Paul Larson and Saharon Shelah- Bounding by canonical functions, with CH

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Bounding by canonical functions, with CH

Paul Larson Saharon Shelah

October 6, 2003

Abstract

We show that the members of a certain class of semi-proper iterations

do not add countable sets of ordinals. As a result, starting from suitablelarge cardinals one can obtain a model in which the Continuum Hypothesis

holds and every function from 1 to 1 is bounded on a club by a canonical

function for an ordinal less than 2.

1 Introduction

Given an ordinal , a function f: 1 Ord is a canonical function for if theempty condition (i.e., 1) in the forcing P(1)/NS1 forces that j(f)(

V1 ) = ,

where j is the elementary embedding induced by the generic. For each < 1,the constant function with value is the canonical function for . For [1, 2), a canonical function f for is obtained by taking a bijection g : 1

and letting f() be the ordertype of g[]. In this paper we let Bounding denotethe statement that every function from 1 to 1 is bounded on a club subsetof 1 by a canonical function for an ordinal less than 2. It is fairly easy tosee that if the nonstationary ideal on 1 (N S1) is saturated, then Boundingholds. The second author has shown [11] that given the existence of a Woodincardinal there is a semi-proper forcing making N S1 saturated, and it has beenknown for some time that there is a simpler forcing making Bounding hold froma weaker large cardinal hypothesis. The most quotable result in this paper isthat this standard forcing to make every function from 1 to 1 bounded bya canonical function is (, )-distributive (i.e., it does not add -sequences ofordinals), and so this statement is consistent with the Continuum Hypothesis,even in the presence of large cardinals. This is in contrast with saturation, as

Woodin [14] has shown that ifN S1 is saturated and sufficiently large cardinalsexist, then there is a definable counterexample to CH. We give a more general

This research was conducted while both authors were in residence at the Mittag-Leffler

Institute. We thank the Institute for its hospitality.MSC 2000: 03E35, 03E50, 03E55. Keywords: Iterated Forcing, Canonical Functions,

Continuum Hypothesis.The research of the second author was supported by the Israel Science Foundation,

founded by the Israel Academy of Sciences. Publication number 746.

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theorem stating that the members of a certain class of semi-proper iterations are

(, )-distributive. This class includes the standard forcing to make Boundinghold, and is general enough to show that a generalization of Bounding for certainsets of reals is also consistent with CH, answering a question in [14].

The key construction used in the proof generalizes the notion of-propernessfrom Chapter V of [11] to semi-proper forcing. Briefly, a forcing is -semi-properif for any -chain of countable elementary submodels of length , there is acondition which is simultaneously semi-generic for each model in the sequence.The problem in applying the method to show that a given improper iteration is(, )-distributive is that for a given model N in the sequence, N[G] canbe a proper superset of N , where is the length of the iteration and G isgeneric for some initial segment, so that new steps appear. For the forcings inthis paper, however, we have a good understanding of how to enlarge each suchN, as well as how to produce the appropriate tower of models to overcome this.

This can be generalized further, getting the consistency of certain forcingaxioms, using ideas from [11], Chapters V and VIII, and [13]. The reader isreferred to [10] for more on this topic and on RCS in particular.

Interest in this question derives also from the study of Woodins Pmax forcing[14], which produces a model in which CH fails and all forceable 2 sentencesfor H(2) hold simultaneously. It is not known whether all such 2 sentencesforceably consistent with CH can hold together with CH. The generalized formof bounding in this paper is a candidate for showing that this is impossible.Candidates for the other half of the incompatibility appear in [12], and Woodinhas suggested others concerning models of determinacy.

2 Skolem Hulls

Given a structure M with a predicate


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Lemma 2.1. Say that M is a structure and


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(c) for all < , 1PQ = Q,f

.

The stationary sets in our iterations must also satisfy certain extendibilityconditions with respect to the countable elementary submodels of a sufficientlylarge initial segment of the universe.

Theorem 3.2. Let : < be a continuous increasing sequence of stronglimit cardinals with supremum . Fix a regular cardinal > (2)+, and let


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Since in the end we show that P does not add -sequences, our iteration

is actually by Countable Support. Nonetheless, at this time we do not have aproof of (the corresponding version of) Theorem 3.2 which avoids RCS.For the rest of this paper, sets denoted by , : < , ,


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Theorem 4.2. Suppose that P , Q : < is an RCS iteration with RCS

limit P such that each Q is forced to be semi-proper and each Q forces thecorresponding P to have cardinality 1. Then the following hold.

1. P is semi-proper.

2. For all regular cardinals > 2|P|, for all countable X H() withP X, for all < in X ( + 1), if

(a) p P,

(b) q is an (X, P)-semi-generic condition in P below p,

(c) q forces there exists r in P X such that p[, ) = r[, ),

then there exists an (X, P)-semi-generic condition q P such thatq = q and q p.

In [2], RCS is characterized by the following property.

4.3 Property. IfP : < is an RCS iteration with RCS limit P and p P,then for all q p in P there exist < , r P such that r q and eitherrcof() = or r (, ) p() = 1Q

.

IfP : < is an RCS iteration with RCS limit P then for each conditionp in P there is an associated P-name supp(p) for the support of p, the set of < such that

p() = 1Q

,

as decided by the P-generic filter. One can easily prove by induction on thatProperty 4.3 implies that for each p in P supp(p) is forced to be countable.

4.4 Remark. Property 4.3 is shown in [7] for the version of RCS in [7], for thespecial case where = 1 (so the first possibility for r cannot hold), en route to(essentially) proving Theorem 4.6. Together these two facts give Property 4.3 forthis version of RCS, since in the remaining case (where |P| < cof() for all |P| for all , then there is a sequence ofpairs (i, ri) (i < ) such that

the is are increasing,

each ri Pi forces a bound below on i,

for all i < j, rji = ri,

each ri pi.

Then the limit of the ris is the desired condition. To find i+1, ri+1, find amaximimal antichain A in Pi below ri such that for each a A there is aa < and an ra Pa such that

rai = a,

ra p,

ra decides i.

Then apply Theorem 4.1 to A and the function a ra to find ri+1, and leti+1 = sup{a : a A}, which must be below since cof() > |Pi |.

Similar considerations give the following (see also Lemma 36.5 of [5]). The

point is that the RCS limit of an iteration of cofinality 1 is just the direct limit,and the ordinals of cofinality 1 are stationary below cof() as below. Thenfor any maximal antichain A in P as below there is some < of cofinality 1such that A P is a maximal antichain in P . But then |A| |P |.

Theorem 4.6. Say that P , Q : < is an RCS iteration with RCS limit

P such that cof() > |P | for all < , and such that each Q forces the

corresponding P to have cardinality 1. Then P is cof()-c.c.

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Applying Theorems 4.2 and 4.6, we have reduced the proof of Theorem 3.2

to showing that each P ( < ) is (, )-distributive.Theorem 4.7. LetP , Q

: < be an RCS iteration of strongly inaccessible

length with RCS limit P such that

each 1P forces the corresponding Q to be semi-proper,

each Q makes the corresponding P have cardinality 1,

each |P| < ,

each P is (, )-distributive.

Then for all < , P/P is semi-proper and -c.c. Therefore P is(, )-distributive.

4.2 Semi-generics for sequences

The following are generalizations of ideas from Chapters V, X and XII of [11].

4.8 Definition. Let be a countable ordinal.

1. The set SE Q() consists of all N = N : < such that

(a) each N is a countable elementary substructure of (H(), ,


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2. 1PN[G

P] SE Q()

V[G

P].

Lemma 4.10. Let be a countable ordinal. Say that P is the RCS limit ofan RCS iteration P, Q

: < in a in a set forcing extension V such that

each 1P forces the corresponding Q to be semi-proper, and each Q

forces P

to have cardinality 1. Then the following hold in V.

1. P is -semi-proper.

2. Fix , and fix N SE Q() with P and ,

N. Letp P, q P and < be such that

N,

p is (N, P)-semi-generic,

q is (

N[, ), P)-semi-generic, p q,

q forces that for some r P N, p[, ) = r[, ).

Let [, ) be such that N. Then there exist q P andp

Psuch that

p is (N, P)-semi-generic,

q is (N[, ), P)-semi-generic,

q p,

p p,

q = q,

q forces that for some r P N, p[, ) = r[, ).

Proof. We prove part 2 by induction on , working in V. The first part followsimmediately. We fix the notation that G is the generic filter for P, for < .The case = 1 is given by Theorem 4.2. For the case where = + 1, thereare two subcases. If < , then we may assume by the induction hypothesisthat = and that there is a (N[, ), P)-semi-generic q

P such thatq = q and q p. Then since q forces that N [G ] will be elementary inH()V

[G], where G is the generic filter for P, we can replace q with a q

with the additional property that q forces that q[, ) will be equal to r[, )for some r N P. By part 2 of Theorem 4.2, then, there is a q P suchthat q = q, q q and q is (N , P)-semi-generic. Such a q suffices.

For the subcase = , by the induction hypothesis there is a (N, P)-semi-generic q P such that q = q and q p. Then since q forces thatN[G] will be elementary in H()

V[G], where G is the generic filter forP, we can replace q

with a q with the additional property that q forces thatq[, ) will be equal to r[, ) for some r N P, and we can let p = q.Then as in the previous paragraph, by part 2 of Theorem 4.2 there is a q asdesired.

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The case where is a limit and = +1 is similar, now fixing and induct-

ing on . By the induction hypothesis for we may assume that =

and bythe induction hypothesis for we may assume that there is a (N[, ), P)-semi-generic q P such that q = q and q p. Since Q

is forced to

be -semi-proper, by Theorem 4.1 (or part 1 of Lemma 4.9) there is a ( N , P)-semi-generic q below p such that q = q.

For the case where and are both limits there are two subcases (in either ofwhich we may assume that = ). If has cofinality or 1, or if|P| < cof()for each < , then we fix an increasing sequence i : i < cofinal in , with0 = . If the cofinality of is countable, fix an increasing sequence of ordinalsi : i < N0 cofinal in , with 0 = . Otherwise, let i be any ordinalin Ni greater than sup(

Ni ), again with 0 = . Let p0 = p and let

q0 = q. Alternately choose conditions pi+1, qi+1 (i < ), such that

1. each pi+1 is a condition in P,

2. each qi+1 is a condition in Pi+1 ,

3. each qi pii,

4. each pi+1 pi,

5. for all i < j < , qji = qi,

6. each pi+1 is (Ni+1, P)-semi-generic,

7. each qi+1 is (N[i+1, ), Pi+1)-semi-generic,

8. each qi forces that for some condition r P Ni+1 , r[i, ) = pi[i, ).

For the case where no condition in any P makes cof() 1, we modifycondition 8 as follows:

8a. each qi forces that for some condition r P Ni+1 such that for some < 1Psupp(r) , r[i, ) = pi[i, ).

That such conditions exist is immediate by the induction hypothesis (8afollows from Property 4.3). Then the limit of the qis (call it q

) will be thedesired (N , P)-semi-generic, as long as it is below eachpi. This fact follows fromthe fact that q forces that {i : i < } will be cofinal in

{supp(pi) : i < }.

This is clear if cof() = , and ifcof() = 1 it follows from the fact that eachNi will be cofinal in Ni [Gi ] since qi is (Ni , Pi)-semi-generic. For theremaining case it follows from condition 8a.

If |P| cof() for some < , then we may assume that > and socof() 1 in the P-extension. Then we may apply the previous argument inthe P-extension, along with Theorem 4.1.

Theorem 4.11. Let be a countable ordinal. Say that P is the RCS limit of anRCS iteration P , Q

: < such that each P+1 = P Q

forces |P| 1

and each 1P forces that Q

is -semi-proper. Then for all < , P/Pis -semi-proper for every < 1.

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To show that each Q,f is -semi-proper for all < 1, we show that we

can extend sequences of models in a suitable way.Lemma 4.12. Assume that < , Q, and A : < 1 []


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Let h : z be a bijection, and fix an increasing sequence i : i < cofinal

in with 0 = 0. Now let each

N = Sk(H(),,


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4. q P and q is (N[, ), P)-semi-generic,

5. for all M, q is (No.t.(M), P)-semi-generic,

6. for all M ,

(a) for all No.t(M) 1, M +1 A ,

(b) M V+2 No.t.(M),

7. p P, p q and for all [, ) M, p No.t.(M).

4.15 Remark. Note that if = + 1 and (M, N , p , q) is a (Q,,)-system,then (M, N , p , q) is also a (Q,,)-system, as (the second part of) Condition 4becomes vacuous in the second case, and Condition 4 in the first case completesCondition 5 in the second.

4.16 Definition. Let M (H(), ,


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(and |P | 2 ), the M-genericity of g can be verified in No.t.(M) (M itself

is not in any member of

N, so this is not automatic).The base case = 0 is trivial. For the successor step, we may assume that+ 1 = Then since for the successor case there is no restriction on the lastcoordinate of the M-generic filter (other than M-genericity), g can be extendedin any fashion to an M-generic for P.

For the limit, fix an increasing sequence i (i < ) cofinal in M, and letp = p0. Now in many steps alternately pick

M-generic gi Pi in No.t.(Mi) (applying the induction hypothesisplus the elementarity of No.t.(Mi) plus the fact that No.t.(M i) No.t.(Mi), as in Remark 2.2) with pii gi such that

gi P No.t.(M) for all M i,

gi Pj = gj for all j < i,

pi+1 pi in M meeting the ith dense set in M for P such thatpi+1i gi(such a pi+1 exists because gi is M-generic for Pi).

Since each pi+1 M, its initial segments are automatically in the correspondingNs. Then {pi : i < } generates an M-generic filter g for P, and for eachi, g Pi = gi. Then by the induction hypothesis, g P No.t.(M) for all M , since for all i < with i , g P = gi P.

Lemma 4.18. Fix . If for al l p0 P there is a (Q,,)-system(M, N , p , q) with p0 M and p p0 a potential-M-generic, then the forcing Pis (, )-distributive.

Proof. Towards a contradiction, fix the least for which the lemma fails, andlet p0 be a condition in P forcing that P adds a new -sequence of ordinals.Let (M, N , p , q) be a (Q,,)-system with p0 M and p p0 a potential-M-generic, as given by the hypothesis of the lemma. Then there exists a P-name in M such that p0 forces that will be a new -sequence of ordinals. Letg P M be an M-generic filter witnessing that p is a potential-M-generic.We wish to see that q is below each member of g. Then we will be done, as foreach integer i there is a member ofg intersecting the antichain in P determiningthe ith member of . So we will show by induction on ( + 1) M thatq p for all p g (simultaneously). The cases where = 0 or is a limitand M is cofinal in are clear.

For the successor step from to + 1, f(M 1) is a P-name in No.t.(M)

for a countable ordinal, so by Conditions 5 and 6 of Definition 4.14, q forcesthatM +1 A

f

(M1).

Then q forces that p() = g()(g()). By Condition 7 of Definition 4.14,q( + 1) p( +1), so q forces that q() extends p(). Now fix p g. Sinceq forces that p() extends p(), we have that q( + 1) p( + 1).

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Lastly, for the case where is a limit and M is not cofinal in , note

that has uncountable cofinality, and also that we have shown that each P

, < is (, )-distributive. By Property 4.3 then, densely many conditionsin P (and therefore g P) are conditions in some P , < , and so denselymany conditions in g P are in some P with

M.

Given a countable X (H(), ,


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Given M, by applying Condition 3 in the statement of Theorem 3.2 repeat-

edly, once for each A

with = N 1, we can choose each M+1 to meetConditions 7 and 8. While the model Y resulting from this repeated application

may not be a minimal (22

, +1)-extension of M,

Y = Sk(H(),,

(so in this case = + 1). If is a limit ordinal, we have from the fact thatM =

{M : < } and the induction hypothesis that : < lists the

first ordinals of M in increasing order. Then the definition of finishes theproof of the claim.

Next we claim that = for some < 1, and so < 1. Given thiswe are done, letting M = M and N = N : < . All the conditions ofDefinition 4.14 are satisfied trivially, aside from Condition 6. Condition 6 issatisfied since for each M , there is some < such that = (andso = o.t.(M )). Then N and M were chosen to satisfy Condition 6 ofDefinition 4.14 by Conditions 6 and 7 of the construction, and this relationshipwas preserved for all later M by Condition 8.

Assume to the contrary that = 1. If 1 = , then since every ordinal in

M1 is equal to some , there is a limit ordinal < 1 such that

{ : < } = M .

But then = , contradicting = 1. So we may assume that 1 < . Wewill show that the cofinality of M1 1 is countable, which is a contradictionsince : < 1 is increasing and cofinal in it. If 1 for some with1 M, then M 1 = M 1 , which is countable. If not, 1 = 1 .If 1 is singular, let be the cofinality of 1 . Then < for some with {, 1} M, and there exists a cofinal map f: 1 in M. Since M1 = M, f[M ] is a countable set cofinal in M1 1 . Thelast remaining case is that 1 is a regular limit cardinal. Let be least with1 M, and fix a cofinal sequence in M1 . By Lemma 4.19 and Condition

8 of the construction, this sequence is cofinal in M1 1.

We finish by applying the following lemma to the case = 0, = , fromLemma 4.20. By Lemma 4.18, then, we are done.

Lemma 4.21. Fix . Let(M, N , p , q) be a (Q,,)-system with inM, and assume thatp is a potential-M-generic. Then there exists a conditionq P such that

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1. q = q,

2. (M, N , p , q) is a (Q , , )-system.

Proof. We first note that the lemma follows from the restricted version of thelemma where = . To see this, fix M, N , p , q , , , as given by the hypothesisof the unrestricted version, and note that the restricted version then gives aq P such that q = q and (M, No.t.(M), p, q) is a (Q , , )-system.Now, if G P is V-generic with q G, then M[G] Ord = M Ord (sincep is a potential-M-generic and q p) and the following hold.

< o.t.( M) N[G] 1 = N 1,

N[G] = N[G] : < o.t.(M ) SE Qo.t.(M)()V[G].

Furthermore, in V[G

],

q[, ) P/P

(\ ) M, q[, ) is (No.t.(M)[G], P/P)-semi-generic,

p[, ) q[, ).

Applying the elementarity of No.t.(M)[G] in H()V[G], we see that there

exists a condition q No.t.(M)[G] P/P satisfying these conditions. Bypart 2 of Lemma 4.10, then, there is a condition q P such that q = q andq forces that q[, ) is a

(N[G][o.t.(M ), o.t.(M )), P/P)-semi-generic

condition extending such a q. This q suffices. We have Conditions 5 and 7 ofDefinition 4.14 by the properites listed above for q[, ), Condition 4 by ourextension, and the others by the assumptions of the lemma.

The restricted version of the lemma follows by induction on o.t.((\ ) M).Note that our induction hypothesis entitles us (once we have fixed and ) toassume that the unrestricted version holds whenever < .

Now, when = there is nothing to show. The argument for increasingo.t.((\ ) M) by one follows from the case where = + 1, and this case isalso trivial (see Remark 4.15).

The only remaining case then is when < and = is a limit. Fix anincreasing sequence i (i < ) cofinal in Mwith 0 = , and succesively applythe induction hypothesis (in V, as opposed to some submodel). That is, let q0 =

q, and given qi, let qi+1 Pi+1 be such that qi+1i = qi and (M, N , p , qi+1) isa (Q, i+1, )-system. Then each qi is (N[o.t.(M i),o.t.(M )), Pi)-semi-generic, and the limit of the qis is the desired q

. Condition 4 of Definition 4.14is then easily satisfied (the second half being vacuous), and Conditions 5 and 7,being local properties, are satisfied by the induction hypothesis.

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5 Applications

5.1 Bounding

The following is a a minor modification of a standard fact.

Theorem 5.1. Let be a measurable cardinal, let be a regular cardinal suchthat V+2 H(), and let (2

|P|

)

+

. Let :


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(H(), ,


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[8] T. Miyamoto, Preserving a Stronger Form of Semiproperness under the

Simple Iteration, in preparation[9] Y. Moschovakis, Descriptive Set Theory, Studies in Logic and the Founda-

tions of Mathematics, 100, North-Holland Publishing Co., Amsterdam-NewYork, 1980

[10] S. Shelah, A more general iterable condition ensuring 1 is not collapsed,II, Publication number 311, in preparation

[11] S. Shelah, Proper and improper forcing, Perspectives in MathematicalLogic, Springer-Verlag, Berlin, 1998

[12] S. Shelah, More on Weak Diamond, Publication number 638Available at http://front.math.ucdavis.edu/math.LO/9807180

[13] S. Shelah, NNR revisited, Publication number 656Available at http://front.math.ucdavis.edu/math.LO/0003115

[14] W.H. Woodin, The axiom of determinacy, forcing axioms, and the non-stationary ideal, DeGruyter Series in Logic and Its Applications, vol. 1,1999

Department of MathematicsUniversity of TorontoToronto M5S 3G3Canada

[email protected] of Mathematics Department of MathematicsHebrew University Rutgers University91904 Jerusalem New Brunswick, NJ 08903Israel USA

[email protected]

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Paul Larson and Saharon Shelah- Bounding by canonical functions, with CH

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