saharon shelah- proper and improper forcing second edition: chapter xi: changing cofinalities;...

8/3/2019 Saharon Shelah- Proper and Improper Forcing Second Edition: Chapter XI: Changing Cofinalities; Equi-Consistency Results

1/57

XL Changing Cofinalities;Equi-Consistency Results

0. IntroductionWe formulate a condition which is (strongly) preserved by revised countablesupport iteration, implies H I is not collapsed, no real is added and is satisfiede.g. by Namba forcing, and any Ki-complete forcing. So we can iterate forcingnotions collapsing ^2 but preserving H I up to some large cardinal.

Our aim is to improve th e results of chapter of X to equi-consistencyresults. If you want to add reals, look at Chapter XV. To prove the preservationwe use partition theorems and -system theorems on tagged trees (3.5, 3.5A,3.7 (and 4.3A)). Some of them ar e from Rubin and Shelah [RuSh:117], se edetailed history there on pages 47, 48 and more on mathematics see [RuSh:117],[Sh:136] 2.4, 2.5 (pages 111 - 113).

1. The Theorems1.1 Discussion. In this chapter we list the demands that we would like ourcondition to satisfy, and show how, having a condition satisfying these demandswe can prove our theorems. Then, in the following sections we will formulatethe condition and prove it satisfies all our demands. Lastly we shall prove somemore complicated theorems applying the condition.


2/57

1. The Theorems 533

The Demands. We will have a condition for forcing notions such that:(i) If P satisfies the condition then forcing with P does not collapse KI and,

moreover, (when CH holds) it does not add reals.(ii) If P = RlimQ, where Q is an RCS iteration of forcing notions such that

each of them satisfies the condition then P satisfies it as well.(RCS iteration was defined in X 1. In 1.9 we will recall its basic properties).

Really we do not get (ii) but a slightly different version (ii) , which is asgood for our purpose:(ii)' Assume V satisfies: if Q (P %,Qi ' i < ) is an RCS iteration, Qu+i is

Levy collapse of 2' p2i+1l+l ll to KI (by countable conditions), each Q ^i sat-isfies the condition. Then P$, the revised limit of j, satisfies the condition(see also 6.2A).

(iii) If Q = (Pi,Qi : i < K ) is an RCS iteration as in (ii)', K is a stronglyinaccessible cardinal and \Pi\ < K for i < K then P, the revised limit ofQ, satisfies the ft-c.c.

(iv) The condition is satisfied by the following forcing notions:a. Namba forcing. (See 4.1, it adds a cofinal countable subset to ;2

without collapsing \.) We denote this forcing notion by Nm.b. Any Ni-closed forcing notion.c. P[5], where 5 is a stationary subset of ^ such that G 5 = > cf() =

, and P[S] = {/ : / is an increasing and continuous function froma + 1 into 5, for some a < }. Note that P[S] shoots a closed copyof \ into S hence collapses fc ^

Remark. The condition on P is, by the terminology we shall use, essentiallythe { < |P| : regular > K}-condition; more exactly, the definition of sucha notion is in 6.7, where (ii); is proved. Now (iv) holds by 4.4, (iv)^ by 4.5,(iv)c holds by 4.6, (i) by 3.2 and (iii) automatically follows from 6.3A(1) as inX 5.3, see 1.13.


3/57

534 XL Changing Cofinalities; Equi-Consistency Results

Remark. The preservation theorem in this chapter is in a sense orthogonal to theone of Chapter X, since here we are not interested in semiproperness of forcingnotions (e.g. Namba forcing may fail to be semiproper, but it always satisfiesthe condition in this chapter). In chapter XV we will present a generalizationof the S-condition which also generalizes semiproperness.

Assume we have a condition satisfying all of these demands and let us getto the proofs of our theorems.

1.2 Theorem. If "ZFC + G.C.H . -f there is a measurable cardinal" is consis-tent then so is "ZFC -f G.C.H. 4- there is a normal precipitous filter D on ^such that Si G >".

Remark.(1) S 0 2is{ i, hence Qn+i collapses \i\ to NI , hence all ,H I < < K are collapsed by P. By l.l(iii) (or X 5.3(1)) P satisfies the K-chaincondition hence K , is not collapsed. So clearly Nj', H ^ = K and Vpsatisfies G. C.H .


4/57

1. The Theorems 535

Let F be a normal ^-complete ultrafilter over K (in V), then by (iii)andX.6.5 (see references there), F generates a normal precipitous filter on K in Vp.Let A be { < K : X is inaccessible} (in V) then A F so we are done withthe proof once we show that G A implies has cofinality in V[P\. As Nmsatisfies our condition (by demand (iv)a ) and is inaccessible in V we knowthat the iteration up to stage satisfies the -c.c. (by demand (iii), or by usingX5.3(l) provided that we restrict A to Mahlo cardinals). Hence after forcingwith P\ we have = ^2 and at the next step in the iteration Nm shoots acofinal -sequence into , a sequence that exemplifies cf() = N 0 in Vp', see4.1A. D.2

1.3 Theorem. If "ZFC + G.C.H. + there is a Mahlo cardinal" is consistentthen so is "ZFC -f G.C.H . 4- for every stationary S C S$ there is a closed copyof i included in it".

Remark. Earlier Van-Liere has shown the converse and is a variant of a problemof Friedman, see on this X 7.0.

For the clarity of the exposition we prove here a weaker theorem andpostpone the proof of the theorem as stated above to Sect. 7 of this chapter.

1.4 Theorem. If "ZFC + G.C.H. + there is a weakly compact cardinal" isconsistent then so is "ZFC + G.C.H. -h for every stationary S C S$ there is aclosed copy of \ included in it".Proof. The proof is very much like the proof of Theorem 7.3 of X; the onlydifference is that now we do not have to demand that there will be measurablecardinals below the weakly compact cardinal. We give here only an outlineof the proof. Let / be weakly compact, w.l.o.g. V = L, so by Jensen's workthere is (Aa : a < K , a inaccessible), Aa = (Aa,e : e < n), a diamondsequence satisfying: Aa,e C H(a), and for every finite sequence A of subsets ofH(), and } sentence such that (H(), G , A) N = there is some inaccessible such that A\H(\) = A (i.e. n = lg(A) and ,e = A. H()) and


5/57

536 XI. Changing Cofinalities; Equi-Consistency Results

(f(), G , ^ 4 ) f = V 7 - Now we define an RCS iteration Q = (Pi,Qi : i < K). LetQa = P[a] (as it was defined in 1.1 (iv)c) whenever Aa (Pa,Pa,Sa), astrongly inaccessible and pa GP and pa\[-pa uSa is a stationary subset of SQ(= { < X : cf() H O } )" , and in all other cases we force with the usual LevyHi-closed conditions fo r collapsing 2' P !!~ I~ I Q: L

In the model we get after the forcing K H 2 and every stationary subset of50 includes a closed copy of \. (For checking the details note that our forcingnotion, and any initial segment of it, satisfies our condition thus no reals areadded, H I is not collapsed and in any -stage for inaccessible , the initialsegment of the forcing satisfies the -c.c. so at that stage = %, when we useP[S\] we are forcing with P[S] for 5 which is a stationary subset of SQ). DI.S

1.4A Remark. If K is only a Mahlo cardinal then this proof suffices if we justwant "for every 5 C 5^, 5 or 50 \ 5 contains a closed copy of \ " or even ifwe want "if h is a pressing-down function on SQ, then for some , h~l({a})contains a closed copy of \". See more in7.2.

1.5 Theorem. If "ZFC+ G.C.H. -f there is an inaccessible cardinal" isconsistent then so is "ZFC 4- G.C.H. + there is no subset of H I such thatall -sequences of ^2 are constructible f rom it".

Remark.(1) This theorem answers a question of Uri Abraham who has also proved its

converse.(2) Again we can omit G.C.H. from the hypothesis.

Proof. Let K be inaccessible and let Q (Pi,Qi : i < K ) be an RCS iteration,Q2i = (Nm)yP2, Q2i+ = L e v y ( H 2 , 2 l p 2 * l + l * l) . In the resulting model K = H 2and as the forcing satisfies the tt-c.c. any subset of H I is a member of a modelobtained by some proper initial segment of our iteration, but the -sequenceadded to ^ by the next Nm forcing does not belong to this model so it is notconstructible f rom this subset of H I . DI.S


6/57

1 . The Theorems 537

1.6 Theorem. If the existence of a Mahlo cardinal is consistent with ZFCthen so is "G.C.H. + for every subset A of N

2there is some ordinal such that

= N O but is a regular cardinal in L[A ]".

Remark. Again this is an answer to a question of Uri Abraham and again hehas shown that the converse of the theorem is true as well by using the squareon .Proof. Let K be Mahlo (in a model V of ZFC + G.C.H .), w.l.o.g. V = L anditerate as in the proof of 1.5. Let A be a subset of N 2 in the resulting model.As the forcing notion satisfies the ft-c.c., we can find a closed and unboundedC C N 2 such that for e C we have A 5 G V[P$] where P$ is the lirn of'th initial segment of our iteration. As K is Mahlo, { < K , : X is inaccessible}is stationary in it so there is some inaccessible in C. Such exemplifies ourclaim. P\ satisfies the -c.c., so in V[P] we have = K 2 hence Nm at the-step of the iteration adds a cofinal sequence into , so in V[P], which isour model, cf()-N 0 But L[A] C V[P] as G C (and P ? P = P). D.6One more answer to a question from Uri Abraham's dissertation is to get Vsuch that if A C > 2 and K 2 N 2 then L[-A] has > K 2 reals. We had noted thatfor the statement to hold in V, it is enough to have: L[{ < K r f : cfv S = N O } ]has at least H ^ reals; more explicitly, it is enough to produce a model in whichthere are K 2 distinct reals r, such that for some GCarL we have r() = 0 iffcf((+^)L) = H O (i.e., the cofinality is in V,\+is computed in L). Then theanswer below was obtained by Shai Ben-David using the same method as of1.5,1.6:

1.7 Theorem. [Ben David] The consistency of "ZFC + there exists an inacces-sible cardinal" is equivalent with the consistency with ZFC of the statement:"There is no cardinal preserving extension of the universe in which there is aset A C N 2 such that L[A] satisfies C.H. and


7/57


However, this proof relies on a preliminary version of this chapter inwhich forcing notions adding reals were permitted, which unfortunately seemsdoubtful and was abandoned. The framework given in 1.1, is not enough sincein (iv)no forcing notions adding reals appear, but we can use XV 3 instead(i.e. for unboundedly many i, Qi is Cohen forcing)

Remark. In fact there is no class of V which is a model of ZFC, having thesame N I and ^2 and satisfying C.H.

1.8 Remark. The partition theorems presented later can be slightly general-ized to monotone families (instead of ideals) as done in the first version of thisbook. But this is irrelevant to our main purpose.

We now recall the main properties of RCS iterations. Whenever it isconvenient, we will assume that all partial orders under consideration arecomplete Boolean algebras i.e.are (B \ {0}, >).

1.9 Definition. We say that a sequence (Pa,Q,\ : a < , < ) is an RCSiteration of length ( not necessarily limit ordinal), if :

(1) For all / ? , 1 / 3 is a dense embedding from P * Q into P+\, or intothe complete Boolean algebra generated by P/3+ [We usually do notmention an d identify P * Q with P+}

(2) For all < < , P < P. [W e assume that for all p < P , theprojection of p to Pa exists, and we write it as p\a. We write Pa,for the quotient forcing P/Pa or P/Ga in VPa or t^[G], and letp H - > p\[a,) be the obvious map in VPa.}

(3) Whenever < is a limit ordinal, then Pa = R\im(P : < a).Also, if we write Q (Pa,Qa : < 5), we automatically define PS =

RlimQ (if is a limit) or P = PS-I * Q- (if 5 is a successor), respectively.We say that (Pa : < 5} is an RCS iteration iff (Pa,Qa ' O L < ) is one,


8/57

1 . The Theorems 539

We will not define here what Rlim Q actually is. A possible definition is inchapter X. Here we will only collect some properties of RCS iterations whichwe will use. First, we need a definition:

1.10 Definition. If g is a P -name we say that g is prompt, if I h "g


9/57


It may be more illustrative to consider the following dense subset:

1.11A Remark.(a) For any prompt P^-name a we have PQ, < > P$. Moreover, if \\ -P "gi p such that < ? I h "r = r*".

(2) // {P, Q :< K ) is an RCS iteration, A C a strongly inaccessible cardinal,P does not collapse N I , and for all < K we have |P| < , en Psatisfies the tt-chain condition.

Proo/. (la) Let p G P$, C* as in 1.12(4), and let q >p decide the value of C*,say q \\-p "C* = ". Then q\( + 1) GP+i is essentially stronger than p.(Ib) Not hard.

(2) Easy, since we take direct limits on the stationary set S" { < X :cf()-Ki}, by


11/57

542 XL Changing Cofmalities; Equi-Consistency Results

2. The ConditionIn this section we get to the heart of this chapter, the definition of our conditionfor forcing notions. W e need some preliminary definitions.

2.1 Definition. A tagged tree (or an ideal tagged tree) is a pair (, I) suchthat:(A) T is a tree i.e.,a nonempty set of finite sequences of ordinals such that if

GT then any initial segment of belongs to T; here with no maximalnodes if not said otherwise. T is partially ordered by initial segments,i.e.,7 7 < v iff 7 7 is an initial segment of v.

(B) I is a function with domain including T such that for every G T : 1 ( 7 7 )( = 1^) is an ideal of subsets of some set called the domain of 1^, andSuc(r?) = {y : v is an immediate successor of in } C D o m ( l r7 ).

(C ) For every G T we have Suc(^) 0 and above each G there is somev G T such that Suc(^) $ . \v.

2.1A Convention. For any tagged tree (T, I) we can define |t,|t {{: 7 7 " () GA} : A G 1^}; we sometimes, in an abuse of notation, donot distinguish between I and |t; e.g. if |t is constantly /*, we write /* insteadof I . Sometimes we also write Suc() for {a : ~ (a) e T}.2.2 Definition, will be called a splitting point of (T, I) if Suc() $ \ (justlike v in (3) above). Let sp(, I) be the set of splitting points of (, I).

W e call (T, I) normal if G \ sp(T, I ) => |Suc(?)| = 1 (we may forgetto demand this).

2.3 Definition. We now define orders between tagged trees:(a) (T2,I2) < ( T , l ) if TI C T2 and whenever G TI is a splitting point

of T I then Suc T l(ry) \z() and li^JfSucT^) = l2(^)SucTl(^) (whereI\A = {B : B C A and B G /}) and om(I\A) = A.


12/57

2. The Condition 543

(b) (2,I2) N 0 , and A = 0 if |Suc()|


13/57


14/57

2. The Condition 545

The proof of (2) uses th e fact that if: / : T > P, then th e existence of abranch in { GT : f ( ) GG

p} is absolute between the universes Vp and VQ.

2.6B Convention.(1) In Definition 2.6, the value F gives to Suc() is w.l.o.g. { () : < }

for some , and we do not strictly distinguish between and Suc().(2) The domain of F consists of triples of the form ( 7 7 , it;, /), where is a finite

sequence of ordinals, w C Dom(?7), and / is an increasing function from{\k : k < ig()} into P. The value F(,w,f) has two components: Thefirst is of the form { ~ (i) i G ^4} for some set A of ordinals (b y (1),without loss of generality A = \A\) and the second component is a familyof elements of P above /(?/), indexed by the first component.When we define such a function F, we usually call the first component

"Suc(^/)" (here "T" is just a label, not an actual variable), and we write thesecond component as f\Suc() or (/(^) : v GSuc(?7)} (i.e. we use the samevariable "/" that appears in the input of F) .

2.7 Definition. For a set I of ideals we define similarly when does a forcingnotion P satisfies the I-condition (the only difference is dealing with I-treesinstead of ^-trees), so now

{Suc (ry), !, { /( i /) : v G Suc (7)))-F(,w[n], f\{v : v < })and Suc() = D om ( lT 7). We allow ourselves to omit Suc(^) when it is wellunderstood. (We can let the function depend on \v(y < 7 7 ) too).

2.7A Remark. If / is restriction closed (i.e. I el, A C D o m (/) , A I thenI\A G I at least for some B C A, B $ . /+ and J G I we have I\B = J) then wecan weaken the demand to

S\c() C Dom(l,), Sucr(?) /,)


15/57


3. The Preservation PropertiesGuaranteed by the 5-Condition3.1 Definition.(1) An ideal / is -complete if any union of less than members of / is still a

member of /.(2) A tagged tree (T, I) is -complete if for each e T the ideal 1^ is -

complete.(3) A family I of ideals is -complete if each / I is -complete.

3.2 Theorem. (CH) If P is a forcing notion satisfying the I-condition for an^-complete I then forcing with P does not add reals.

As an immediate conclusion we get:3.3 Theorem. (CH) If P is a forcing notion satisfying the 5-condition for aset S of regular cardinals greater than N I then forcing with P does not addreals.

The main tool for the proof of the theorem is the combinatorial Lemma 3.5from [RuSh:117], fo r which we need a preliminary definition. More on suchtheorems and history see Rubin and Shelah [RuSh ll?].3.4 Definition. We define a topology on H mT (for any tree T) by definingfor each G T the set T [ ] = {y :


16/57

3. The Preservation Properties Guaranteed by the S-Condition 547

2) In part (1) we can let H be multivalued, i.e. assume lim() is \J Ha, each


17/57


first n moves of player I in a play in which he plays according to Ha^}. Nowfor GT

1" sp(), let A = Suct(r/). Then A I, otherwise player II couldhave played it as An. So


18/57

3. The Preservation Properties Guaranteed by the S'-Condition 54 9

Remark. 1) Note that this is stronger than 3.3, as we do not assume C.H. andthat we allow K

0G S. The proof is quite similar to the proof of Theorem 3.2

but here we use a somewhat different combinatorial lemma. Note also that weshall not use this theorem;2) We can generalize 3.2 and 3.6 to I-condition when I is 2-complete, striaght-forwardly, see3.8.

3.7 Lemma. Let (T, I) be such that for some regular uncountable , for every G T either 1 ^ is +-complete or |Suc(^)| < , then for every H : T > satisfying {77 G lim T : H() < a] is a Borel subset of lim T for any successor < , there is < and (T', I), (T, I)


19/57


W e define T" just like we did in the proof of Lemma 3.5, collecting all theinitial segments of plays of player I in the game )Q* when he plays accordingto his winning strategy Ha*. Da.Proof of Theorem 3.6. Just like in the proof of 3.2, having a name of a functionin V[P] mapping into \ we take an S-tree T, define a function h : T > Pusing the F in odd stages and in even stages forcing more and more values forr. Using 3.7 with = NI , we get a condition p GP forcing the function that names to be bounded below i, so we are done. Ds.eSimilarly we can prove:

3.8 Theorem. If P satisfies the I-condition, and is regular uncountable and(V / I) [| \JI\ < or / is +-complete] then lh P "cf() > N 0" If (V/ e I) [/ is+-complete] and = H then P adds no new -sequence from .

3.8 Warning. The statement "in VP,Q satisfies the 5-condition" may beinterpreted as "in VP,Q satisfies the I-condition" in two ways:(a) I-{{A e Vp : A C \,V P \ = \A\< X} : X G 5}(b) I-{{A G V : A C , V N \A\ < } : G 5}

Note that / I is identified with the ideal it generates.However the two interpretations are equivalent if P satisfies the -chaincondition (or is +-complete) for each G 5 (and even weaker conditions) [andthis will be the case in all our applications.]

4. Forcing Notions Satisfying the 5-Condition4.1 Definition. Namba forcing Nm is the set {T : T is a tagged {N2}-tree,such that fo r every G T, for some z / , < v G T and |Suc(^)| = ^2} with th eorder T


20/57

4. Forcing Notions Satisfying the ^-Condition 551

write h or hNm for the generic branch added by Nm, i.e. l h " : > %n, andfor every p in the generic filter, t(p) = trunk(p) C /", where the trunk of Tis G T of maximal length such that v T & ^g(^) < ^g(r?) = i / |Suc^(?7)| {1, ^2}-For / an ideal of ^ N m ( / ) is defined similarly (notused in this chapter).4.1A Claim. Nm changes the cofinality of H 2 to N 0 (/^vm exemplifies this).

Remark. In X 4.4 (4) the variant of N am b a forcing N m ' is the set of all trees ofheight such that each tree has a node, the trunk such that below its level thetree-order is linear and above it each point has ^2 many immediate successors,the order is inversed inclusion. Namba introduces Nm in [Nm]. Both forcingnotions add a cofinal -sequence to 2 (Nm by 4.1A, N m ' by X 4.7(2)) withoutcollapsing K I , and (if CH holds) neither of them adds reals (Nm by 4.4,N m 'by 4.7(1), (3)), but they are not the same.

4.2 Claim. (Magidor and Shelah). Assume CH. If h[h'\ is a Namba sequence[Namba'-sequence] then in V[h] we cannot find a Namba'-sequence over F, norcan we in V[h'\ find a Namba-sequence over V.Proof. Trivially we can in Nm and N m ' restrict ourselves to conditions whichare trees consisting of strictly increasing finite sequences of ordinals. First welook in V[ft'], let h' be the Nm'-name of the Namba' sequence, and let / be aNm'-name of an increasing function from to ^ Let T G Nm ' and suppose \\-Nm . "Sup Rang(/)-^". Now it is easy to find1 in Nm', 1 > , suchthat for each m,n < the truth values of "f(n) = '(m)", and "/(n) < '(m)"are determined by 1 (i.e., forced), (possible by X 4.7(1), as forcing by N m 'does not add reals.)Let A = {k < : for some m, T1 forces that for every i < k wehave f ( i ) < h'(rn) < /(&)}, so A is an infinite subset of


21/57


(*) T1 I h N m ' "fr every F G V, an increasing function f rom < 2 to ;2, thereis 4< such that for every > 4+ 3, /(**) > F(/(fc^_

2))".

Why? This is because for every F GV (as above, without loss of generalitystrictly increasing) and T2 >1 (in Nm') if I Q is the length of the trunk of T2then for some T3 >2;

(**) T3 lh Nm, " if I > 4> + 3 then f(k) > F(/(fo_2))"Why? Simply choose 3-{ G2 : if g(j) > + 1 > 4then j() >

F((l - 1))}, clearly 3 G Nn T3 >2 and3 satisfies (**).So (*) holds, but it exemplifies / is not a Nm-sequence, i.e., T1 I l- N m/ "/

is not a Namba-sequence". [Why? Because l l ~ N m " for some function F G Vfrom ;2 to 2 for arbitrarily large I < we have ( f c ^ ) < F(/(/^_2))"as if T G Nm and for simplicity each G T is strictly increasing we letF : < 2 > 2 be such that F() = min{5 : if G T > then for somei/ , < 3 z / G >5}, and let -{ G : if t < g(?) and ?|t Gsp() and\{m a)(3p G fc7)(/> G 2)}. So F is


22/57

4. Forcing Notions Satisfying the 5-Condition 55 3

nondecreasing and f] I h "/(n/ai) < 77(^+1) < F((2i-)) and < 5 there is always some such that a < < and 7 7 {/?) T1'. Now for some G we pick 5 G C (72 W such that^ < and construct an


26/57


27/57


(b) For every 70 G IQ there is I\ G Ii such that \JAeIl A C (J^e/o anc*

Then P satisfies the Ii -condition.Proof. Trivial. 4.7

5. Finite Composition5.1 Theorem. Let QQ satisfies the I0-condition and let Q be a Qo-name ofa forcing notion such that the weakest condition of QQ forces it to satisfy theIi-condition. Let(a) be the first regular cardinal strictly greater than the cardinality of the

domain of each member of I0(b) be such that =\ \Q Q\(c ) assume I hg 0 "Ii is +-complete"(d ) let I be I0 U l

Then P = QQ* Q\ satisfies the I-condition.

Remark. Note that E I G V (w e will not gain much by letting Ii G VQo.)

Proof. Once again we have to define the function F and then prove it doesits work. We will need a combinatorial lemma and its proof will concludethe proof of the theorem. For f ( ) G P we denote by /(?) its Qo-part andby fl() the Qi-part (it is a Qo-name of a condition in Qi), let F be thefunction exemplifying Qo satisfies the Io-condition an d -F1 be the Qo-name ofthe function exemplifying Q\ satisfies the Ii-condition.

We divide the definition of the F to even and odd stages. In even stages i.e.,when \w\is even, we will refer to the QQpart of P and use F. More precisely, let(B, l, ( T V , : v GB) ) = F (, (f(\t) : < g ()) ) where wl = {i Gw : \(w\even}. Now let Suc() = B and for / G B, fl(v) = fl(), /(/) = rv. Inodd stages we essentially do the same for the Qi-part but we need a little


28/57

5. Finite Composition 559

modification; Fl is just a Qo-name of a function and we may not even knowthe domain of the ideal 1 ^ it give (= the Suc(^)), so first we extend f()to a condition q ' Q satisfying f() < q ' Q G Q o an d forcing a specific value fo r\ (hence for Suc()) as defined by F1, and then proceed like in the Q0 part(of course we change each f() there to fl() and so on) and we let the QQpart /(/) fo r each ' Suc() be the q ' Q we have picked (the Q part willbe defined by F l(, (fl(\ : < g()))) (if we want to allow F to have justSuc() 0 mod 1^, act as in the proof 6.2).

Before we can show that this definition works we need a definition and acombinatorial lemma.

5.2 Definition. For a subset A of T we define by induction on the lengthof , res(?,A) for each e T. Let res({),A) = {). Assume res(7,A)is already defined and we define es((),A) for all members r / ( ) ofSuc(^). I f G A then r e s O ? () , A) = res(r/, A ) ( ) and if A thenres(rf (),,4) = es(,A)~(0). Thus res(T,A) d= {es(,A) : T} isa tree obtained by projecting, i.e., gluing together all members of Suc (?7)whenever A.

5.3 Lemma. Let , be uncountable cardinals satisfying there exist T', (T, I) Qo as a function to , (remember |Q0|


29/57

560 XL Changing Cofinalities; E qui-Consistency Results

res(f, A). Let T* = {resT(r/, A) : GT"}, it is an I0-tree (since the "even"fronts of the original tree now become splitting points) and / induces afunction / f rom it to Q0 i.e., v = es(,A) implies fQ(v) = f() (by theconclusion of 5.3, / is well defined).By the definition of F for even w\'s and the assumption that F exemplify theIo-condition we can find an r0 G Qo and a Qo-name of a member of limT*such that r0 I h g 0 "for every k < we have f(\k) G GO" where GO is the )0-name for the generic subset of Qo" Let T+ d= {p G T" : res(p, A) = res(r/, A)};this is a Qo-uame of an I-tree and by the definition of F in the odd stages (i.e.Fw when \w\is odd) there are a Qo-name i famember of Q\ and a Qo * Qi-name v of an u -branch of T+ such that T O I h g 0 [r* H - Q J " v G limT~ l~ is suchthat fl(v\k) GGI for every k < "] where Q is the name of the generic setfor Q and v is forced to be a name in Qi of a member of limT+. The conditionin P = Qo * Qi which witnesses that the I-condition holds is of course (r 0,r),since (r0,ri> lh "" limT, and for all fc G;, / (z/t fc) = / 0(res( /A:), A) GG0,and/( / t fe)G. D5.We now pay our debt and prove Lemma 5.3; the proof is in the spirit of theproofs of the previous combinatorial Lemmas 3.3 and 3.5.

5.5. Proof of Lemma 5.3 Without loss of generality ~(a) G T => a K. Let {(ha,ga) : < } be a list of all the pairs (h,g)such that g is a function f rom >K to {0,1} and h is in a function from >to (by the assumption


30/57

5. Finite Com position 561

when GA, and otherwise picking as the only member of SUCT* () an elementof Suc() that is not in any of the A

a's that are defined for player II at that

stage by his winning strategy for Da (this is possible as we assume that Aimplies I is +-complete).

The map - res(?7, A) is 1-1 on T*, so there is a pair (hao,gao) in ourlist such that for each G T* we have H() = hao(es(, A)) and gaoM 0if f GA. Now we define a play in the game 3ao: player I plays choosing onlymembers of T* while II plays according to his winning strategy for 3ao, but insuch a play, player I surely wins and we get the desired contradiction.

So there exists some < X for which player II has no winning strategyin dp, but the game is determined hence player I has a winning strategy ford . Now let T1 be the tree of all sequences that can appear in a play whereplayer I used this strategy. T1 satisfies the requirements (similar to 3.5). Thisfinishes the case where is a successor.If is singular, then


31/57

562 XI. Changing Cofinalities; E qui-Consistency Results

Proof. As o = 1 clearly TO = will satisfy the condition for k = 0. We applyLemma 5.3 to T0 (with = i, = o, / = /i, A = ^40) to get a subtreeTI satisfying the condition also for k = 1. We continue by induction. In the/c-th step, we apply Lemma 5.3 to T / - with = fc +i , = & , / =

Finally, let T* = VZV Clearly * satisfies (*). Note that () G*, and if7 7 G T* A f c , then by the conclusion of lemma5.3,

so Suc *(7) = Suc Tf c(r?). Hence * is a tree and (T0, 1 ) 6

6. Preservation of the E-Condition by Iteration6.1 Definition. We say that Q = (Pi,Qi : i < ) is suitable for (Ii>j , ^ j , i}J :(i, j) GVF*) provided that the following hold:(0) W* C {{z,j) : i < j < ,i is not strongly inaccessible } and {{i 4- 1, j) :i +1


32/57

6. Preservationof the I- Condition by Iteration 563

For each i < , let Fi be a P^-name of a function witnessing that Qisatisfies th e E ^-condition. W e will act as in the proof of Theorem 5.1, but nowwe have countably many F^s rather than two.We can a priori partition thetasks, so let = \Ji and\by {~ (i) : i < } but we shall ignore such problems].

We now have to prove that P, I, and F satisfy Definition 2.6. So let (,I),Jk(k < ) and / : -> P be as in Definition 2.6 and (T, I)


33/57


6.2A Remark. Note that here as well in the next theorem we need that thelij's are well separated (compare 6.1(4), (5)) i.e. some have small underlyingsets, others have large completeness coefficients (e.g. in the previous theoremwe required that ideals 1$ are on sets C ^, and ideals in I i + had to be \i+-complete, + > ^.To satisfy these requirements we will in applications onlywork with iterations in which in every odd step some large enough cardinal iscollapsed, see l.l(ii)'.

6.3 Lemma.(1) If Q = (Pa,Qa : a < ) is suitable for {I ,/3, )/3, > / 3 : a < < ,anon-limit), and I = U{^,/3 ' O L < < \ and a non-limit} then P = P limQ satisfies the I-condition.(2 ) We can replace \ by any such that K 0 < cfv


34/57

6 . Preservation of the I-Condition by Iteration 56 5

define it as any ( I , ( f ( ~ ( ) ) : G B)) such that / ( r ? { ) ) forces a value to

N ow let \w\G JiNaturally we shall use one of the Fj jg, but we have to determine which one.

By the way we are defining F, we can assume that for k < ig(), f($k + 1))determines (i.e. forces a value to) m(f($) for I, m


35/57


that (Tt, I)


36/57

6 . Preservationof the I-Condition by Iteration 56 7

< ?P W I ". However, we have f(\(m + +l)\ [7 \ \ ~ P m(f(\^)) = 7" forsome 7, so 7 < m+ +2( ) = *(ra 4- + 2) < a*Since also p H -p # "f(\(m ++ 2)) fy G Gp# ", we conclude that

for every ,m < .So p essentially forces Wf(\t) G P* Hence clearly p lh p "f(\t) G

/nr )5&PI f we do not have CH , we modify the proof as follows: let g : > be such thatg(m) < m and (Vra)(3ra)[0(m) = n]. Next, for each < let (pf : < )list all finite sequences of the form ((ik,k) ' k < fc*}, A > = 0 , / ?& < /?&+!


37/57


{ V Un and letting w = [t < ig() : \t G \Jm 1,fc < g(r/) , let / 3 ( f c ) -Min{7 -f J k : < k = / ( r / f ) G P7}. Now we shall


38/57

6 . Preservation of the I-Condition by Iteration 56 9

use JF>(


39/57


Since we are using RCS iteration see (1.12)(1), letting * = supn/3n, we canafter many steps find a condition p G P $ * which is stronger than every pn,and a P0*-name of a branch in T*, defined by v = \Jn n.This is as required.

6.4

6.5 Lemma. Suppose Q = (Pa,Qa : a < K ) is suitable for {I > /3,,/3>,/3 :(a,) G W*), K is strongly inaccessible, \\JI\ + a, + a, < K for any(,/3) GW*, and / GI,0, Q is a P-name of a forcing notion satisfying thelU-condition and let I0 = U ,/ 3 ! cf ^> / is |cf+|-complete}

and assume:(a) for every / G I: either / is /^-complete or / is /^-complete and normaland K \ A* G/.

(b) for some /* GI and \JI* K and all singletons are in /*.Then P * Q satisfies the I-condition.Remark: In Gitik Shelah [GiSh:191], () + (b) were weakened to: each / G I is^-complete (or see XV 3).Proof. Let Faj witness "Pj/3 = P/Pa satisfies the I^-condition" and F(a P^-name) witness "Q satisfies the I^-condition" and Uij, fc,mi,j,k,m


40/57

6. Preservation of the I-Condition by Iteration 571

We let \ = /* and for v G Suc() we have f(v) > f() be such that:if i < g(), () G A*, then for some < (e)J(/)\() G P

a(possible

by 6.3(2)). We denote the minimal such by a i. Thus, < v() and

Case ii. \w \ + 2 is divisible by 3. We act as in the proof of 5.1, i.e.,we useour winning strategy for the game on Q: let w* = { Gw : \w\l\+2 is divisibleby 3}, and we let {/, (rv : v G JB = F(,w\ (f(\t)() ' i f()\, q lh p "/ = \" for some 1^, and letF(,w, (f(\t) : t


41/57


Assume for simplicity CH. So by Lemma 3.5A, we can find T1, (T^, I)


42/57

6. Preservationof the I-Condition by Iteration 57 3

(i) If G T2,g() - 1 G 2, then has cofinality > K 0 hence 7^ = U{7r/,m :7 7 i >lg(), m divisible by 3} is <

.

Now if G T 2, g ( 7 ) G 2 then the function > > 7^- is a regressivefunct ion on a subset of K which do not belong to 1^. Hence for some 7, {y : z / =r / ( ) , 7 / = 7}isnotinl^So without loss of generality

(j) If G2,^g(/) G2 then for some 7^ < K :

So w.l.o.g.(k) If 77 GVg(?) GB2, < v G T2 then (I^ | |J1^ | < 7, = ^ > 1^ =l,2,]Let Q 0 an(3 { m : 1 < 7 < ;} be an enumeration in increasing

order of {I < : i - 1 G2} For any GT2,tg() = lmwe define asSup{7i : 77


43/57

574 XL Changing Cofinalities; E qui-Consistency Results

Now look at ^ > / ( ) and p\ctQ (w.l.o.g. p\ctQ = p\). As in case (iii) ofthe definition of F we work hard enough (repeating previous proofs) there is aP/ 3 ()-name ^ an d Q Q G P(} such that :

#{) " ~ P / 3 "/o^oO Qo for every f as our strategy will often haveusedF and res(7, AO) < () [G GT2 then we can (asin 5.1 and above) find q GPv and Pv -name j]v such that: qv is compatiblewith every member of G() and

9 / H ~ / 3 0 "/i/ ^i/^) ^^ for every - , and for every r y G T2, [res(y,*)


45/57


6.8 Conclusion. We can satisfy the demands of 1.1 by "P satisfies { : ^2 < < |P|, regular }"-condition.

Concluding Remark. We could have strengthened somewhat the result 6.6,butwith no apparent application by using a larger A*.

7. Further Independence ResultsI n this section we complete some independence results.

7.1 Theorem. The following are equiconsistent.(a) ZFC + there is a Mahlo cardinal.(b) ZFC + G.C.H.+ F r + ( N 2 ) (where F r + ( N 2 ) means that every stationary

S C S$ { < ^2 ' cf () = N O } contains a closed copy of i).

Remark. Our proof will use, in addition to the ideas of the proof of Theorem1.4 also ideas of the proof of Harrington and Shelah [H rSh:99], but,fo r makingthe iteration work, we build a quite generic object rather than force it (as in[Sh:82]).

2) In b) we can also contradict G.C.H . (using XV 3 for a) = > b)).Proof. The implication b) = > a) was proved by Van Lere, using the well knownfact that if in L there is no Mahlo cardinal, then th e square principle holds fo r^2- So the point is to prove a) = b). As any Mahlo cardinal in V is a Mahlocardinal in L, we can assume V L, K a strongly inaccessible Mahlo cardinal.

We shall define a revised countable support iteration Q = (Pi, Qi : i < ),\Pi\ < N + i If * isnotastrongly inaccessible cardinal Qi is the Levy collapseof2Hl to N I by countable conditions (inVPi). If i is strongly inaccessible then Qi isP[Si] (see 4.6), Si is a P^-name of a stationary subset of 5 = { < i : c f ( J ) = K 0in F} (note \ \ - p "i = ^2" by 1.1(3)), where i will be carefully chosen as


46/57

7. Further Independence Results 57 7

described below. Note that N ^ = N ^ , K = H J f " (again we use 1.1(3)) so Pcollapse no cardinal > A C and in VP the GCH holds.

We let P K = RlimQ. In VP we define an iterated forcing Q* = (P?, Q* :i < A t " 1 " } , with support of power N I , such that (in VP) for each z , 5* is aP^-name of a subset of 50 which does not contain a closed copy of > , and:

Q* {/ G (yp*)p** : / an increasing continuous function from somea < 2 into L 2 \ S* and if a is a limit ordinal, Ui


47/57


Let (A\ : < K inaccessible) be a diamond sequence. Let N be the uniquetransitive subset of H() isomorphic to (, G ) where G = {(, ) : a < , < , ( 0 , , / 3 ) A\}, if there is one. Let g : > N be the isomorphism. Letfor i A T * and #*(^) = T V * (for i < ), Q = Q\, < 7 * ( Q * > )-Q*,S*(l)-,0V) =P,


48/57


49/57


Now we define by induction on 7 GN [G\ ] (*) +1) a set C7, a sequence(C/3,7 : < 7, G N

A[A]n

(*'

+ 1)>

and sequences (p7j : i G C7) satisfyingthe following:

(i) Each C7, and each C} is a closed unbounded subset of , C, CC^ncv

(ii) p7>i G f [G ] P7 ', and is increasing with i.(iii) pi \ = p, for < 7, i GC/3,.(iv) if (p7j : j < i), (


50/57

7. Further Independence Results 581

C/ 3 7 = C. For r < we let C 0 / > 7 = C / ? / , / ? C7. So for i GC7, we will haveP , \ = P,, so we only have to define p7)(/3). We will do this by induction oni: If i is (in C7) the successor of j, then we let p^() be a condition extendingpj() such that there is G C7 with ht(p7 j (/?)) <


51/57


A known forcing is7.3 Definition. For any set S C ^ define the forcing notion Club(S') by

Club(5) = {h : fo r some < ^ we have:Dom(/) = a + l,Rang(/) C 5, h continuous increasing}

This forcing notion "shoots a club through 5". See on it in [BHK] and more in[AbSh:146].

For h e Club(5) let(/) = max Dom(/) (h) = h(a(h))

7.4 Lemma.(a) If 5 S Q is stationary, then Club(5) does not add new -sequences of

ordinals.(b) If the set

5 d= { GSi S : 5 contains a club subset of }is stationary, and CH holds, then Club(S) does not add i-sequences ofordinals to V.

7.4A Remark. Instead CH it suffice to have for some list { : a. < ^} ofsubsetes of < 2 that { e Sf : there is a club C of such that CCS and < = > C G {d : < a}.Proof: We leave (a) to the reader as it is easier and we will need only (b).To prove (b),le t p G Club(5), a Club(5)-name such that p l h " r is a functionfrom \ to the ordinals". Let N -< (#(N3), e) be a model of size N I which con-tains all relevant information (i.e., {5,p,r} C T V ) , is closed under -sequencesand satisfies N < 2 G 5. We can find such a model because we have C H and5 is stationary. Let C C 5 be a club set.


52/57

7. Further Independence Results 583

N ow we can find a continuous increasing sequence (Na : a


53/57


Con(a)^Con(b):So without loss of generality V = L, K is 2-Mahlo.We define an RCS-iteration Q (Pi, Qi : i < ), where:l)if i is not strongly inaccessible, Qi is Levy collapse of 2Hl to H I by countableconditions.2)if i is strongly inaccessible but not Mahlo, Qi Nm'.3)if i is a strongly inaccessible Mahlo cardinal, Qi P[Si] where Si = { < i : strongly inaccessible }.

It should be easy for the reader to prove that P RlimQ satisfies theft-chain condition, and the S-condition, 5 = { : H I < < K, inaccessible (inL)}.Lastly in Vp* let P* =Club({ < K : X inaccessible in L}). So our forcing isP * P* V.Now P* is not even HI-complete, but still P was constructed so that P* doesnot add Hi-sequences by 7.4(b), and Vp**p* is as required. D7 H I , Fr^() holds (and we can ask also GCH).Proof. W.l.o.g. V \ = GCH. Let K will be the first strongly inaccessible cardinalft which is ft+7-supercompact. Let j : ft > H(), j() G f(||+6) be aLaver diamond under this restriction. Let (P~,Q~ : i < ,j < K) be anEaston support iteration. Q~ is j0(j) if j(j) = ( j t ( j ) : ^ < 2} and J0(j)is a P^-name of a j-directed complete forcing notion, j strong inaccessible,(V < j)(|Pcl < j), and the trivial forcing otherwise. Let Vb = V, V\ = VQK .Clearly V 1 = 0{ * we have:(*)(9 if 5 C {5 < : cf(5) = H O} is stationary then for some 5* < we have:

cf ((5*) = K, and S 5* is a stationary subset of 5*. (seeFact in X 7.4)In V2 = VR we have *;* = + and define Q, P, Q*-( ,9*: i < *;+),

P*+ as in the proof of 7.1 except that for < K strong inaccessible, Q issuggested by j(5) when j() = (j^() : < 2} as above, j() a (P1* P5)-


54/57

7. Further I ndependence Results 58 5

name of a forcing notion satisfying the 5-condition (in the universe VP+l X Pofcourse). So we force with R' = R*P

K*P*

+(really we can arrange that P

G V ,

P* is an .R x P^-name). Looking at the proof of 7.1, the only point left is toprove (*)0 for = cf(0) > ft*. If > ft*, as the density of P * P*+ (e F2) isft*, any stationary 5 C { < : cf() = N O } from V / contains a stationarysubset from V^. We can use V % N (*)0, so we are left with th e case 0 ft*.

I f in V2, 5 is a P * P* + name, p e P * P*^, p I h "50 C {5 < ft+ : cf() =stationary", let

Sl = { < ft+ : c f ( < f )-N O and p I/ "5 SQn}.For J GSi choose p5 GP * P*+, P p$ GP * P*} is a club of ft+in V2 It is enough to show that0 ty =f {5 < ft+ : cf (5) = ft,e 7 and p l l-p K *p j "^2 5 (which is a

(P * PJ* )-name) is a stationary subset of 5" } is a stationary subset of ft"1".[Why? As then instead of guaranting 52 will continue to be stationary, weguess such name and related elementary submodel in some < ft and in Qatake care of 52 having a closed subset of order type \ .]Let GR C .R be the generic subset of R.

In Vi we can find < ft* such that V\ N "ft < < ft*, is stronglyinaccessible" and letting Rg = Levy (ft, < ) R, GRS = GR R, in ^[G#Jwe have Q, Q*\ hence P * P , (pi : i e 5 ) and S is stationary(and of course V [ G R S] t = 5 = ft+). Also in Vi[G#6], the forcing notion P * P*satisfies the -c.c. (just as in F[G/?], P * P* satisfies the ft*-c.c.). So for aclub of i < 5, i G S J implies p Ih "5* d= {j e Si 5 : P J e GP*p*} isa stationary subset of 5*" (as in Gitik, Shelah [GiSh:310]). Choose such i(*).I f this holds in V i[ G #] too, then we are done so assume towards contradictionthat this fails, so moving back to V Q for some q e P ", r G R and p we have

l h p * " i *


55/57


we have(

Let M x ( # ( ( * ) + + ) , ) be such that ||M|| = + , tf(+ ) C M, x ={,*,,^,9,(r)p),/G,Plt * P;} C M and *M C M .

Let (M1 ,x') be isomorphic say by g to (M, x), M transitive, so g : M >M1. We can find 23 - < (#(;+6),G) to which x' and M belong, such thatletting = 03 K we have: 0 is strongly inaccessible fl+^-supercompact, 2S ~(#(0+6), G)and j(0)-(jo(0), j(0)) #(0++) issuch that: J0(0)-g(Rs) andj((9) is the g(P~ *5* P/)-name i.e. (P * Q^ * P5)-name of g(P) * club(5*).We can finish easily. D7>5

8. Relativising to a Stationary Set8.1 Definition. For a set I of ideals and stationary W C \ we define whendoes a forcing notion P satisfies the (I, W)-condition (compare with 2.6). Itmeans that there is a function F such that (letting J =the bounded subsetsof ):

, I), / satisfies the following properties:

(*) (a) (T, O i s a n l(b) f .T P.(c ) v < implies P |= /(z/) < f ( ) .(d) There are fronts Jn(n < cj) of T such that every member of Jn+ has a

proper initial segment of Jn and:(a) If GJn then (Suc(y), I,,, (/( i /) : i / G Sucr(/)-F(, w[] , { /( i /) :

/ / < 7 7 ) ) (where y [ r y ] = {:\ UnJn}).

(/?) U Jn is the set of splitting points of (, I)(7) If n is odd, GJn then \ = J^f.( ) If n is even, 7 7 G Jn then 1^ G I .


56/57


57/57


(a)(, I) < (T, I)(b)/7 G T5, r? G Un Jn^t sp(, I ).(c)r/ G limT5 = > > 5 = sup(Rang(ry) )& ^Rang(?7).This last statement holds also for branches of T $ in extensions of the universe,being absolute. Choose G VFC*, an d apply Definition 8.1 to (T$, I ) (standingfor 1" there), and get q as there. N ow q I h "{/(/ |^) : < } C GP, r y G limT"(for some P-name 7 7 ) . I n particular q I h "p G Gp" (a s p^ = p) so w.l.o.g. p < q.Also by (ii) (), (iii) above f lhp "sup(Rang(ry) i) = sup { r?r(2n+i) : n

saharon shelah- proper and improper forcing second edition: chapter xi: changing cofinalities;...

Documents