Canonical Functions, Non-Regular Ultrafilters and Ulam's Problem on ω1

Canonical Functions, Non-Regular Ultrafilters and Ulam's Problem on ω1Author(s): Oliver Deiser and Dieter DonderSource: The Journal of Symbolic Logic, Vol. 68, No. 3 (Sep., 2003), pp. 713-739Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/4147708 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 68. Number 3, Sept. 2003

CANONICAL FUNCTIONS, NON-REGULAR ULTRAFILTERS AND ULAM'S PROBLEM ON coi

OLIVER DEISER AND DIETER DONDER

Abstract. Our main results are: THEOREM 1. Con(ZFC + "every function f : wo -* o1 is dominated by a canonical function") implies

Con(ZFC + "there exists an inaccessible limit of measurable cardinals"). [In fact equiconsistency holds.] THEOREM 3. Con(ZFC + "there exists a non-regular uniform ultrafilter on wco") implies Con(ZFC +

"there exists an inaccessible stationary limit of measurable cardinals"). THEOREM 5. Con (ZFC + "there exists an owl-sequence 3r of wol-complete uniform filters on wo s.t.

every A C wo is measurable w.r.t. a filter in 7 (Ulam property)") implies Con(ZFC + "there exists an inaccessible stationary limit of measurable cardinals").

We start with a discussion of the canonical functions and look at some combinatorial principles. Assuming the domination property of Theorem 1, we use the Ketonen diagram to show that wo2 is a limit of measurable cardinals in Jensen's core model KMO for measures of order zero. Using related arguments we show that w2 is a stationary limit of measurable cardinals in KMo, if there exists a weakly normal ultrafilter on wol. The proof yields some other results, e.g., on the consistency strength of weak* -saturated filters on ol, which are of interest in view of the classical Ulam problem.

Introduction. The phenomenon of equiconsistency in set theory is remarkable with regard both to the contents and the methods. Regarding the contents the following hypothesis is supported by a large number of instances: For every (natural, combinatorial) set theoretic proposition there exists a large cardinal axiom that is equiconsistent to the proposition (over ZFC). [We set I1 <con,, 2 for propositions '1 and T2, if "Con(ZFC + T2) implies Con(ZFC + WI )". <con

is a partial order. P1 and T2 are called equiconsistent, 1 =con' 2, if '1 <con T2 and T2 ~ Pcon T1.]

The astonishing empirical fact is that <con restricted to the large cardinal axioms is a linear order. So the large cardinal axioms give us a canonical scale to measure the strength of set theoretical propositions, giving them an unexpected linear structure with all the advantages of an equivalence relation with canonical representatives.

Regarding the methods two completely different but likewise sophisticated tech- niques are used for measuring the consistency strength of a proposition '. Given ' we look for large cardinals 01 in core models. On the other hand, given a large cardinal axiom )2 one uses iterated forcing to construct a model of ZFC that satisfies '. This yields ~1 < con on 2, and we say that "1) is a lower bound for T", etc. If one arrives at D1 - 2, then the consistency strength of YI has been determined to be D1. How to choose 41 (and (D2) is not obvious, and a proof of ,1 <,con needn't be easier provided than a proof of (' <co , Y for 01 <co,, ~

'co.n .

Received March 22, 2001; revised May 20, 2002.

? 2003, Association for Symbolic Logic 0022-4812/03/6803-0001/$3.70

713



714 OLIVER DEISER AND DIETER DONDER

A generally intelligible paper dealing with "large cardinals" and "equiconsistency" is Woodin [19]. Concerning inner models we refer to Jensen's paper [11], which provides a non-technical survey of the subject.

col is a central object in infinitary combinatorics. The question whether there can be a a-additive measure on the real numbers was asked very early in the history of set theory and was denied under (CH), i.e., for the cow case (Banach, Kuratowski, 1929 and Ulam, 1930; the proof of Banach-Kuratowski uses dominating functions on cw). "Ulam's problem" and "non-regular ultrafilters" on cl may be regarded as two different interpretations of "Can owl be almost measurable?". Ulam's problem reads: "Is there a set - of size oi consisting of cl -complete filters s.t. every subset of

oi is measurable w.r.t. 7 ?" [concerning the motivation of the conditions imposed on 9 see chapter 2]. A non-regular ultrafilter on o)l has the following "additivity" property: For every cl-sequence in U there exists an infinite subsequence with a non-empty intersection. Both propositions imply the existence of many measurable cardinals in the core model K as we shall show.

Further the ol case is of particular interest since cil is a point of discontinuity. Combinatorial principles on w2 are much stronger than their wol-variants, which has to do with condensation properties of the core model.

Some very natural combinatorial principles concern the canonical functions. In the cwl case the canonical functions are a modulo club increasing o02-sequence of functions h, : co --+ cl. They are defined by recursion, but have a surprising direct characterization [see chapter 1].

The canonical functions appear in the Ketonen diagram when one looks at the lengths of the models in it. Thus dominating properties of the canonical functions translate into properties of the diagram, which turn out to be very useful in various constructions.

Let

(D = "There exists an inaccessible limit of measurable cardinals."

We show that the consistency strength of the domination property of the canonical functions is at least D (Theorem 1). This is in fact best possible: Larson and Shelah have recently shown that the domination property and the Continuum Hypothesis hold in a forcing extension of a model in which (D holds [15].

We also discuss more generally dominating properties ofiol -complete filters (The- orems 1' and 2). The argument in the proof of Theorem 2 concerning the foundation of the ultrapower will also be used in the proofs of Theorems 4 and 5.

Regarding the existence of non-regular ultrafilters on col we get:

P = "There exists an inaccessible stationary limit of measurable cardinals."

as an improved lower bound (Theorem 3). In view of the Ulam property - more exactly: the existence of a sufficiently good

weak*- saturated filter - we again obtain T as a lower bound (Theorems 4 and 5). The first result on the strength of "there is a non-regular uniform ultrafilter on

iol", (NR) for short, is J. Ketonen's well-known result that (NR) implies that 0# exists [14]. A useful invention of Ketonen's is the diagram used to prove that result (which we call the Ketonen diagram, see chapter 3). The notion of a canonical function however was developed later.



ULAM'S PROBLEM ON ol 715

Jensen showed in [6], that (NR) >con "there is a measurable cardinal", and this was also the best known lower bound for the Ulam property. This has been proven by the second author in [4, 5].

The actual upper bound for both (NR) and (UP) is "there exist ow-many Woodin cardinals". The existence of a non-regular ultrafilter on wi and of a sequence having Ulam's property follows directly from the existence of an wol-dense ideal on col.

We use the notion of a weakly saturated filter on wol in our investigations of the Ulam property (see chapter 2), since it is not clear how to use (UP) in a more direct fashion. Taylor has shown in [18] that the Ulam property implies that a weakly saturated filter exists [Taylor uses "splitting hypothesis" for "not weakly saturated"]. We shall show (and need) a somewhat stronger implication [Lemma 2.6].

Chapter 1 and 2 are combinatorial in character. In chapter 3 we introduce the Ketonen diagram for K o02 and fix often used extension properties of the diagram. The underlying core model of chapter 4 and 5 is Jensen's core model KMO for measures of order zero [10, 21, 3, 2]. We assume that the reader is familiar with mice and iterations in the KMo-context.

The notation is standard. Concerning dual ideals and filters we write I(F) and

F(I), respectively. F+ consists of the F-positive (or F-stationary) sets, i.e., if F is on n, then

F+ = {X C nlX nY for every Ye F}

(= {X C

nI_

- x ? F}).

If I is an ideal on r, then

F(I)= {X - X I},

I+ = F(I)+ = 9() - I.

IfF is a filter on . andA c F+, we set F[A] = { Y C n| Y D AnX for aX e F}. Thus F[A] is the proper filter generated by A. If I = I(F), then IIA = I(F[A]) =

{X C n|jA n X E I} is the restriction of I to A. If M - (JE, Eca) is a premouse we set

Ecv = {XI(X, wv) E E} for wv o)a,

Kov = "the unique n such that sup(X) = r, for all X e E,,v" for E,,~ ~ 0.

If E,, 0, then v - ov. If E, 0, then E, is said to be (a partial or full

measure) on iv. Hereby E, # 0 is called a partial measure on r, if v < (tK+)M

and a full measure on ri otherwise.

Generally if a E M and a+ doesn't exist in M, we set (a+)M = M n On. A non-trivial mouse is a mouse M such that E,,(M) # 0 for some 0ov < On n M.

The existence of a non-trivial mouse is equivalent to the existence of 0#. For f 5 a let

Ml = (JE,Eop) ,Mfl = (Jf, 0), (Mll)- = Jjf.

We use <c to denote the pre-wellordering of mice and weasels (and we use <* for the ordering on functions induced by club filters, cf. 1.1.)




Thus if M, N are mice or weasels, we write:

M <c N iff there is a common iterate of M and N, which is a simple iterate of M,

M ~& N iff M <' N and N <c M iff there is a common simple iterate, M <cN iff M <c N and non(M ~c N).

?1. The canonical functions. Let n > ol be a regular cardinal, and let 9, the club filter on n, i.e.,

S= {X C Il there is a closed and unbounded Y C ,r s.t. Y C X}.

DEFINITION 1.1. For f, g E "n set

f <* g iff f < g} := {a < ti

f(a) < g(a)} e ', f -* g iff {f = g} := {a < 'if(a)

= g(a)} E c.

f <* g is defined analogously.

W is co-complete, so a rank function I| fI is defined: We let Ilf = |If ', i.e.,

IJfl = sup{llgl + l Ig <* f}. We define the canonical functions (h,lv < r+) g " as follows.

DEFINITION 1.2. For limits 2 < r+ fix a sequence (2ili < cf(A)) such that

sup{2i i < cf(2)} = 2. Now set:

ho = 0,

hv+l = h, + 1, h = supi<cf(A) hA,, if 2 limit and cf(A) < K,

hA = diagsupi< hA,, if 2 limit and cf(2) = ,

which means:

ho (a) = 0 for a E n,

hv+l(a) = h,(a)+l for a E K, hA(a) = sup{h,(a)ji < cf(A)} for a < K, if 2 limit and cf(A) < K, hA(a) = sup{hA,(a)li < a for a < K, if 2 limit and cf(2) = K.

We shall use the operations "sup" and "diagsup" (w.r.t. to fixed 2i-sequences) in the sequel also for other sequences of functions in "r.

LEMMA 1.1 (Elementary properties of the canonical functions). (a) (h, v < .+) is <*-increasing.

(b) For all v < K+ we have: (i) IIhvll > v,

(ii) hv <* f for every f E " K s.t. Ifll > v.

[thus Ihvll = v]. (c) The conditions (i) and (ii) in (b) characterize the canonical functions modulo

club, i.e., for all (glv < K+) satisfying (i) and (ii) we have hv =* gv for all v < K+.

(d) The canonical functions are uniquely determined modulo club, i.e., they do not depend modulo club on the choice of the sequences (A2i i < cf(2)).




PROOF. We show (a) by induction on v < n+. This yields (b, i). We now show (b, ii) by induction on v < n+. Now (c) follows easily, since if (i) and (ii) holds for both sequences, then we have Ilfv II* Ilgv II and Ig, 1| <* If, II, thus f, =* gv. (d) follows from (c). -1

The operations sup and diagsup are not norm-continuous on sequences, i.e., in general

Ilfll > sup{llf ,llv < A} for <*-increasing sequences (f,lv < A) , A < r. in " s.t. f = sup{fvlv < A}. Similar for diagsup if A = r.

For the canonical functions we have, however: Let F be a normal filter. Then

IhvllF = v for all v < i+. (where |II IF is defined like II 1F, replacing <* by the following relation <F on "': f <F g iff {f < g} = f{a < lf(a) < g(a)} E F.)

We note a simple lemma in this context.

LEMMA 1.2 (Mutually cofinal sequences for normal filters). Let F be a normalfil- ter on n.

Let A < n+ be a limit and let (fili < cf(A)) and (gi |i < cf(A)) <F-increasing mutually cofinal sequences in 'n, i.e.,for every i < cf (A) there is a j (i) s.t. f i F

gj(i), gi F fj(i). Let

f = supi<cf(A) fi, ifcf() # ,

diagsupi<sfi, ifcf() = .

Let g2 be defined analogously. Then fr =F gA, i.e., {a < sjfA (a) = gA(a)} E F.

PROOF. cf(A) < r: Let Ci = {a < nIfi(a) < gj(i)(a)} for i < cf(A) , C -

Fi<cf(A) Ci. Then fA(a) < gA(a) for a E C, thus f <F gg. Analogously one shows that gA <F fA.

cf(A) = n: Let Ci be defined as in the first case for i < r. Let C' be club in I s.t. j"a C a for every a E C' and set C = Ai<KCi n C'. Then C E F, since F D F. Then fA(a) = supi<a fi(a) : supi<a g(i)(a) supi<agio(a)

= gA(a) for all a E C, thus fA F gA. Analogously gA <F fA- -

For n-complete filters the lemma holds in general only if cf(A) < r. Back to the canonical functions! Every h, for v < r+ can be defined directly: LEMMA 1.3 (Non-recursive construction of the canonical functions). Let v < ,+

and (X~la < t) be a sequence in 9 (v) s.t.

(i) the sequence is C -increasing and continuous, (ii) I XvI < K,

(iii) v = Ua<K X,. We set gv(a) = o.t.(X) for a < K. Then gv =* hv, i.e., gv is "the" v-th canonical

function. PRooF. by induction on v < n+. v = - : Clearly go y ho. Induction step v.: For fl < v and a < K set

g() = o.t.(X ). gf (a) = o.t.(X ).




(Xfl a < n) satisfies (i)-(iii) for every fl < v. Thus by I.H. Cp = {gqp = h# } E c for < v.

Case v = v' + 1: Here we have g,(a) = o.t.(Xv) = o.t.(X2') + 1 = gv,(a) + 1 for a > a*, where

a* < n is sufficiently large. Thus gv(a) = gy,(a) + 1 = hv,(a) + 1 = hv(a) for a E Cv, - a*.

Case v = A limit, cf(A) < .: In this case gA(a) = o.t.(Xi) - sup{o.t.(Xa')ji < cf(A)} = sup{hA,(a)ji <

cf () } = hA(a) for ac EF i<cf(A) C, E C ,.

Case v = A limit, cf(A) = n- Since I X| < n there exists for every a < r a minimal i~ s.t. X C Ai,. W.l.o.g. we

assume that (Aii < ,r) is <-increasing and continuous. By (i) and (iii) we conclude that (ia la < n) is <-increasing, continuous and cofinal in A. Thus there is a D E W s.t. i, - a for all a E D. In particular X -

= X~ A = X o for a E D. Therefore

g (a) = o.t.(X') = o.t.(Xa") = sup{o.t.(X)li < a} = sup{h, (a)|i < a} = h2(a)

for a E Limn Dn Ai<, C2, E ,. Thus if f, : - v is surjective for all v < r+, then the functions defined by:

hv(a) - o.t.(f "a), a < K, v < n

are "the" canonical functions. This alternative definition is the reason why the canonical functions appear in model-theoretical constructions, cf. the Ketonen diagram.

?2. Some combinatorial principles. Let q: -- (n, + 1) defined by

O(a) = (a)' for a < n.

It is easy to see that h, <* for all v < ni+. We now define the following domination properties of the canonical functions:

(D1) "For every f <* 0 there is a v < n+ s.t. f <* hv,

i.e., f(a) < hv(a) for club-many a." (cofinality)

(D2.) "For every f <* 0 there is a v < n+ s.t. non h, <* f, i.e., f(a) < hv(a) for stationary-many a." (unboundedness)

The function is superfluous for successor cardinals n and "f <* 0" can be replaced by "f E '"r". (D2,) means, that there is no f <* 0 s.t. 11f1[ = .+, and is equivalent to the weak Chang Conjecture (wCC) . (Dl ) means much stronger, that

cf ((f f : r, -+ nf <* 0, <*)) -

K'.+

(DI,,F) and (D2,,F) are defined analogously for filters F, i.e., replace <* by <F

in the definitions (and "club-many" by "F-many" , "stationary-many" by "F+- many").



ULAM'S PROBLEM ON cow 719

REMARK. The Transversal Hypothesis for , is the proposition:

(TH*) There are regressive functions f, : n - nt, v < n+, which are pairwise almost disjoint, i.e., {a < , lf,(ca) = f,(a)} is thin for all v

- ku. If , -= wi this is equivalent to the existence of w2-many f, : wcl - cw, which are pairwise almost disjoint ((TH*,,) does not mean I"' o/=*| > 092). For the exact consistency strength of -(TH* ) see [8]. It is easy to see that -(TH* ) implies (D2,,).

Unfortunately, the (usual, historical) definitions of combinatorial principles are not uniform in the sense that (P) >co,. n(P) would always hold. To keep the strength in mind one may use that combinatorial principles on "K/=* get the stronger in consistency strength the more they compress K"/=*, e.g.,

o.t.({f/U f <u diag}) =

- if U is a normal measure on rI. On the other hand there exists a <u-increasing sequence in "n of length 2" = I"n even for normal measures U on r.. The effect of (D 1) on the continuum function is open as well as the consistency of (Dl,,2), see [20].

We shall from now on concentrate on the case n = wi and study (D1) = (D1,, ). On the consistency strength of (D2K) see [7] and [8]. We show:

THEOREM 1. (on the consistency strength of the domination property of the canonical functions on owi)

(D1) >con "there is an inaccessible limit of measurable cardinals".

More exactly we show:

(DI) implies "w24 is an inaccessible limit of measurable cardinals in KMo". Here and the sequel let K = KMo be the core model for measures of order

zero, see [10, 21]. Central for our proof is a natural system introduced by J. Ketonen consisting of - in modern terms - embeddings between mice, in which the canonical functions appear as the lengths of the mice. We first note the relation between (DI) or (D1F) and various saturation properties.

LEMMA 2.1. Let 7,, be saturated. Then (D1) holds. PROOF. Let G be a generic ultrafilter for P = ( ,C) and let j : V -- M

be the canonical embedding into the transitive collapse of the generic ultrapower n :ultV{,G]( G) r M. Then crit(j) = oi and sat(P) =

o2 yields j(o,) = o2V-

0M. For all v < coV we have r([h,]) = v. Since j(o() = V there exists for all

f :cOl - +o1 a v < co s.t.

ult V[G](VG) [f] [h,],

thus {a < oI f(a) < h,(a)} is stationary. The argument in fact shows that

D = {S E P I f is dominated on S by a canonical function} is dense in P. Now another application of saturatedness gives the conclusion. -t

In the same way one shows: If F D ,, and F is saturated, then (D1F) holds. We now define a weaker form of saturatedness and show combinatorically that

this version is also sufficient to get (DIF).




DEFINITION 2.1 (Weakly saturated). Let F be an wi-complete filter on wi. F is

weakly saturated, if: There is no G c F+ s.t.

(i) IG=W- 02, (ii) for all X, Y E G, X 4 Y we have IX n YI

- w.

If F is a weakly saturated filter then F is uniform and whenever F' D F is a

w01-complete filter, then F' is weakly saturated. We note:

LEMMA 2.2. (1) If there is a weakly saturatedfilter F on wl, then there is a weakly saturatedfilter F' D 'i,.

(2) If F is weakly saturated and normal, then F is saturated. (3) If F is weakly saturated and F, , C F, then there is no G C F+ s.t.

(i) IGI = 02, (ii) for all X, Y E G, X Y we have that IX n YI is thin.

PROOF. (1) We set S = {f : coi - Co wif -l{a} is countable for all a < Co}. Let f E S be (<F IS x S)-minimal. Then

f*(F) = {X C ol If - X E F}

is weakly saturated (this holds for all f E S) and contains no thin sets. [well-known; suppose A E f*(F) is thin. We set g(a) = sup(a n (wo - A)) if a E A, g(a) = a otherwise. Then g E S and g(a) < a for all a E A, thus g(f(a)) < f(a) for all a E f -"A E F. But this contradicts the minimality of f, since gf E S, gf <F f.]

But then

F'= (f*(F) U ,l) = {X Cw IX = Y n Z for a Y E f*(F) and a Z E , }

is as desired. (2) For (A, a < 092)

C F+ define (Aj la < 02) by

Aa = Aa - Vf.<aAp = Aa - Vfl<a (Aa n Af) = Af<< (As - Al).

Since F is normal we have A, E F+, if An, Ap E I(F) for all / < a. Suppose that F is not saturated. Let (A, a < W02) C F+ be a counterexample. Then

(A ,a < 02) C F+. But for y < a, A, n A , A, n (A, - V<,Aft) is countable, contradicting the fact that F is weakly saturated.

(3) Analogously to (2) . If (Aala < 02) C F+ and As n Ap is thin for f < a,

then (A, a < C02) C F+, since I( ~,) C I(F). -1 LEMMA 2.3. Let F 2 F, be weakly saturated. Then (DIF) holds.

PROOF. Suppose not. Then there is a f : owl -- i1

s.t.

Av = {a < oIhv(a) < f(a)} E F for all v < 02.

Let g, : f (a) --+ be injective for a < (0l. Since F is Wi-complete there is for all v < 02 a nv < oW s.t.

B, = {a E Avlga(hv(a)) = n,} E F+.

Let D c0 2 be unbounded in 02 and n < W s.t. n, = n for all v E D. But then by (3) of the lemma (Bv iv E D) is a counterexample to the weak saturatedness of F D ,, since Bv n B, is thin for v v, v, E D, contradiction. -d




The notion "weakly saturated" is interesting due to its connection with a classical problem of Ulam. We start with a definition.

DEFINITION 2.2 (9-measurable). Let 7 = (Fala < 0) be a sequence of filters on col. X C oi is F-measurable, if there is an a < 0 s.t. X E F, or i1 - X E F, (i.e., X E Fa U I(F)). We set

u<O

We study the following covering property (Ulam property):

(UP) There is a sequence 9 = (F, la < oi) of iol-complete uniform filters on ol s.t. 9- covers 9(coi), i.e., every X col is F-measurable.

(A,a n E w, a < iol) is an Ulam matrix, if

(i) UnE An,a = ol

- a for all a < Oli, (ii) (An,, la < cil) is pairwise disjoint for all n < cw.

Using an Ulam matrix one shows that there can be no non-trivial col-complete measure on ol (and no ciol-complete col-saturated uniform filter on col). An Ulam matrix can be obtained from surjective functions fp : co - (fl + 1) for # < wi by

An,a = {f < ol

ffp(n) = a}.

The length ai of the sequence in (UP) is minimal. Ulam knew that a finite amount of measures does not suffice. But even countable many measures cannot cover 9 (cI):

LEMMA 2.4 (Erd6s-Alaoglu). Let - =- (F, In E o) be a sequence of ol-complete uniform filters on co1. Then there exists a non-3r-measurable X

_ col.

PROOF. For n E (o let (An,a~a < wo) be a partition of col s.t. A,,, E F,+ for all a < cwl. For n E o set:

f(n) = min{m E colAm, E F, for olW-many a < ci}.

Then f (n) < n for n e o and there is a f < col s.t. for all n E w:o:

(+) Am,a ( F,+ for all a > fl and all m < f(n).

Let y1 = ft. Define recursively for n c c:

y, = "the least y > yn-I s.t. Af(n),y E F+'". We set B, =

Af(n),Yn - {Af(m),ynIm E o, f (m) < f (n)}. Then

(i) Bn E F, for all n E co, (ii) Bn, Bm = 0 for all n, m E ow s.t. n # m.

(i): We have Af(,),Y E F,+ and if f (m) < f(n) we know using (+) that Af(m),m C

I(F,) since ,n > ft. (ii): If f(m) = f(n) theclaimfollowsfrom 7n # 7m, since

(Ajf(n,t < t1)} is a

partition. If f(n) , f (m) then (ii) follows from the definition of B, and Bm. Now split Bn for n E o into two parts Bn,o and Bn,1 s.t. Bn,o, Bn,1 E F,+. We set

X = Unco Bn,o. Then X 0 Fn and

ol - X

( F, for all n E c. o.




More generally it was shown by Gitik in [9] that countably many cOl-complete uniform filters on r, covering Y(K), > woi, can be reduced to one such filter covering ,(n). If one demands normal filters in (UP), then the consistency strength is known since

(UPnormai) iff "there exists an col-dense ideal on wlo" [18]

(=co, "there exist (o-many Woodin cardinals").

Useful for applications and of interest in its own right is the following simple application from [18] of an Ulam matrix (which will be used in Lemma 2.6).

LEMMA 2.5 (Partition lemma). Let = (FllP# < w1) be a sequence of cwl- complete uniform filters on owl. Then there is a partition (Y-" In (0) of{F Ifl < cil }, so that for all n E o there is a not n -measurable X, C ol. More precisely we have: For n E w let " =n (F, la < On), gn < CoI. Thenfor each n E co there is a sequence (B, la < E"), B"

C_ ol ofpairwise disjoint sets such that Bn E F '

for all a < O". PROOF. Let (Any In E co, 7 < woi) be an Ulam matrix. Let

f(fl) = min{n coA,,Y E Ff' for ol-many y < oi}

for P < o1. We set Z, = f-l"'{n} for n E w. Fix n E o. Let (fla < o") be the increasing enumeration of Z,. Let (y, a < O") be strictly increasing s.t. B, := An,y, E F, for all a < O". Split Ba into two parts B,.o, B .1 E

F, and set

X - U<o.t.(,) Bo. -1

Using a projection argument similar to that used for weak saturatedness one shows: If there is a sequence 5 as in (UP), then there is a sequence

'7'= (FJa <wol)

as in (UP) s.t. F' D F,, for all a E wol. We now study the consequences of (UP) w.r.t. the existence of a weakly saturated filter.

DEFINITION 2.3. (f-weak*-saturated) An Iol-complete filter F D ,,

is f-weak*- saturated fora p < o2, if for all (Xy, < W2) C F+ there is a T C 02 s.t. o.t.(T) and

]yET XY is stationary in wo.

(A refinement of that hierarchy at limit points is possible, but we content ourselves with the version given.) Now (UP) implies for all P < ol the existence of a fl-weak*- saturated filter on olI. The argument given is based on the proof of Theorem 4.4. in [18], where the existence of a weakly saturated filter was shown.

LEMMA 2.6 ((UP) and weak*-saturatedness). Suppose (UP) holdsfor the sequence (F, aj < cwl) and that F2, D , for all a. Let fl < cw. Then there exist AC ol and a < wo s.t. F,[A] is #-weak*-saturated.

PROOF. Suppose not for a P < cl. For n E co let n = (Fnja < On), (BJ a< K E")

be as in the partition lemma. We construct a C-decreasing sequence ( Y nInE (0), Yn

_ ow s.t. for n E a) the following holds:

(1) X,n : Yn - Y,n+ can be split into two not -n"-measurable parts.

(2) B' E Fm[yn,]+ for all m E o, a < 0m.



ULAM'S PROBLEM ON wl 723

We set Yo -= ol. Induction step from n to n + 1 Let Yn be constructed. By I.H. (2) Fn [Bln Yn] is a proper filter and by assumption

not #-weak*-saturated for all a < On. Thus let

(Xa Y < (02) C B n Yn

be a sequence in F,[Yn]+ s.t. yrX

is thin for all T c 02 s.t. o.t.(T)> P W.l.o.g. X, n (a + 1) 0 for all a < ?n

and 7 < 02. Now set for y < c2

Z = U XY. a<O"

Then Z, C Yn and Z, E F,[ Yn]+ for all a < On. Further for all a < On and all T C C02 s.t. o.t.(T) f P we have:

(+) ( Zy =() U Q) XY is thin, YET a<On 7ET

since n, r Xo is a thin subset of wo - (a + 1) and 0"n i. But then

(++) There is a Zy s.t. Y, - Zy E F7m[B' n Y,]+ for all m E co, a < Om,

since by (+) we have Zy E Fm [Bm n Y,] for less than #-many y < o2 for each Fm, m E co. [Here we use Fp i D , for all f < cwl].

We set Y,+l - Y, - Zy. Thus X, = Zy. Then (2) holds by choice of Z,. BecauseifBm ?

Fm [ Y,]+ foranm ca, a < om, thenBmn( Y,-Zy) E I(Fm), thus Z, E Fm[B'm n Y,], contradiction.

Further we have (1), because X, = Z - U0<o-

X7 and ;X C B", Xy E F+ for a < On. Now split X7 into Fa-positive XY, Xy . Then Ua<O, xy is not

9n" -measurable. Now the existence of a not 7t-measurable set follows from (1) by an easy splitting

argument (since (Xnln E o_) is pairwise disjoint), contradicting our assumption (UP) for r.

-. Weak saturatedness rejects even very weak forms of the Transversal Hypothesis

(TH*). Regarding our investigation of non-regular ultrafilters and Ulam's problem, the following form is of interest:

For a filter F D ,, on iol let

(THy) "there are partial functions (fvV < W2), fv : dom(f) --+

w, s.t.

(i) dom(f,) E F for all v < C02,

(ii) {y E dom(f ) n dom(f,)f,(7) = fu(y)} is thin for all v 54u." Clearly (TH*) is equivalent to (TH

*,),

and (TH*) implies (TH},) for F C F'. As for saturatedness one can consider more than two f,'s in (ii): For f < (02 and

a filter F D F, on wo let

(THF) "there are partial functions (f,Iv < 02), fv : dom(fv) o, s.t.

(i) dom(f v) F for all v < C2,

(ii) for all A C 02 s.t. o.t.(A) = fl we have:

{y E ( dom(fv)lf(y) =

f,(7y) for allv,uu c A} is thin".

vEA




(TH) is the version without filters (with total functions) and is equivalent to

(TH,*) for fl < wi. Again we have: (THF) implies (TH,,F,) for f fl',

F C F'.

LEMMA 2.7. Let F be fl-weak*-saturated. Then -'(TH#,F,)for all F' C F.

PROOF. It suffices to show: --(TH*,F). Suppose not. Let (f, v < C02) as in

(TH8,F), f, : X, - w, X, E F. Since F is wi-complete there is an n* E o and an

A Cw2 s.t. JA = W2 and

X' = f,'"{n*} E F+

for all v E A. But then (X'Iv E A) C F+ is a counterexample to the #-weak*- saturatedness of F, contradiction.

COROLLARY 2.8. (UP) implies -(TH*) for all # < oi. PROOF. Let fl < cwl. By Lemma 2.5 (UP) implies the existence of a fl-weak*-

saturated filter F. Using the lemma above we get -(TH*,o

), thus -(TH*). -

?3. The Ketonen diagram for K 102. For 0 < v < C02 let g, : wi -- v be surjective. Let G = {(v, a, fl)g, (a) = fl}. Let D

_ 092 -01 be club in w02 s.t. (K Iv, G nKv) -<

(KIw02, G) for v E D. For v E D and a < cw let

Xav = Hull (a; (K I v, gv ))I

i.e., the smallest submodel M of the structure (K v, gv) s.t. a C M. We set for vED

C= < o l XV nol = Then C, is club in w, for all v E D and X, n On = g"a for a CC. For vCD

and a E C, let 7r v V~v a xa

be the inverse collapsing isomorphism, Mv transitive. Then Mv is a mouse for all v E D, a E Cv and we have:

(1) {aIh,(a) = Mv n On} E o, for all v E D, (2) crit(n7r) = a and (3) 7 v(a) = wi for all v E D and a E Cv.

Here (1) follows from the alternative characterization of the canonical functions:

M, n On = o.t.(Xv n On) = o.t.(gv'a)

for all a E Cv, v E D. Since rng( 1) C rng(irv ) n On for at a2, al, a2 E Cv we

also have mappings

(7r, ',a2 . MV' q --+ Mc ,ll < a2,aI, a2 C)

s.t. 7r' 7= ~ V 7 2

al a2 a1,a2

The diagram below visualizes the situation concentrating upon the ordinals. The ordinals of the mice Ma are drawn with a "rectangular break" at a, the ordinals of K v with a break at col. That ordinals appear repeated in the diagram is not dangerous. All Ma are countable. Their lengths build "the" canonical functions. We suppose in the following that hv (a) = Mv n On holds for all v E D and a E C,.



ULAM'S PROBLEM ON l 725

(ol z v (02

a/

a hT(a) h,(a)

0

Often used are the following extension properties of the construction.

LEMMA 3.1 (Extension properties of the Ketonen diagram). (a) Let z, v E D and z < v. Then there is CT, club in wo s.t. for all a CT, we have: (i) hT(a) < hv(a),

(ii) nr = nrIMa, (iii) Ma = M~ h (a), (iv) nr(h(a)) = T

(fp) Let E D, cf(A) = o and let (Ai Ii < 6) C D be increasing, continuous and cofinal in A for some ordinal 8. (The standard case is 6 = wo.) Then there is C - Cl((Al|i < 6)) club in wCl s.t.for all a C. we have: (i) (hA, (a) |i < 6) is increasing, continuous and cofinal in h2(a),

(ii) ri1M' =ir i for all i < 6, (iii) Ma = U-i< o

M', MaIhA, (a) = Mi for i < 6, (iv) n (h, (a)) = At for all i < 6.

(7) Let A E D, cf(A) = Col and (Aili < wi) C D be increasing, continuous and cofinal in A. Then there is C. = C ((Aii < i)) club in wi s.t. for all a CA we have: (i) (hA, (a)li < a) is increasing, continuous and cofinal in

hao (a) for all limits

a , (ii) iIMi

i7 for all i < a,

(iii) MA = Ui<aMi< , Ma|h (a) = MA for all i < a. Thus (M, j aE C,) is the diagonal union of (M i a E CA, i < co).

(iv) n (hi (a)) = Ai for all i < a.

PROOF. Straightforward.

In the following a prime ' at a club CT,, or CA in col indicates that we are referring to this lemma (w.r.t. to a given A-sequence in the limit case, which should be clear from the context). It is convenient to stipulate that all C' c Lim.

We shall also need the ultrafilters induced by the embeddings 7nt. So let

U ({X E (a) n Mlaj E 7rv(X)} for vE D, Cv.




?4. The domination property. In this chapter we shall prove Theorem 1.

! A global assumption is --Os, since otherwise much stronger large cardinal axioms hold in K. (E.g., each uncountable cardinal in V is a measurable stationary limit of measurable cardinals.)

We start with two general lemmata.

LEMMA 4.1 (Well-founded vs. ill-founded). Let M be a mouse and a : M - M . Let crit(a) = a and U = {X E (a) n Ma E a(X)} be the induced ultrafilter. Let a+ :- (a+)M < On n M and N a mouse s.t. N1(a+)N - Mlla+. Further let Ult(N, U) be not well-founded. Then M <c N.

PROOF. Suppose that N <c M. Coiterate N and M (lengthened) to No M with indices (vi ii < 0) and let (Ni, Mi i < O) be the coiterations. By hypothesis vo > a+. By assumption the N-side is simple. If (f, n E co) c N is a counterexample to the well-foundedness of Ult(N, U), then

(n7NNo(fn)I an E w) C No is

a counterexample to the well-foundedness of Ult(Ne, U). On the other hand all

Ult(Mi, U) are well-founded for i < O: Let ai : Mi -*E* Mi, i < O, ao = a, be the level embeddings of the usual copying construction, where we copy the iteration

(Mi i < O) to a new iteration (Mi ii O0) of M = Mo using a = ao. Using vo > a+ we conclude that crit(ai) = a and that

U = {X E c(a) n Mija e ai(X)} is the ultrafilter induced by ai. Thus Ult(Mi, U) is well-founded. But No = Mo, contradiction! -1

LEMMA 4.2 (Countable mice beyond the Ketonen diagram). Suppose that

((w1)K < (02

and assume w.l.o.g. that D c (02 -- ((0-)K + 1). Then the following holds:

(#) For all a E Cmin(D) there is a mouse N s.t.:

(i) N I < 1,

(ii) for all v E D there is a club C' s.t. Me <C Na for all a E C'. PROOF. Let v - min(D). We write a+ for (a+)K. We set 7n = n , M = M ,

U -= Uf for a E C,. We define Na as follows: Let a E CV. If a is singular in K set Na = Kly for a y < w01 such that

Kly "a is singular". If a is regular in K but (a+)Ma < a+, then choose a y < (1 s.t. Kly "(a+)Mc is singular" and again set

Ns - Kly. Otherwise (a+)M = a+

and thus in particular a+ < 0w. Then:

(+) Ult(K, U,) is not well-founded.

PROOF OF (+). Otherwise (J, , U,) is a beaver, hence a strong mouse (we assume --,Os). Thus Ea+(K) = Ua. But Ua is the ultrafilter induced by no : Ma - K| and thus Ea+ (KIK) = 0, hence also Ea+ (K) = 0, contradiction!

Let (f, In cE ) C K be a sequence s.t. f, : a -* On and

{y < alfn+l() < fn(y)} e U0

for all n E co. Let Y =- Hull((a+ + 1) U {f,,n E )};K). Let N, be the transitive collapse of Y, and a : N --* Y0 be the matching isomorphism. Then Ult(Na, Ua) is not well-founded, since for (g,nn E (0) C N s.t. a(g,) = fn we



ULAM'S PROBLEM ON co 727

have g, : a --+ On and the sequence is a counterexample to well-foundedness. This

completes the definition of the mice Na. Now let v E D and C, : C , be club in wi with the extension property (a) of

Lemma 3.1. Thus 7r jh,(a) 7- r, M| hV(a) Ma and U - U. for all a E Cv. (i) is obvious. (ii) is obvious if a E C~ is singular in K or (a+)M = (a+)M < a+ Note that (a+)Mc

- (a+)M holds, since (wo)K E rng(7za) and

7rc c rX Thus the remaining case is (a+)M = (a+)Me - a+. But then Ma <c N, follows

from Lemma 4.1 above since M Ilat N a I+ and Ult(M , Ua) is well-founded, but Ult(Na, Ua) is not well-founded. This proves (#) .

We now show that (D1) implies (#). In fact:

LEMMA 4.3. Suppose (D2). Then (w )K < 02.

PROOF. Suppose not. Then (wo))K _ 02 and for all v E D and a E C, we have:

(+) M k "a is the largest cardinal".

Define f : wi -* ol by

f(a) = "the least y s.t. K7

= a is not a cardinal", if such a y exists,

(a+)K + 1, otherwise.

[Note that f(a) >2 w for at most one a.] By (D2) there is a v E D s.t.

E {= a < w llf(a) < h,(a)} is stationary in -1. Then for each a c E:

(++) M2 is not an initial segment of K.

Let v* = min(D - (v + 1)) and let C,. be club in wol as in Lemma 3.1. Let

a E E n C..

Let u be the first index in the coiteration of Mv and K. Then

a < p < M n On. But then u is a cardinal in Mv*, contradicting (+). -

Thus we have: (DI) -+ (D2) - (wo)K < 02 -* (#), where (#)is as in Lemma 4.2.

Now suppose that (DI) holds. Then (#) is true, too. Wl.o.g. Mv <* Na for all v E D and all a E C,. Coiterate Mv and Na for v E D and a E C,. Let

E) < woi be the length of the coiteration,

(4v,i i < Ov) be the indices of the coiteration,

(r.iji < O) be the coiteration points.

Further let

(Ma,Nv,i, Ii < O) be the coiterations,

(r;ij i < j < O) be the coiteration maps of the Mg-side,

(a:i.ii < j < Ov) be the coiteration maps of the Na-side.

We shall use these objects and notations from now on. We note:

(A) If z < v, z, v E D and a E C,,, then the coiteration of Ma and Na is an initial

segment of the coiteration of M and Na (since each M -A KIJC2 satisfies

ZFC-). Suppose towards a contradiction that C02 is not a limit of measurable cardinals

in K. Let f0o < 02 s.t. K - "all measurable cardinals < 092 are < fio". Wl.o.g. D C

02 - (fl-

+ 1). Then for v E D and a E C, there exists a fa < Ma n On s.t.

Mg )- "there are no measurable cardinals > ".




We define by recursion

(vy |y < 092) C D < -increasing, continuous and cofinal in

w92, (fy ' < (02) s.t. fy :

0O1 --+ 01 for ' < (02

as follows. To start let vo = min(D) , fo = h0. Successor step from ' to ' + 1 Let f, and v, be constructed. We set

vy+l = "the least v E D s.t. fy <* hv".

( On n N,eo; (where v =v-y+l) ifa E Cy+,

fY+(a)-0 otherwise.

In the limit step 1 set

vA = sup vy,

Sf supi< o fa if cf(A) = 0,

diagsupi<, f , if cf(A) = 01.

We used (DI) in the successor step. We have:

(+) fy <* h~y+, <* fy+l for all < w2.

By the lemma on mutually cofinal sequences we have in particular:

(++) f2 =* hv, for 2 limit.

Using (+) we can define club sets (By, < W02) in 01 by recursion as follows. To start set

Bo = 01.

In the successor step from y to ' + 1 set

By+l = By n {ja < I0flfy(ca) < hV,+, (a) fy+l(a)}. In the limit step A set

B -

fni<,O B , if cf(A) = o,

Ai<o, B2 if cf(A) =

ol1. Now let v = v,,, v* = min(D - (v + 1)). Let C' be club in

w9l w.r.t. the sequence (vy |7 < (01) as in (') of Lemma 3.1 on

the extension properties of the Ketonen diagram. Finally we set

C = C n C,. B,. v V, W

From the boundedness of the measurable cardinals in the mice Mv we gain the following information concerning the coiterations with N,. Let a E C C Lim. Then:

(a) (M' I' < a) is strictly increasing and continuous w.r.t. initial segments. Fur- ther

Ur<, M, = Mv = M* Ih,(a).

(b) viv = n,v* for all y E a U {(01} and all i < . We write N,i for n~,i.




(c) Ov is a limit and there is an io < O' s.t.: iv;i,

( ,i) = ,

for all i, j s.t. io < i < j < Ov. Further the sequence (Na,ilio < i < OV) is continuous. We fix io(= io(a)). [io exists, otherwise one could construct a degenerated iteration.]

(d) Let Iv = { K,i io < i < 0 v}. We have r,,e;

= hv(a) and:

For all A E 9(hv(a)) n M e;*

there is an q(A) < h,(a) s.t.

Iv - q(A) C A or

Iv - q (A) C h,(a) - A.

(e) 0~ is a limit for all limits 6 E a. If io < ?O we have

h (a) = f6(a) = sup ,i i<O,5

=,o= On nMMv,

and nv* (K,

) = a, aThus set f,Or a Lim2 (i.e., a is a limit of limits)

Thus set for a E Lim2 (i.e., a is a limit of limits)

I*v = ({Ko 16 < a, 6 limit,io < 0 }, cr a

Then I* c_ I c On n M, is unbounded in I , and I* U {h,(a)} is a set of

fixed points for the mapping nv"* Thus if a E C n Lim2, we have: For all A E c (h(a)) n M ' there is an q*(A) < h,(a) s.t.

I*V - q*(A) C A or

I*a - qI*(A) C h, (a) - A.

It follows from this that there are unboundedly many (inaccessible) cardinals in M for a E C. Thus the same is true for each K v', where v' E (D U {w2}). In particular we have that

(v) n K = 9(v) n Kv*.

Let

D* {= {v < w(,6 limit}. Our goal is:

(0) D* induces an wo1-complete K-ultrafilter on v.

Then v would be measurable in K, contradicting flo < v < 02- It suffices to show:

(0') For all AEC (v) n K thereis a <vs.t.D* - C AorD* - C v-A.

Since cf(v) = wi and D* is unbounded in v we then have that

U= {X E (v) nKID* - C Xforac < v}

is an wli-complete K-ultrafilter. Now let A E (v) n KIv*. Then there is an ao < Wo1 s.t. For all a E C - ao there

is an A, E Mv* n Y (h (a)) s.t. n * (A,) = A.




Then for a E C n Lim2 - ao there is an = rq*(A,) < h,(a) s.t.

I*- C Ac or

-*i Chv (a) -Aa.

Let 5 < v s.t. S = {a E C n Lim2 - aol < he(a)} is stationary in Col. [We have q <* h, where r : ol -+ ol, r(a) = q for a E Cn Lim2 - a0. Thus

lrqI < |lhj = v. Then any < v s.t. It11 < will do.] Further let w.l.o.g.

S' = {a E SIl*v -

A* } c, -qc C A ,)

be stationary in ol. Then in particular nr, (I*' - r,) CA for a E S'. Thus in order to get (0') it suffices to show:

For v5 E D* s.t. < v5 there is an a E S' and a , E I* l s.t. n7r () = v. But 7 * (h, (a)) = v6 for all a E S' - (6 + 1). Thus it suffices to show:

hv,(a)On n M,,6cI*' for anac E S' - (6 + I).

But since 6 is a limit this holds for even all a E S' - (6 + 1). -] The same proof shows: THEOREM 1'. Let F be a normal filter on o1 and suppose that (DIF) holds. Then

C02 is a limit of measurable cardinals in K. Without normality we cannot take v = v,, having cofinality col as basis for our

proof (a as above s.t. S is in F+ needn't exist). It turns out that a v s.t cf(v) = w will do, but we have to supply a new argument concerning the foundation of Ult(K, U), where U is the induced ultrafilter.

THEOREM 2. Let F D F, be an ol-complete filter and suppose (D1F) holds. Then

092 is a limit of measurable cardinals in K. PROOF. We construct (vy 17 < c02), (f I7 y< 002) as before, where now

vy+I = "the least v E D s.t. f, <F h,".

Then we have

(+) fy <F hvy, <F f y+l for all y < 02.

The lemma on mutually cofinal sequences now holds in a restricted form only:

(++) f; =F h for all 2 < o2 s.t. cf(A) = o.

(Here the woi-completeness of F suffices. Since F contains the club-filter F, we have that (h,|v < 02) is <F-increasing.) Now set v = V,2 and choose an arbitrary v* ED - (v + 1). Let C', C/,. as usual and set

C = C' n nBo2.

D* = {v 6 < (o2,6 limit }. As before we conclude:

(0) D* induces a K-ultrafilter on v.

Let U = {X E g(v) n KID* - C X for a < v} be the induced ultrafilter. To reach a contradiction it suffices to show that:

(#1) Ult(K, U) is well-founded.




To this it suffices to show:

(#2) Ult(Klo2, U) is well-founded.

PROOF THAT (#2) IMPLIES (#1). Suppose not. Let (f, In E wo) be a counterexample to the well-foundedness of Ult(K, U). Let M be the transitive collapse of

Hull({fnIn E w} U v; K).

As usual U is an M-ultrafilter and Ult(M, U) is not well-founded. If Ult(K wo2, U) is well-founded, we have that KIwo2 <c M by Lemma 4.1. But IMI < w02, contradiction, since N <c KIw2 for all mice N s.t. INI < 02. (This is shown as "K is universal": If

KIOw2 <c N, then there is a and an wo-complete F on K s.t.

(KIl(,+)K, F) "F is a normal measure on K" and E(,)K = 0, contradiction.) For a E C, a E Lim2 we set:

a = {X E 9(hv(a)) n Mv* II*V - r C X for an r < hv(a)}.

By (e) we conclude

(Mv, U) b " U1 is a normal measure on h,(a)".

Further we have

(#3) Ult(M'*, U0) is well-founded.

PROOF. Suppose not. Let U0 = E,,, (Nv,;). Then

Ua = {X E 9(h(a)) n Me |Iv C - c X for an 4 < h,(a)},

and by Lemma 4.1 Ult(M, ,oQ U,) is well-founded.

Let (f,,n E o) be a counterexample to the well-foundedness of Ult(M , U).

Let

r = 7E MM

,

and

gn = r(f,)

for n E o.

For y E I*' U {h,(a)} we have 7r(y) = y and hence X E c iff n(X) E GU. But then (gln E w) C Me;

is a counterexample to the well-foundedness of

Ult(Mv,\e, U0), contradiction! We now show (#2). Suppose not. Let (f,ln e wo) C KIw2, f, : v -- 02

for n E c w, be a counterexample to the well-foundedness of Ult(KIw2, U). Since

v* E D - (v + 1) is arbitrary w.l.o.g. f, E KIv* for all n E o. Thus Ult(KIv*, U) is

not well-founded. For a sufficiently large a E C we have {f,In E ow} c rng(7' ). Fix a and choose gn E M'* for n E w s.t. 7r<* (gn) = f . Then since n' (h,(a)) = v we have that gn : hv (a) -+ hv (a) and that (g In E co) is a counterexample to the well-foundedness of Ult(M , U0):

For n c w set

xn = {Y < vifn() > fn+l(')),

Yn = {y < h,(a)lgn (') > gn+,(Y)}.




Let n E ow. Then n?v (Yn) = X , U. Thus there is a < w2 s.t.

r (Y) _

{v6 < 6 < o2 limit} = { n (h,(a))] < < ow26 limit}.

Now there is an q < h,(a) s.t.

{h, (a)1 < 6 < w2,g limit) I* - r,

thus I*' - q C Y,. Therefore Y, E Uo for n E co, contradicting the fact that

Ult(M , , ) is well-founded by (#3). -1

?5. Non-regular ultrafilters and Ulam's problem. We improve the actual lower bound "there is a measurable cardinal" (Jensen, [6]) of the consistency strength of "there is a non-regular uniform ultrafilter on o)" [(NR)]:

THEOREM 3. Suppose there is a non-regular uniform ultrafilter on oi. Then o V is an inaccessible stationary limit of measurable cardinals in K.

We begin with some definitions.

DEFINITION 5.1. ((p, r,)-regular filters, regularity sequences) Let M be a set, U an ultrafilter on M and u,' be cardinals s.t. p .. U is (p, ,K)-regular if there is a sequence (X |a < K) C U s.t.

X a = for all A [] cEA

Such a sequence (Xla < rn) is called a (p, .)-regularity sequence.

DEFINITION 5.2 (Non-regular and weakly normal filters on wc(). Let U be a uniform ultrafilter U on coi.

(i) U is non-regular if U is not (co, col)-regular, i.e., whenever (X Ia < ol) C U there is an infinite A C (ol s.t. fl CA XA $ 0.

(ii) U is weakly normal if for all regressive functions f : () -+ o1 there is a y < cl s.t. {a < olIlf(a) < y7} U, i.e., the identity on ol is a least unbounded function modulo U.

Weakly normal ultrafilters are non-regular: LEMMA 5.1. Let U be a uniform ultrafilter on oI and suppose there is a least

unboundedfunction modulo U. Then U is non-regular. PROOF. Let f be a least unbounded function modulo U, and let (X, Ia < ol) C

U. We set

g(p) = sup{a < f (l)I E Xl}.

If (X,Ala < col) is a regularity sequence, then g(fl) < f(fl) for all # < ol. But one can easily verify that g is unbounded modulo U, contradicting the minimality of f. Thus U is non-regular. -

We note the following well-known result (Kanamori, Ketonen, [12]): LEMMA 5.2. Let U be a uniform ultrafilter on col. Then:

(a) U is weakly normal iff U D O, and U is non-regular. (b) If U is non-regular, then there is a least unboundedfunction f modulo U. Further

f*-(U) = {X C w If -'"X C U} is weakly normal.




We now show that the canonical functions are dominating modulo U for a weakly normal ultrafilter U. To do this, we consider the weak version (TH*) of (TH*) introduced in the second chapter.

LEMMA 5.3. Let U be a weakly normal ultrafilter on o1. Then --( TH*U). PROOF. Suppose not. Let 17 be as in (TH*). Choose f E7 and (fIa < w0i) C

JV s.t.:

(a) fa f# for a fl, (b) fa <u f for all a < 0i.

For a < 0() we set

Aa = dom(fa) - U {y E dom(fa) n dom(fg) If,(y) = f(7)}. fl<cr

Then A, E U since U D ,. We set g, = f ,Ao for a < co1. But then {y E dom(g,) n dom(gp)lg, (y) = gp(y)} = 0 for all a < f < 0io. Finally set for a < 01:

Xa = {y E dom(g,) n dom(f) g, (y) < f(7y)} Then (X[ a < 0ol) C U is a regularity sequence, since if Y E [0)10] then for all

a, fl E Y s.t. a fl: (i) g(y) gp(y) for all y E dom(g) n dom(gp),

(ii) gc (y) < f (y) for all y E dom(g.) n dom(f). Thus nfl x = 0 (since f(y) is finite for each y c dom(o)). Thus U is regular, contradiction.

COROLLARY 5.4. Let U be a weakly normal ultrafilter on i1. Then (Dl u). PROOF. Otherwise there is an f : i - , -+ s.t. h, <u f for all v < 0)2. Thus

Yv, {y < w0lhV(y) < f(y)} E U for v < (02. For 0) < a < )1 let b, : a --+ be a bijection. For a < c0 set b, = idIa.

Then (h'|v < 092) satisfies (i) - (iii) in the definition of (TH;*), where

h'(y) = bf(y)(h,(y))

for y c Y, and v < 02.

Contradiction! -A

The lemma on mutually cofinal sequences is not true any more for lack of completeness and normality of ultrafilters on wi. But we have a substitute for weakly normal ultrafilters at limits of cofinality w)1:

LEMMA 5.5 (Continuity lemma). Let U be a weakly normal ultrafilter on 0)i. Let (fy l < W02) be a <u-increasing sequence of functions f :

01 --- oli. Further

let A < 02 s.t. cf() = iol and let f = diagsupi<,,, f,. Suppose that for all r < oi we have supi<, f~i <u fy, for a y, < A. Then: f/=u is the supremum of {fy/l=ul < } w.r.t. <u.

PROOF. Suppose not. Then there is a g s.t. g <u f and fy, < u g for all y < A. Let X =a < tg(a) < f(a)}. Since f(a) sup,<,, f, (a) thereis for all a E X an r(a) < a s.t. g(a) < fr(a),(a). Thus since U is weakly normal there is an X' c X, X' E U and an r < col s.t. r(a) < r for all a E X'. Thus g (a) <u supi<, f ,(a)

_u fy,, contradiction. -A




COROLLARY 5.6 (Continuity lemma on the canonical functions). Let U be a weakly normal ultrafilter on wl1. Let v < w2 andcf(v) = s1. Then: hy/= v is the supremum of {h/,l=ul < v} w.r.t. <u.

PROOF. We have U D F,, and the canonical functions are independent modulo club of the choice of (jlii < cf(A)). -1

For an arbitrary _ v-increasing sequence the continuity lemma needs not to be

true, even if the limit points of the sequence are constructed using the sup and diagsup operations; these operations may depend on the choice of the sequence (Aii < cf(A)).

Now we prove Theorem 3. Thus let U be a weakly normal ultrafilter on Col. We use the constructions and notations in the proof of Chapter 4 up to the point (A).

(D Iu) suffices for the proof of (wt-)K

< 02 [Lemma 4.3], so there exist mice N, beyond the Ketonen diagram as in (#) of Lemma 4.2. (One might also use that (NR) implies that 02 is inaccessible in K, since (NR) refutes the Kurepa Hypothesis, which in turn implies that 02 is inaccessible in K.)

Suppose that 02 is not a stationary limit of measurable cardinals in K. Thus we can assume w.l.o.g. that D C {v < w21v is not measurable in K}. As

before we construct (vy |y < 0)2) <-increasing and continuous in o2 and a sequence

(fy|I < 0)2) of functions fy : I -+ w01.

To start let vo = min(D). Successor step from y to y + 1 and construction of f,: Let v, be constructed. We set

OnnNf',NO + 1, ifaE Cv,,

fy(a)-0, a, otherwise.

Vy+l = "the least v E D s.t. fy <u h,".

Limit step A: We set

vA = sup vy. Y<A

We used (D1 u) in the successor step. We have:

(+) hvy , fy and f, <u h,,+, for all y < w02-.

For the following we set for 2 limit

Ssupi<w fi , if cf(A) = 0),

diagsupi<w fA,, if cf(A) = cl.

Then:

(++) For all A < 02 s.t. cf(A) = 01 we have that fx =u hA,.

PROOF OF (++). Let l < 0)1. Let 7 < A s.t. supi<q, i < y. Let C' =

Cv (A) as in Lemma 3.1, where A is a sequence s.t. {vA, |i < r} C A. Then for all a E C' we have that

supi<r/ fAi(a) < f, (a). Further C' E U. By the continuity lemma

fA/-u is the supremum of {f,y < A} w.r.t. <u, thus f =u h, by (+) and the continuity lemma on the canonical functions.




Let D' = {vl cf((A) = w} c D. For v2 E D' we define Xv E U and gv, : Xv,

o-1 as follows.

Let v = v2 E D', v* = min(D - (v + 1)). We set

Xv = C'((v,Ii < w i)) n C'v,.

n a < wlf2(a) = hv,(a)} U For a E X, set

,v = v ( Oa) ,

i, (a) = "the least i < 0 s.t. pv (a) E rng( jv~i ,

(note that E? is a limit)

gv(a) =

N, .(,) n On.

By definition of X,, we then have g,, <u h,, for all vA E D'. By the continuity lemma on the canonical functions there is for all vA E D' an r(X) < s.t. g,, <u hv~, . Thus fix < wo2 and D" C D' - v stationary in C02 s.t. r(X) = c for all vA E D". We set

No N v fora E Cv . For v E D" let X' = C,, n {a E Xlgv(a) < h, (a)} E U. For a E X' we then

have

uv (a) E rng(nig,N ). t , O'

Thus define for v E D", : w -- wi by:

(7

oNv'o v )

ifa E X,

=0, otherwise.

Then -, <* -, for all v, r E D" s.t. v < z. We use this to show:

(#) (gJv E D") is eventually constant mod (U).

PROOF OF (#). Suppose not. Then there is a D c D", D unbounded in C2 s.t. g, <u g, for all v, C E 15 s.t. v < T. For v E /

let v* = min(D - (v + 1)) and

= < g~('0 <ggQ)( ) U

Let g -

glY, for v E D. Then {gv E D} is almost disjoint: Let v, z E D, v < z. Then > v* and since g, * <*, we have by definition of Y, that {a E Y, n Y,~g,(a) = g,(a)} is thin. But

v < u k, where k(a) - , n On for a C,.

Thus (TH*) holds, contradicting the fact that U is weakly normal and Lemma 5.3. This proves (#).

W.l.o.g. g, =u g, for all v, T E D". Now fix v = vA E D" s.t. v is a limit point of D" (D" is stationary in 092). Thus

w.l.o.g. (vA, 16 < coi) C D" (and therefore cf(Al) = w1

for all 6 < oli). We write v5 for v~, 6 < ol, where A,,, = A. Let v* = min(D - (v + 1)). For 6 < o let

Y,5 = {a < 0119g, (a) = 9,(a) o} E U.

Then Y, 5C X' c_

Xv,= n C'~ C. n {a < wollf,(a) = hv,(a)}. Finally set

Z = Yr, C n C,.

n {a < if011f(a) = hv(a)}.

Then:




(a) For a E Z(M 16 < a) is strictly increasing and continuous w.r.t. to initial

segments. Further U<a Mv = MMM M hv h(a). (b) Kv6i = Kyi for all E aU {w } and all i < 0? whenever a E Z. We write r,i

for K,i

? (c) O,

is a limit for all a E Z. For a E Z let

Ieve= {K,iJ7TaO v (Kaj) = h,(a) = < Ov}

It follows from Ya E U for 6 < wi and the weak normality of U that

ZI = {a E Z sup(I) = hv (a)} E U

Note that Ic is closed in hv (a) for all a E ZI. Further hv (a) = On n MM , for all a E Z1. (d) For all a E ZI and A E 9(h(a)) n M

V;. there is an q(A) = ;q(A) < hv(a)

s.t.

I - r(A) C A or

I, - _(A)

9 hv(a) - A.

(e) We set Z2 = {a E Z1 a E Y for unboundedly many 6 < a}. It follows from the weak normality of U that Z2 E U. For a E Z2 let

W= 6 < ala E Y}. Then

sup(Ws) = a and eO is a limit for all 6 E W, a E Z2. Further

for 6 E W, a E Z2 we have: hv(,(a) = fA,(a) = o,O

= On n Mv-> and

TO," (Eo ) = ov~ E Iv. Thus for a E Z2 set

I {

= (),eo E 16 E

Ws}. Then I*' C I, c On n M, is unbounded in Iv and I*v U {hv(a)} is a set

of fixed points of the mapping n*i *

a;0,O,* "

For a C Z2 we have: For all A E 9(h(a)) n Mc* there is an q*(A) = q (A) < h,(a) s.t.

I* - q* (A) C A or

I*Va - q* (A) C hv(a) - A.

Let

D*= {val < ol}.

[Recall that we write v6 for vA .] To reach a contradiction it again suffices to show:

(b') For all A E ,P(v) n K there is a 5 < v s.t. D* - Aor D* - C v- A.

Thus let A C 9(v) n Klv*. Wl.o.g. there exists for all a E Z2 an A, E M * s.t. n7 (Aa) = A. Then for a E Z2 there is an r77 = r*(Aa) < h, (a) s.t.

ia -a A or

*V~ - 7 C h, (a)) A,



ULAM'S PROBLEM ON w1

737

The continuity lemma applied to h, yields a < v s.t.

Z3 = {a E Z2 a < he(a)} E U Further suppose w.l.o.g. that

Z4= { E Z31I*v- - A} E U

Then in particular 7 "(I*' - rq) C A for a E Z4. To show (0') it suffices to prove:

For v6 E D* s.t. < v6 there is an a E Z4 and a r E I*' s.t. 7r () = v*.

But 7 v (hv (a)) = v5 for all a E Z4 - (6 + 1). Therefore it suffices to show:

hv6(a) = On n MC EI* vfor an a E4(+).

But this holds even for U-many a E Z4, since hv, (a) E I*v if a E Y n Z2. -

Building on this proof we can infer the same implication concerning measurables in K if we assume the existence of a sufficiently good weak*-saturated filter on (01.

THEOREM 4. Suppose there is an co2-weak* -saturated filter on wi. Then w2V is a stationary limit of measurable cardinals in K.

PROOF. Let F be an w2-weak*-saturated filter on co1 (thus F 2 , ). In particular F is weakly saturated, thus (DIF) and -' (THy) hold [Lemma 2.3 and 2.7].

We redo the proof up to (++). Here we get using the wi-completeness of F:

(++) For all 2 < w92 s.t. cf() = ow we have fA =F hyv.

Let D' = {vi Icf (A) = w} C D. Construct gv, for vA E D' as before, where now for v = VA

X, = Cv((v,

i < w))n C'.,.

n {a <wi )llf(a) = h,,(a)} E F.

Then g,, <F h,, for all v2 E D'. We now only have: For va E D' there is an r() < 2 s.t. {a < wig, (a) < h r(,)(a)} E F+. Let < i1 and D" C_ D' - v be stationary in (02 s.t. r(2) = for all vA E D". Then for v E D"

X;= C/', n (a E Xvlg(a) < h1v(a)} F+. We now define partial functions 9, : X' -- wi for v E D", where 9,(a) is defined as before for a E Xv.

Then:

(#) (~, v E D") is eventually constant mod(F).

[Otherwise we can show (TH*+)

as before, where (TH*+) has the expected mean- ing, i.e., we consider functions with domains in F+. Now we argue exactly as in Lemma 2.7 to show that F weak*-saturated implies (TH*+), contradiction.]

Thus we have for w.l.o.g. all v,z E D", v 5 z that

{a E dom(g9) n dom(g',) g, (a) Z 9, (a)} E I(F).

Let ( 7, l < W2) C D" be strictly increasing with limit A = 2A,2 s.t.

(##) X x'

is stationary in wi. Y<C02

[Such a sequence exists by assumption. This is the only passage in the proof where "F weakly saturated" doesn't seem to be sufficient.]




We write vy for v , y < 02, and let v = v,2 . We set

D* = {vy < 2, y limit}.

As before one shows using (#) and (##):

(0) D* induces a K-ultrafilter on v.

Let U be the induced ultrafilter. The well-foundedness of Ult(K, U) follows as in chapter 4. Thus v is measurable in K, contradicting v E D. -

Applying Lemma 2.6 we get the following corollary improving the lower bound "there is a measurable cardinal" of [5]:

THEOREM 5.

(UP) _con

"there is an inaccessible stationary limit of measurable cardinals."

REFERENCES

[1] C.C. CHANG and J.H. KEISLER, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, 1990.

[2] 0. DEISER, Commutativity in the core model theory MO, submitted.

[3] , Untersuchungen iiber das Kernmodell fiir Mafle der Ordnung Null, 1999, Dissertation an der Ludwig-Maximilians-Universitiit Miinchen, Munich.

[4] H.-D. DONDER, Families of almost disjoint functions, Contemporary Mathematics, vol. 31 (1984), pp. 71-78.

[5] , On wo -complete filters, Logic Colloquium '90 (Helsinki 1990), Lecture Notes in Logic, vol. 2, Springer, Berlin, 1993, pp. 62-65.

[6] H.-D. DONDER, R.B. JENSEN, and B. KOPPELBERG, Some applications of the core model, Set theory and model theory, Proceedings, Bonn 1979, Lecture Notes in Mathematics, vol. 872, Springer, Berlin, 1979, pp. 55-97.

[7] H.-D. DONDER and P. KOEPKE, On the consistency strength of 'accessible' Jonsson cardinals and of the weak Chang conjecture, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 233-261.

[8] H.-D. DONDER and J.-P. LEVINSKI, Some principles related to Chang's conjecture, Annals of Pure and Applied Logic, vol. 109 (1989), pp. 39-101.

[9] M. GITIK and S. SHELAH, Forcing with ideals and simple forcing notions, Israel Journal of Mathe- matics, vol. 68 (1989), pp. 129-160.

[10] R.B. JENSEN, Measures of order zero, handwritten notes, 1989. [11] , Innere Modelle and grofle Kardinalzahlen, Jahresbericht der Deutschen Mathematiker-

Vereinigung Jubildumstagung, 100 Jahre DMV (Bremen, 1990), B.G. Teuber, Stuttgart, 1992, pp. 265- 281.

[12] A. KANAMORI, Weakly normal filters and irregular ultrafilters, Transactions of the American Mathematical Society, vol. 220 (1976), pp. 393-399.

[13] - , The higher infinite, Perspectives in Mathematical Logic, Springer, Berlin, 1994.

[14] J. KETONEN, Non-regular ultrafilters and large cardinals, Transactions ofthe American Mathemat- ical Society, vol. 224 (1976), pp. 61-73.

[15] P LARSON and S. SHELAH, Bounding by canonical functions, with CH, to appear. [16] S. SHELAH, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer,

Berlin, 1998.

[17] A.D. TAYLOR, Regularity properties of ideals and ultrafilters, Annals of Mathematical Logic, vol. 16 (1979), pp. 33-55.

[18] , On saturated sets of ideals and Ulam's problem, Fundamentae Mathematica, vol. 49 (1980), pp. 37-53.

[19] WH. WOODIN, Large cardinal axioms and independence: The continuum problem revisited, The Mathematical Intelligencer, vol. 16 (1994), no. 3, pp. 31-35.



ULAM'S PROBLEM ON w 739

[20] - , The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and Its Applications, vol. 1, Walter de Gruyter, Berlin, 1999.

[21] M. ZEMAN, Inner models and large cardinals, de Gruyter Series in Logic and Its Applications, vol. 5, Walter de Gruyter, Berlin, 2002.

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Canonical Functions, Non-Regular Ultrafilters and Ulam's Problem on ω1