proof The forcing notion QA is defined by

conditions are pairs r, w such that r is a finite function, dom(r) 2< ,rng(r) 2 and w [A]< ,the order is such that r1, w1 r2, w2 if and only if r1 r2, w1 w2 and

( dom(r2) \ dom(r1))([( w1)( )] r2() = 1).

The QAname r is such that

QA r =

{r : (w)(r, w QA)}.

It should be clear that QA is -centered, QA = A + 0 and

QA r : 2< 2.

Moreover for each 2 and r,w, QA:

(m)(r, w QA r( m) = 1) iff w.

Consequently A = A[r].

Corollary 3.5 If A 2 is not analytic thenQA cannot be completely embed-ded into a ccc Souslin forcing. In particular, if c > 1 then there is a -centeredforcing notion of size 1 which cannot be completely embedded into a ccc Souslinforcing notion.


23/23

8

re

vision:1996-06-22

modified:1996-06-22

[RoSh:508] October 6, 2003 22

References[BaJu] Bartoszynski, Tomek and Judah, Haim, Set Theory: On the

Structure of the Real Line, A K Peters, Wellesley, Mas-sachusetts, 1995.

[Ci] Cichon, Jacek, AntiMartin Axiom, circulated notes (1989).

[DF] van Douwen, Eric K. and Fleissner, William G., Definable ForcingAxiom: An Alternative to Martins Axiom, Topology and its Appli-cations vol.35(1990): 277289.

[Je] Jech, Thomas, Set Theory, Academic Press 1978.

[JR1] Judah, Haim and Roslanowski, Andrzej, Martin Axiom and the con-tinuum, Journal of Symbolic Logic, vol.60(1995): 374391.

[MP] Miller, Arnold and Prikry, Karel, When the continuum has cofinality1, Pacific Journal of Mathematics, vol.115(1984): 399-407.

[Sh:480] Shelah, Saharon, How special are Cohen and Random forcings, IsraelJournal of Mathematics, vol.88(1994): 153174.

[Sh:f] Shelah, Saharon, Proper and improper forcing, Perspectives inMathematical Logic, Springer, accepted.

[To] Todorcevic, Stevo, Remarks on Martins Axiom and the ContinuumHypothesis, Canadian Journal of Mathematics, vol.43 (1991): 832

851.

andrzej roslanowski and saharon shelah- simple forcing notions and forcing axioms

Documents