Saharon Shelah- Proper and Improper Forcing Second Edition: Chapter VII:Axioms and Their Application

8/3/2019 Saharon Shelah- Proper and Improper Forcing Second Edition: Chapter VII:Axioms and Their Application

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VII. Axioms and Their Application

0. IntroductionIn the first section we introduce the -e.c.c. (ft-extra chain condition). We provethat if we have an iteration of length < K of (< ct>)-proper forcing notionswhich do not add reals, and if, moreover each forcing used is B-complete fo rsome simple Ni-completeness system D, then the limit satisfies the -c.c. . Thishelps us e.g. in iterations of length ^ of forcings among which none add reals,but each adds many subsets of H I.

In the second section we deal with forcing axioms; essentially our knowl-edge is good when we want 2H = 2Hl = ^2 and reasonable when we want2 = H I and even 2N H 2. In the third section we discuss applications ofthe forcing axiom which is consistent with CH as just mentioned. In the fourthsection we discuss the forcing axiom which is consistent with 2 = 2** 1 = ^2?and in the fifth section we give an example of a CS iteration collapsing H I onlyin the limit. See relevant references in the sections.

1. On the >Chain Condition, When RealsAre Not AddedWhen we prove various consistency results by iterating proper forcings, weoften have to check that the H 2-chain condition holds.


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1 . On the -Chain Condition, When Reals Are Not Added 373

Remember, we deal with a CS iteration Q (P^, Qi : i < a) where Qi is aPi-name of a proper forcing (in V

Pi), Pa = {/ : Dom(/) is a countable subsetof and i G Dom(/) implies 0 I h p . u f ( ) e Qi" , i.e. /(i) is a P^-name of a

member of Qi}. So, P is proper by III 3.2.Here, we concentrate on the casewhen no real is added, in fact when we have a sufficient condition for it. Thecase without this restriction will be discussed again in VIII 2 .

Remark. Note that even if \\ -pi "IQ 1 ]-NI " , still may be |P2| = 2Nl , as theremay be many Pi-names of elements of Q\.

1.1. Lemma. If K is regular, (V < )" < K and Vp" N "|Q| < K " then Psatisfies the /-chain condition.Proof. See III 4.1.

1.2. Definition.P satisfies the ft-e.c.c.(ft-extra chain condition) provided thatthere is a two place relation R on P (usually > # < ? is intended to mean that "pand q have a least upper bound") such that:A) for any pi G P (for i < K) there are pressing down functions fn : K > K (i.e.

(V)/n() < 1 4- ) for n < such that: Q < i,j < K and /\npn& f\nr>qn

1.2A Remark. This is very similar to the condition used in [Sh:80] (andsimilar to a work of Baumgartner, see VIII 1.1, 1.1A(1)). The real differenceis the absence of N I -completeness. The fact that there (in [Sh:80], clause (C )there is a parallel to clause (A) here) we use only one function and closedunbounded C, and demand i, j G C, cf( i ) = cf(j) = H I, is just a variant formwhich was more convenient to represent there. The role of p,q is just toshow that {pn : n < ] and {qn : n < ;}, each has an upper bound (so


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374 VII. Axioms and Their Application

for a Ni-complete forcing they are not needed, hence, also g ( Q ) < ft is notneeded). Even closer is Stanley and Shelah [ShSt:154], [ShSt:154a]. We canask in (A) only that there are p( (for i < K ) such that P \ = pi < p\ and[0 and get the /t-c.c. of Pao. In fact, to prove 1.3, we shallrepeat the proof of V7.1, after an appropriate preparatory step.

2) We can weaken Definition 1.2 so that the proof of 1.3 still works, e.g. bystrengthening the hypothesis on the pn,qn in clause (B). For example,we could have demanded that qn+ G Z[q n], P n + 3[pn] where fo rr G P we have that I[r] C P is a dense subset of P (or even


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1 . On the rc-Chain Condition, When Reals Are Not Added 375

Proof. As in the proof of Lemma III 4.1 we can conclude that if the Lemmaholds for each o < ft, then it also holds for o = ft. So, w.l.o.g. 0 < ft. LetRi be (a Pi-name of a 2-place relation ) exemplifying Definition 1.2 for Q;. LetP a Pa0 (a < ft) be given. We now define, by induction on n < , countablemodels N (for all < ft simultaneously) such that:

i) N S x (tf(),e),Pao NS,Q e N Pa eNa, \\N\\ = K 0 , and N.ii) N -< + 1 , and the additional conditions below are satisfied.

For n 0, choose any N satisfying (i).If we have defined N% for n, let N Po-{p t : , / ^ : i < 0 > : < ; ) ) .c ) 7 V 7 n ft c j.d) A^ (0 -hi)- N? (0 + 1) = {


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1.4. Claim. Suppose that 0 < < < &, Ni(i < &), N? G A 0, N? G 0,are as above, r G P(0, G * C (A^ u A^) P(), G * N? is generic fo r(N?,Pa()) (i.e. G* ^ is directed and if I G ^> and J is pre-dense in PK),then J G* A ^ 0) , and h maps G* A ^ onto G* ^. In addition,every element of G* is < r, r is (]V ,P)-generic for any satisfying = 0 ore


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2. The Axioms 377

2. The AxiomsAXIOM I.

1 ) 2*-2Hl-K 2 and:2) if |P = N2,P proper, J C P pre-dense (for i < \ ) , then there is a

directed G C P, i < W l G n u 0.2') Moreover if |P| = N2,P proper, ! C P pre-dense (for i < 2), P =

Ui


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Case I. If i is not strongly inaccessible, Qi is the Levy collapse of (2' Pl) vto N

:Qi = { f : |Dom(/)| =

0, Dom(/) C c^, Rang(/) C (2\p*\)vP*}.

Case II. If i is strongly inaccessible and, in VPi, there is a proper forcingP, with universe i and 1$ C i (for j


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2. The Axioms 379

2.2. Theorem.1) CON(ZFC -f K is 2*-supercompact) implies CON(ZFC 4- Axiom II +

G.C.H.).2) In both cases (2.1, 2.2(1)) we can relativize to 5 (S C \ stationary,

costationary). If in Axiom II we assume \P\ < 2K l , no large cardinality isneeded.

Proof.1) Similar.2) Just note that the same iteration works. D2>2

2.3. Discussion. In almost all the applications we need a weaker version ofthe axioms for whose consistency we do not need a large cardinal.

Usually our task is to show that for every A C i ( N ) with \A\ = H I, thereis B C f f ( N ) such that

where for Axiom II, is any first order (or LiUl) sentence, and for Axiom I, should have quantifications on A, B only.

So we iterate 2 times only, each time forcing a B for a given A, tillwe catch our tail in 2 steps (we can have \A\ N 2 and visit A VPa forstationarily many a < ;2,cf() = \).

So, only Consis(ZFC) is needed.For Axiom I part (3), K inaccessible suffices (see III 4.1,2,3), whether it

is necessary is still not clear. We can get Axiom I without inaccessible in thecases above, when we are able, provably, to find QA in an intermediate V. ForP2 to satisfy the N 2-c.c. we need \QA\ N I With K inaccessible, QA shouldjust have a cardinality smaller than A C .

For \P\ = H I, if P V Pw2, P is proper in VPz, and for arbitrarily large < , Q = Levy ( H i , 2Hl ), then P is proper in VPa for every large enough,(by III 4.2 though not vice versa). So Axiom I 2) for \P\ = N I comes under theprevious discussion.


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There may be applications where we really have to use information onVPa(a < % ) to build Q, mainly like in Axiom I, when we want to builda forcing giving B for a given A (see above) and we want to use CH or 0^for the building. But with Axioms I, II (or the versions we can get with aninaccessible) we can use the axiom: collapse 2 , building a forcing and look atth e composition (see 3, Application G ).

Sometimes we use properties of Pa (like ^-boundedness) which we usuallydemand from each Q^, and from P in the axiom, and we have to prove that Pasatisfies it, (see Chapter VI).

However still Axioms I, II look like a reasonable choice. We shall use them,and can remark, for suitable applications, that only CON(ZFC) is needed.

As we mentioned (see III 4.3) CON(ZFC-h K inaccessible) implies th econsistency of

AXIOM Ia:If \P\ M I , P does not destroy stationary subsets of \ and 1$ C P pre-dense(for i < u i), then there is a directed C C P, such that

We can ask whether we can get something like Axiom I for 2^ = N 3 .Roitman (see [B]) proved that this is difficult, by proving that:

2.4. Theorem. (R oitman)

1) If Q (Pn,Qn : n < u;} is a CS iteration, Qn nontrivial (i.e. above everyelement there are two incompatible ones) then LimQ does not satisfy the2 -chain condition.

2) If Q = (P,Q i : i < ) is a CS iteration, Qi nontrivial, 2N > N 1 ? thenLimQ collapses 2N to NI.


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2. The Axioms 381

2.5. Question.1 ) What kind of axioms can we get with:

A) 2*-N 3?B) with Ni


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p G P,Q,Ii are P names, p I h ), Ii (for i < i) form a counterexample}.Choose the first (by the canonical order of L) member (p,Q,J). Now useindescribability.2) By Laver [L], w.l.o.g. ft remains supercompact if we force by any ^-completeforcing. Force by {Q : Q is a CS iteration of proper forcing of power ^g(Q) < ft}> ordered by being an initial segment. The generic object is,essentially such Q of length ft, so force by limQ. U2.7Probably better is the following:

2.8 Definition. Let ft be a supercompact cardinal. We call / : ft > #(ft) aLaver diamond if for every cardinal and x G ff(), there is a normal fineultrafilter D on S


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3 . Appl ica t ions of Axiom II (so CH H olds) 383\P K*Q \Definition 2.8 to x Q and such that Q G H(X) and even 2 < , and

get D as there. Choose a G AD(X) such that (Ii : i < ) and (S : < \)belong to , (,G) is isomorphic to some (if(), G) and letting MCa be theMostowski Collapse of (i.e. the unique isomorphism f rom (, G) onto ( H ( ) , G )and = a K G ft, we have f() = MCa(Q). Note MCa(Q) is a P-name of aproper forcing etc.Easily, Q = f*() in VF , and Q is isomorphic to Q,so we can finish.

2.10 Definition. (1) Let


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3.3 Application C: Uniformization.Axiom II implies: If $ is an increasing -sequence converging to 5, for any limit < \ then for every F : \ thereis a g : \ -> such that: (V < )[ limit -> (3n)[n >F(J) & n = g((k)}fo r all but finitely many fc < ]]

/. Let 7 7 5 for limit 5 < \ be given. Let PF = {/ : Dom(/) is an ordinal< i, Rang(/) C , and for every limit Max=)nf^(), such that A = U^(Rang(77|) \ a) has order type andis disjoint to Dom(/), and find a q >p, q is ( A T , PF)-generic, g f A is constantly771.

More formally, (seeV. 5.2, 5.3), we shall define D(jv,prw,p)> asafilter onA0 = {G C P N : p G G,G is directed not disjoint to any I G JV , J C P, Jpre-dense} such that it depends only on the isomorphism type of ( A T , P,p). Let

The filter will be generated by A JJ j p , where for n < , an -sequenceconverging to 5, A p = {G C AQ : for some fc < , and fc >n, for every q G Gand e ^. (For a definition - see below).But we know (2H < 2H l) =^ ^ (Devlin and Shelah [DvSh:65], or here AP1).

3.4A Definition. The statement J is defined as: For every G :>2 > ; thereis an F : X such that for every p G2 we have {i < : G ( ^ t z ) = F(i)} isstationary.


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3. Applications of Axiom II (so CH H olds) 385

Question. Does G.C.H. -* ^? (See the next chapter.)

3.5 ApplicationD*. Axiom II implies: If ( : < i, limit} is as above, andc e , then there is an / : -> such that (V)(3n)[/(^(n)) = c(n)}(this was proved in U. Abraham, K. Devlin and S. Shelah [ADSh:81]).

There an application of this to a problem of Hajnal and Mate on thecoloring number of graphs is given.

Proof. Easy by now. 1 3 . 53.6 Application E. Fleissner showed:

^ = > (topological statement A) = > not there is a tree = ( g : < i),with Dom(7$) = , for each 5, and s(n) (for n < ) are increasing with Jn(n), such that:

(V/i : i > ) ( 3 f : \ > )(3/' : i -> ;)(V limit 5 < i)

(3 m* < ;)(Vn > m^)[/ i ( J) < f(s(n))


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< 8 > r (V/ : -> ) [(Vt t)(3a; < t)(Vy)[z < y < - /(y) < /()](3p : -* )(V 6 t)(3x < ) (Vy)[a: < y < -

(This is sufficient for the existence of some examples in general topology, seethe end of the application and 3.25 below; we can add (Vy)(/(y) < g ( y ) ) . )Again CON(ZFC) suffices (as Claim 3.19F10 holds), i.e.we prove:Claim. [Axil] Every Aronszajn tree T satisfies 0 (remembering that Axil isconsistent with G.C.H. and implies that every Aronszajn tree is special, we getthe desired answer).

The proof is quite similar to that of Application A, see V 6.1. However,here we have to do more; an incidental point is that here we have to find ag : T > , not from a club of levels but from all of them.

Let T be an Aronszajn tree and /* : T be such that:

(V Tt)(3x < t)(Vy)(x < y < t -> /*(y) < /*(*)).

Let F = {(g,C) : for some countable ordinal i we have Dom(g) T


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3. Applications of Axiom II (so CH H olds) 387

a) 7 C U{( )m(/) : > Min(C)} and(,...,o m (/)) G / = > /\

k[l,m(

b) if Min(C) < < / ? , and (i , . . . , m ( / )) G (T0)mW 7, then

(of course for T/j, f is the unique 6 Ta such that 6 Max{/*() : ^ < t < be} < .

Notation. If G ( )mW, then () d= .

3.10 F2 Definition. For ,6 / , < 6 holds i/ G (T())m(/), 6 6( ()r(J),()


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a) i(g) G C(I\ i(g) > (J) and C \ 0(/) C C(J).Subdefinition (F3i). We say that g is /-good for / i f '() < i(#) and

< y < a -> /( j/) such thatb G (:7m(/) / and (g1, &) is /-very good for b. (Hence (g1, C f) will be /-verygood for .)

Let { (/ fc , ! fc ,fc ) : k < } be a list of all triples as above. As (g,C) is//--very good for jt, there is a set of pairwise disjoint sequences {bkte t

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Saharon Shelah- Proper and Improper Forcing Second Edition: Chapter VII:Axioms and Their Application