partition properties and well-ordered sequences

Annals of Pure and Applied Logic 48 (1990) 81-101 North-Holland

81

PARTITION PROPERTIES AND WELL-ORDERED SEQUENCES

Steve JACKSON Mathematics, 253-37, California Institute of Technology, Pasadena, CA 91125, USA

Communicated by T. Jech Received 13 April 1989

0. Introduction

We work throughout this paper in the theory ZF+ AD + DC, with the additional assumption V = L(R) used at times in extending our results to general Suslin cardinals. We assume the reader is familiar with the basic aspects of determinacy theory and, at times, with the theory of L(R) under AD (as can be found in [13]). We have, however, abstracted what we need from this theory in the next section. The reader not familiar with this theory may take these facts on faith, or restrict our discussion from general Suslin cardinals to more familiar territory such as within the projective hierarchy.

We concern ourselves with the possible lengths of well-ordered sequences; either of sets of reals from a given pointclass, or of subsets of a given ordinal. To motivate the discussion, consider the problem of determining the possible lengths of one-b-one sequences; either of subsets of a given cardinal K, or of subsets of reals from a given pointclass. In the first case, the problem bears directly on the GCH in HOD. Thus, if a(~) = the supremum of the lengths of the l-l sequences ,of subsets of K, say, then 2” as computed in HOD, ( = the sets hereditarily ordinal definable with x as a parameter) must have value <8(~). This follows immediately since HOD, k AC. In fact as (assuming AD + V = L(R)) every such sequence is in HOD, for some x E R’, a(~) = sup,,n [(2”)+ of HOD,]. The natural conjecture here is that 8(~) = K+ for all K < 0 = the supremum of the lengths of the prewellorderings of R. The result is not difficult to verify (see Section 5) for K a Suslin cardinal. For K a Suslin cardinal, this result is due to Moschovakis [12]. Also, Martin verified this for K = w,. Our Theorem 5.1 generalizes this, giving a new proof for K = w, and establishing the result for all K < w,.~ as well. When the sequence consists of subsets of R from a given pointclass, the problem is related to that of determining the possible number of equivalence classes of an equivalence relation from a given pointclass. For

example, (see Section 4 for a discussion), from a result of Harrington [l], it

follows that if every one-to-one sequence of E: sets has length <w2, then every &equivalence relation either has perfectly many or <w2 many equivalence

0168~0072/90/$03.50 @ 19SElsevier Science Publishers B.V. (North-Holland)

82 S. Jackson

classes, working now in ZFC + AD ‘(Iw) Harrington and Sami [3] have, in fact, .

established this result for ni equivalence relations by a different route. This seems to be some evidence for this result about E$ sets being true. While we are not able to establish this result for E:, in Corollary 4.3 we get it for 4. Kechris had already shown (see [8]) that there was no o,-sequence of disjoint & (in fact E:) sets-a result which generalizes to Ei,, and Sk,,, as well.

A related question (the relationship is made more precise in Section 4) concerns the possible lengths of increusing subsets of R from a given pointclass (increasing with respect to containment). A result stated in [6] claimed that for K

a Suslin cardinal, there is no increasing K+ sequence of sets each in S(K) = the rc-Suslin sets. The proof there, however, contained an error. We amend the proof here, but require K to be either a successor or a regular limit. In view of this result, a natural question is where in the Wadge hierarchy do we pick up a pointclass from which we do get a K + increasing sequence? In Theorem 2.1 we show that this pointclass lies considerably beyond S(K). Specializing this to the projective hierarchy we get that the least pointclass r for which we have a r prewellordering of length S:, lies considerably beyond J$,, Z&.

In the first preliminary section we define the basic notions we will be using and recall some general facts about Suslin cardinals. We also define the pointclass r, for Suslin K of cofinality # o, which measures where in the Wadge hierarchy we pick up a K+ increasing sequence of pointsets. In Section 2 we show that r, is closed under negation, countable unions and intersections. In Section 3 we pose some questions concerning the closure of r, under longer unions and intersections, and prove some results about longer unions and intersections. We also in this section amend the theorem of [6] referred to above. In Section 4 we use partition properties to prove additional results about well-ordered sequences of pointsets. The key result here is a theorem of Martin about the non-existence of long sequences of distinct sets in S(K) fl S(K) (where S(K) = the rc-Suslin sets) for Suslin K with K+ having a certain partition property. Finally, in the last section we also use patition properties to study sequences of sets of ordinals.

1. Preliminaries

We let w” be the set of ‘reals’, i.e., the set of infinite sequences of natural numbers. To each A E o” associate a two player game where I and II alternatively play natural numbers, thereby jointly producing a real X:

I X(O) X(2) * * *

II x(l) x(3) - - - .x = (x(O), x(l), . . .)

We say I wins the game if x E A. The notion of a winning strategy is defined in the usual manner. The axiom of determinacy, AD, asserts that for every A c_ tom, one of the players has a winning strategy for the associated game. The axiom of

Partition properties and well-ordered sequences 83

dependent choices, DC, asserts that every ill-founded binary relation has an infinite descending sequence.

We work throughout in the theory ZF + AD + DC. We refer the reader to [12] for more details on the development of descriptive set theory within this theory. In extending some of our results to the case involving a Suslin cardinal of uncountable cofinality, we will also require the assumption V = L(R), which we

indicate. For sets Xi,..., X,,, a tree T on X1 x X2 x . . . XX,, will be a subset of

(X,XX,X*..XX,)“” closed under subsequence. As usual, we identify (X, x

X2 x . . . x X,)‘” with X;“’ x X,‘O x * . . x X,‘O, so we may think of the ele- ments of T as n-tuples of sequences from the Xi’s. For the tree T, [T] denotes the set of paths through T, i.e. [T] = {(tl, . . . , t,) E Xy x X,0 x * * * x X,0 :

VIE o(t, 1~ rz rm,. . . , t, 1 m) E T}. Also, p[T] denotes the projection of T = {tl E Xl” : 3t,, . . . , f, (tl, . . . , t,) E [T]}. Throughout this paper, the Xi will

in fact be ordinals. We say the set A s coo is K-Suslin if there is a tree T on w X K such that

A = p[T]. We say K is a Suslin cardinal if some A c cow is K-Suslin but not A-Suslin for any k < K. We let S(K) denote the pointclass of K-Suslin SetS.

Much is known about Suslin cardinals under AD, and even more under AD + V = L(R). We refer the reader to [13] for a treatment of the theory of Suslin cardinals under AD + V = L(R). For the convenience of the reader we collect some facts about Suslin cardinals needed for this paper. We number the facts here in the manner in which we will refer to them later.

Assume AD + DC and let K be a Suslin cardinal. Then S(K) is not self-dual, and K+ is regular (cf. [9, Theorem 1.21). Assume now that K is a successor, i.e. K = A+ for some cardinal 3c. Then:

(Al) A is a Suslin cardinal. (A2) ,$(A) has the prewellordering property via norms of length K. S(K) =

?,$(A) and S(K) has the scale property.

(A3) 9~ E s(L) n g(n), where 9$ is the closure of the open sets under complements and well-ordered unions, intersections of length <A.

(A4) The o-closed unbounded filter ~1 on K is a normal K-complete ultrafilter with j&c) = K+, where j, denotes the embedding from the ultrapower by measure p.

We assume now that K is a limit Suslin cardinal of uncountable cofinality. For the first three facts below we require AD + DC and for the last we also need V = L(R), although AD + DC alone will suffice to establish this for K < K' = the first admissible over L(R). We have:

(Bl) K is a limit of Suslin cardinals. (B2) S(K) has the scale property. (B3) There is a o-closed unbounded set of cr< K such that (Y is a Suslin

cardinal of cofinality w, and if A, denotes the pointclass of countable unions of sets from lJB<N S(p), then A, has the scale property with norms /3 < (Y of length sm. Further !?& c S(a) n .$(a).

84 S. Jackson

(B4) There is a measure p on K extending the w-closed unbounded filter and in the ultrapower by ,u, the function f(a) = my+ represents K+ (cf. Theorem 4.9 of [13] and its proof for (B3) and (B4). (B4) in this case is due to Kechris).

Unless otherwise specified, the results of this paper require AD + DC for the case K a successor Suslin cardinal, and the additional assumption V = L(R) for the limit case of uncountable cofinality, although as already remarked, AD + DC alone will suffice for K’s not too large.

We state an amended version of the main theorem of [6] (whose proof contained an error-see Section 3 of this paper):

Theorem. Let K be a S&in cardinal, either a successor or a regular limit cardinal. Then there in no strictly increasing sequence of sets (A, : (Y < K+) such that each A, E S(K).

As an immediate corollary of this (taking K = 6:,-J we have:

Corollary. There is no &,, increasing sequence of sets, each in Zi,.

For K as in this theorem, one of our interests here is how far we can extend this result beyond the pointclass S(K). Specifically, we make the following definition:

Definition. For K a Suslin cardinal, either a successor or a limit of uncountable cofinality, we let r, be the pointclass of A c o” such that there is no strictly increasing sequence of sets (A, : cx < K+) with each A, 6w A.

Here + denotes Wadge reduction. Thus the conclusion of the above theorem may be restated as S(K) E r,. We investigate the size and closure properties of r, in this paper. In particular, we show that r, is considerably larger than S(K). It’s precise determination, however, is still open.

Remark. For K a Suslin cardinal of cofinality o, it is easy to see that the above theorem fails, so r, is only of interest for cof(K) # o.

2. Closure of r, under countable unions, intersections and complements

We make a preliminary definition (recall that K is a Suslin cardinal, with COf(K) # 0):

Definition. We say the tree T on w X K is good if sup{ 1 T,I : T, is well-founded} = K+, where lTxl denotes the rank of TX, and TX is the section of T at X, i.e.,

T, = {(ab, . . . , an-d: (ho, . . . , X,-I), (a,, . . . > a;t-I)) E T).

Partition properties and well-ordered sequences 8.5

Note that good trees always exist. To see this, first note that K+ = the supremum of the lengths of the K-Suslin well-founded relations. Also, the proof of the Kunen-Martin theorem (which asserts that every K-Suslin well-founded relation has rank <K+) gives a tree T on o X K such that for every K-Suslin well-founded relation, there is an x with TX well-founded and lTxl > IRI. So T is good.

In studying r, it is convenient to introduce an auxiliary pointclass r;:

Definition. I=,, is the collection of all A c w” such that there are good trees 7’,, T2 on o X K and a0 < K+ such that for all limit ordinals p 2 LX,, with cof(p) > o, A G,,, Cg, and 1A sw C?, where CF = {x : TX is well-founded and (TX I < p}.

Clearly l=,, is a self-dual pointclass. We first show that r; is closed under countable unions and intersections. For this, it is enough to show that given Ai, K with T good and Ai~w Cg for sufficiently large /3 of cofinality >w, that we may find a good T with ni Ai G,., Ci for all large enough /I of uncountable cofinality, and similarly for IJi Ai. We consider the intersection case first, and fix Ai, T as above. We define T on o X K as follows: We fix a standard bijection of (o~)~ with o*, with x E w” coding x0, x1, x2,. . . E coo, and the maps x+xi continuous. We then set T = {(s, t) :s E [co]“, t E [K]" for some n, and 3x 2 s

(r = (j, ab, . . . , CX~_~) for some j E o and (xirn - 1, (LYE,. . . , a~~-~)) E TJ}.

Intuitively, T is the ‘union’ of the T. Now, for fixed large enough /I < K of uncountable cofinality let ndi be a

continuous reduction of Ai to C$ (recall that K+ is regular). We let JC(X) be the real coding n,(x), n,(x), . . . , etc. So n(x) is continuous and (x(x))~ = nj(x). If x E ni Ai, then (n(X))j = nj(X) E C? for all j. Hence, n(x) E CF as ITI1 = SUpis (Ir,,iJ + 1). AS cof(/I) > w it follows that ni Ai +, CF.

We now consider the union case. We fix a bijection between K<~ and (K<~)<~

SO that each tuple (so, sl, . . . , s,,_J where Si E Km+ is coded by the single sequence (so, . . . , s,-~) E K~. We then define T = {(s, t) :s E [co]“, c E [K]" for some n and 3x1s (t= (to,. . .,f,,,-l) for some tiEKm-i, and vo<jGrn-1 (Xi rm - j, tj) E q)}. For /3 < K sufficiently large of uncountable cofinality, we fix continuous reductions Ici of Ai to C$. As before, let n be continuous with n(x) coding n,(x), n,(x), . . . , etc. Now, T, is well-founded iff at least one of Ti,zi is well-founded, and in this case

lTrl= inf i: &i is well founded

(I &I + (i - 1)).

Since b is a limit, it easily follows that JG reduces UiAi to CF. Hence, r is self-dual and closed under countable unions and intersections. Now, the above proof actually shows a little more. Suppose A E I=; and define

U, z w o x w” by: (w, x) E U, e (w codes a continuous function from o” x ow) A (w(x) E A). Then U, E p; as well (more precisely, the set 17’ defined by

86 S. Jackson

U’(t)@ U(&, z,) is in r=). [To see this, let TI E (w x K)‘~ be a good tree with A cw C$ for large enough p. Likewise, let T2 be a good tree with P cw Cy for large enough /3, where P = {w; w codes a continuous function from w” to o”}, as P is arithmetical. We then have that for large enough j3 that U, q,, C$, where T is constructed as in the first half of the proof above as the ‘union’ of Ti and T2,

and T; is simply obtained from TI. Specifically, let T; = {((zO, . . , z,,_~),

(%, * * * 7 CX~-~)): 32 extending (zO, . . . , z,-~) with z, = w, z1 =X say, and

(&I, - . . 7 z,_~) computes w(x) ]m, and (w(x) ]m, (a,,, . . . , CV,_~)) E T. Similarly for lU,.] Note that (by a diagonalization argument) U, &,lA.

We now state a lemma which abstracts part of the proof of the main result of

M-

Lemma. Suppose A E o”, Tisagoodtreeon WXK, and (A,:cY<K+) isan increasing sequence of sets each reducible to A. Then there is a B E S(K) such that for c.u.6. many /3 < K+, C$ is reducible to A fl B and 1A fl B.

Proof. As the argument is similar to that of Lemmas 2.6-2.9 of [6], we provide a sketch. We fix A, T, and A, as in the statement of the lemma. We let r, denote the pointclass of sets Wadge reducible to A. We view strategies x (for II) as giving continuous functions x : w o + w w in the usual way. We say x codes the set U, = {y :x(y) E A}. Thus, for every B E r,, B = U, for some x E w”. We play the game:

I x

II Y, 2, w

Here, I plays x E ww and II plays (a code of) the reals y, z, w E ow. If one of TX /a; TY r(~ is ill-founded for some (Y < K, and (Y is the least such, then II wins if TX rcx is ill-founded. If both TX, TY are well-founded, then II wins provided

IT,1 > ITxI, z codes the set A,?,, and w E+, -UB<,~,A~. As in [6, Lemma 2.71, a boundedness argument gives that I cannot win the game, hence II wins, say by strategy o E o” (we use here A3, A4 or B3, B4). A similar boundedness gives a c.u.b. C G K+ closed under a, i.e., if LYE C and lTxl < (Y then IT,1 < cy too, where a(x) = (y, z, w). We fix /I in this c.u.b. set, a real i coding A, and a real rig E A, - lJvGBA,. We then have that for all x:

XECF e (TX is well-founded) A w E U,

G (TX is well-founded) A W $ U,,

where a(x) = (y, z, w). As ‘TX is well-founded’ is in S(K), the lemma follows. 0

We now state the main theorem of this section:


Theorem 2.1 (AD + DC + V = L(R) in the case K a limit Suslin cardinal). Let K be a Suslin cardinal, either a successor or a regular limit. Then r, = rK. In particular, r, is closed under negation, countable unions and intersections.

Proof. We first show rK G r,. Let A E r but suppose there is an increasing sequence of sets (A, : a < K+) each reducible to A.

Now S(K) E rK as otherwise, from the previous lemma, we would have for c.u.b. many /l< xc+ that CF is in S(K), where T is any good tree on o X K. This, however, contradicts the result of [6] ( see Section 3 of this paper) that for sufficiently large /3 < K+, an S(K)-complete set is reducible to Cg (this is where we use the additional assumption that K is regular in the limit case). From the closure properties of rK and the remarks after the proof of the lemma, we have that the pointclass S(K) v r,, is contained in l=,,, where U, = {(w, x ) : (w codes a continuous function from w w to w “) A W(X) E A}. It follows that for some good tree on o X K and sufficiently large @ < K+ that S(K) v r,, CW Cg. However, from the lemma above, for a c.u.b. set of p <K+, CF<, S(K) A pa. This is a contradiction since a set in S(K) v r,, parametrizes S(K) v r,, and hence can not be in (S(K)V G)".

We next show r, G rK. Let A E r,. If T1 is a good tree on w X K, then since A E r,, for all sufficiently large fi < K+ we must have C? 4 r, and so any set in r’, is reducible to C?. Since S(K) c pK as well, from the proof of the closure properties of p’ it follows as above that for some good tree T and sufficiently large /l that S(K) v FuA SW (2;. From the lemma, COGS S(K) A r,. This contradicts S(K) v i;, &S(K) A &. 0

As a corollary to this theorem and its proof we have:

Corollary 2.2. r, = rK = {A E w o : for some good tree T on w X K and sufficiently large /3 of uncountable cojinality, A sw Ca = {A E w‘” :for all good trees T on w X K and for all suficiently large p < K+, A, 1A SW CBT).

Proof. Suppose T is a good tree on w X K and A SW C$ for all large enough /I of uncountable cofinality. From the proof of the closure properties for r, it follows that for some good tree T1 on w X K and all large enough /I of uncountable cofinality that S(K) v ruA cW C?. If, however, it were not the case that for all large enough j3 of uncountable cofinality that 1A sw C$, then by Wadge we would have a K+ increasing sequence of sets each in r,. By the lemma, this gives a c.u.b. set of p for which C? sw S(K) A fA, a contradiction. This establishes the equality of the first three sets above.

Finally, if A E r, and T is any good tree on w x K, then for all large enough fi < K+ we must have A, -A SW CF as otherwise we would get a K+ increasing sequence of sets each in r, or each in fa, contradicting A, 1A E r,. Cl

88 S. Jackson

As a special case of Theorem 2.1 we obtain:

Corollary 2.3. There is no a1 Zn increasing sequence of seb, each in %?(I&,) = the smallest pointclass containing .?&, and closed under complements, countable unions and intersections.

3. Longer unions and intersections

In this section, we pose some questions concerning the closure of r, under longer unions and intersections and prove some related results. We also amend the statement and proof of the result in [6] that S(K) G r,. We first introduce some pointclasses.

We recall that for 6 a cardinal, B6 is the smallest pointclass containing the open sets and closed under well-ordered unions and intersections of length ~6.

We introduce an effective version of this. Let r be a pointclass. Suppose T is a well-founded tree on 6 with each node a = ((yg, . . . , an) E T labelled with an index a, E (0, l}, and a set A, E r is assigned to each a which is a terminal node of T. By induction on T we define A, for all a in T by:

A,=U {A,,: a’ is a one point extension in T of a} if a, = 0,

= n {A,, : a is a one point extension in T on a} if a, = 1.

Definition. We say that A E eff B,(r), the effective B6 sets starting with r, if for some well-founded tree T on 6 with indices a, and sets A, E r as above, A = A@.

We may further stratify the sets in eff B6(r) according to the rank of the tree T. The first w levels arise frequently enough to warrant a separate notation.

Definition. A E eff U:(r) if there are sets A,, ,,..., anj E r for al, . . . , a,, < 6 with A = AB = IJlrlCGAn, where A,, = n,,, A,,,,,, etc. Similarly we define eff n;(r), starting with A = AB = nluICGAn,. We also define the non-effective versions, for example, A E U:(r) if A can be written as A = kJorCGAa where each A, E n;-‘(r), etc., where U:(r) = n”,(r) = r.

We are primarily interested in the cases where 6 = K and r c S(K) or r G S(K). We state some conjectures. Recall that K is a Suslin cardinal of cofinality # w.

Conjecture 1. %IK E r,.

Conjecture 2. BK = eff ciq(r) for any r E S(K) (equivalently, for r = clopen sets).


A particular instance of this is:

Conjecture 3. UC(S(K)) = eff CI”,(S(rc)) and f%(s(~)) = eff R(~(K)).

We note that if IZ is even ~;(S(K)) = n:-‘(S(K)) and for n odd ~“,(S(K)) =

IJ;-~(S(K)) as S( ) f 1 d d K is c ose un er K-unions which follows easily from the coding

lemma. In the event Conjecture 2 fails, we may replace Conjecture 1 with

‘eff %&(S(K)) G r,.’ Motivation for this comes from the observation that if T~(CIIXK)<~ is a good tree, then each Cz= {x : T, is well-founded of rank < a} E eff 9&(r) where r = clopen sets. Thus, there is a K+ increasing sequence of sets of each in eff 9&, and it seems reasonable that this is best possible.

The next two results verify some instances of Conjecture 3.

Theorem 3.1 (AD + DC + V = L(R) ‘f 1 K is a limit). Let K be a Suslin cardinal

with COf(K) # co.

(a) Suppose S(K) is not inductive-like, i.e. S(K) is not closed under VR. Let r be the scaled pointclass closed under Vu such that S(K) = ZIRr (cf. [13, Theorem 4.3b]). Then n;(S(K)) = eff n:(F).

(b) Suppose S(K) is inductive-like. Then ~:(S(K)) = eff ~:(S(K)).

Proof. We prove (b), the proof of (a) being essentially the same. So, assume A=U ,<,A, is given, where each A, can be written as a K-intersection of sets in S(K). Let U~o”x w” be universal for S(K). Let P E S(K) be complete and [I.)] a norm on P mapping onto K. Let P, = {X E P: llxll = a}. If a sequence (B ~ : LY < K) of sets each in S(K) is given, we say the real z E o” codes the intersection B = n,,, B, if Vx E P 3y E w” U,((x, y )) and Vx E PVy (UJ(x, y))+ U, = B,,J. That is, z codes via U a choice subfunction of the function which assigns to x E P the set of y coding B,,,,,. By the coding lemma, such a z exists for any such (B, : a < K). We now consider the game where I plays x, II plays z and II wins provided x $ P or x E P and z satisfies:

(i) Vx’ E P (llx’ll s llxll * 3ylu,(x’, y)) (ii) Vd EPVy (Ib’ll s x A lU=(x’, y) + y codes a sequence (ByB: p < K) II II

with f&, Bj = AIIx,II). Since Z: subsets of P are bounded in the norm, it follows from the coding

lemma that II wins the above game, say by strategy o. We then have for all WEOY

WEA e ~~<K(wEA,)

@ 3ff<KvXEP,vyvp<K(lU,(,)(x,y) j WEB;)

e ~~<KV~<K{VXE~,V~(~U,(,)(X,~) j WEB@}

If we let A,,, = {w :VX E P, VY W&X, Y) + w E ByS)}, then A,,, E S(K) and A = lJ,<, f&&p, so A E eff L&s(K)).

90 S. Jackson

Theorem 3.2 (AD + DC). Let K be a S&in cardinal. Then ~&(S(K)) =

effUXS(K)).

Proof. We assume n is even. We take n = 4 for notational simplicity. We fix A E ~:,(S(K)). Let U E w” x ww be universal for S(K). Let WO E o” be the standard set of codes for countable ordinals, a #-complete set, and let B c w” x o w x w o be universal for El. Let II-11 be a #-norm on WO. We define inductively on m the set of codes for a set S E IJ~,(S(K)) or n:,@(K)). If, say, S E ~:,(S(K)), then we say o1 is a code of S if for some sequence (S,, (Y < wl) with each S, E n:,-'@(K)) with S = U,<,, S, we have:

(i) Vxi E WO 31 y B&l, y), (ii) Vxi E WO Vy (B,,(xi, y) + y is a code of S,).

The case SE n:,@(K)) is treated similarly. To say “y is a code of S,” for S, E S(K), we simply mean U, = S,.

Using induction, the coding lemma, and uniformization for J$, it follows easily that every such S has a code. We fix a code crl for A. We use the notation: if u is such that Vx E WO 3! y A&, y) then f(u, X) denotes the unique y such ‘that

A& Y )e We recall now a notion and some results of Kechris (see [S]). For each

countable ordinal 5 < w1 there is a notion of category on PE = {x E WO: 11-x 11 = g}.

To be specific, we may identify {x E WO: llxll s g} with 5” = the set of all o-sequences of ordinals <g. Since f is countable, the usual definition of category on o0 carries over to E”. Also, Ps is identified with the set of all onto sequences from E”-a comeager set. This defines the notion of category on 5”. We recall from AD that the well-ordered union of meager sets is meager-which clearly extends to category for PE for any 5 < oi. We let “V*x E PE” abbreviate the expression “for non-meager many x E PE”.

We have for all w E o”:

w EA e 3c-q < o1 V/*x1 E Pa, (“x is in the set coded by f(a,, xi)“)

e 3x, < wi v*xi E Pa, vC.rz < Wl v*x* E Pa2

(“x is in the set coded by f(f( oi, x1), x2)“)

e 3.x1 < WI v*x1 E Pa, VW* < 01 v*xz E P,, 5x3 < Wl v*x3 E P,,

vCr4-C 01 v*x‘J E Pa4 (321 32,323 32, B(a,, Xl, ZJ A B(z,, x2, ZJ

A B(Z2, x3, z3) A B(Z3, x4, 24) AXE U,,)

e 3al<o,vCY3<013~3<01v~4<<01[v*X1EP,,...

v*x‘j E Pad 321. * *3Z4(B(% XI, ZI) A * * - A B(Z3, X4, Z4) AXE u,,)].

We have used the additivity of category in obtaining the last equivalence. Now, S(K) is closed under the V* quantifier (which follows easily from additivity if K is a limit Suslin cardinal, and in the successor case follows as for the projective


hierarchy-see [12]). Hence, the last expression above shows that A E

eff U4o,G%>). 0

The above theorem lends credibility to Conjecture 3. One could attempt to generalize this proof to unions and intersections of lengths 6 > o1 using the notion of generic codes as developed by Kechris and Woodin [ll]. However, the more general notion of category lacks the additivity (i.e. the union of meager sets is meager) necessary for the proof.

We take this opportunity now to amend Theorems 2 and 3 of [6]. We state the amended theorem:

Theorem 3.3. Let K be a Suslin cardinal, either a successor or a regular limit.

Then S(K) E r,, i.e., there is no K+ strictly increasing sequence of sets each in

S(K).

Proof. We consider the case S(K) not inductive-like as otherwise the theorem follows from Theorem 1.2 of [6]. Let F be the scaled pointclass closed under V” such that S(K) = 3°F. Let P E r be complete, and 1(-J] a p-norm on P. Also, let U E w w x o o x o o be universal for binary relations in r and it?, M be universal for unary, binary relations in r, S(K) respectively. We define the set u E ~Ox~Ox~Oby:

qz, x, Y> e x> Y E p * Vx’, VY’ [Ilx’ll = lbll A IIY’II = IIY II

+ U(z, x’, y’)].

Now, 0 E r E S(K) as r is closed under disjunctions by [14] (cf. Theorems 2.1 and 3.2).

We fix A E S(K) and write A = n a<K A, where each A, E i? We let cr E ow be a code (via M) for a S(K) choice subfunction for f : K+ R where f(a) = the set of codes (via &f) for A,. We define a relation R by:

R(z, n, ~1, ~2) - ~1, ~2 E P A oI(z> YI, ~2) A 3~ 3~2

W,(YI> WI) A Mo(~2, ~2) A x E &I Ax E M,,.J.

Clearly R E S(K), so let s c (w X w X w X 0 X K)<O be the tree of a scale on R. Also, R is easily seen to be in S(K) by rewriting the last conjunction using “VW1 VW,“.

We construct our ‘Kunen tree’ T now as follows: let a : w x w x o x co-+ o be a bijection monotone in all arguments. We set T = {(s, q) : for some n, s E eY,

tl = (rlo, * * * , qn-l) E K” A 32 3X gy,, . . . ) Y, ((2, x> extends s A W, x, yi, Y,+~)

holds for all 1 s i s n - 1 A Q(,,~,~,~) =yi(j)provideda(l,i,j,O)<nr\forlSiS

n - 1 (z ]ZP X ]l7 Yi+l rr, (rla(2,i,i+l,O)7 . * * j q,(z,i,i+l,l))) ES for all 1 such that a(2, i, i + 1, I) <n}.

T is thus the Kunen tree for the relation R. It is clear, then, for all z, x that Rt,,,) = {(yI, y2) : R(z, x, y,, y2)} is well-founded iff T(_) is well-founded. Clearly

92 S. Jackson

T is good as ITtz,x,I 5 the length of R,,,,,, and we may find r relations well-defined on the ordinals as coded by P having lengths arbitrarily large below K+.

We define an ordinal cu, < K+. We fix ordinals /&, & < K. We let

Here R,,,,, l/l1 denotes the restriction of R,,,,, to codes for ordinals <fir. Each

VB,.B~ l S(K), as each VB,.B~ can be constructed from <K unions and intersections of sets each in S(K) A S(K) c S(K). The fact that S(K) is closed under <K

intersections follows from the coding lemma (see [2] for a similar proof). Now, for each (2, x) E Vs,,s2, R(,,,), and hence T(_) is well-founded. Since VB,,B2 E S(K), it follows by a standard tree argument that sup IT(z,x,l : (z, x) E

Vsl.s2~ = WI, P2) < K+. We let

a$ = (,,s;l& Wb m) < K+.

We now choose z such that a, is well-founded of rank LY > ao. Let f(x) = (z, x). Then for x EA, IR(z,xjI = lU,l> a, and so ITfcxJ > a. If x $A, then for some

/%, Pz< K, f(x) E VS,JJV and hence ITf(x,I < a. Hence, f’reduces A to Cz, and we have shown that A sw C,’ for all a 3 ao. The remainder of the proof is now as in

PI. •I

4. Sequences of distinct sets from a given pointclass

Recall that K is a Suslin cardinal, either a successor or a limit of uncountable cofinality. Also, r, = {A : there is no K+ increasing sequence of sets each reducible to A}. A related pointclass is r: = {A: there is no K+ sequence of distinct sets each reducible to A}. Clearly r: E G, and r: is closed under complements.

Conjecture. r: = r,.

We are unable to prove this conjecture, although we give a result due to Martin, in this direction. Martin’s result implies that if K+ has a certain partition property, then A= S(K) fl S(K) E r;. We first define the relevant partition property.

Definition. Let 6 be a cardinal. We say the partition ‘9’: [S]“+ (0, 1) of the increasing functions from 6 to 6 is clopen if for all f E [ 61’ there is an (Y < 6 such that Vg E [&I6 (f ]a = g ra j S(f) = 9(g)). We say 6 has the cfopen strong partition property if for all clopen partitions 9 : [ 61’ + { 0, 1) there is an X G 6 of size 6 and i E (0, l} such that S(f) = i for all f E [Xl”.


We will establish below a result of Martin that for K a successor Suslin cardinal with the strong partition property, K+ has the clopen strong partition property. We note that from a theorem of Kunen, K+ cannot have the full strong partition property (see [lo], Corollary 17.41).

Theorem 4.1 (Martin). Suppose 6 has the clopen strong partition property and T is a pointclass such that there is no 6 increasing sequence of sets each in n;(r),

f%(f) or Uk(f? f or some 6’ < 6. Then there is no 6 sequence of distinct sets

from T.

As a corollary, we obtain:

Corollary. Suppose K is a Suslin cardinal with cof(K) # w and K+ has the clopen

strong partition property. Suppose aLso that U”,(S(K)) c T, (for n even, and

n:(S(K)) E r, f or n odd). Then nE_‘(S(K)) E: r: (UE-‘(S(K)) G r: for n odd).

Proof. Apply Theorem 4.1 with r = n:-‘(s(K)), and 6 = K+. Our hypotheses and the closure of r, under negations give that n:(A) = n:(A) where A = r or

i: Hence n:(A) G n:@(K)) G G. Also, UZ(p) c U~K(S(K)) G G. So, the hypothesis of Theorem 4.1 is satisfied, so r c r:. Cl

Concerning the clopen strong partition relation, we have:

Theorem 4.2 (Martin). Assume AD + DC. Suppose K is a successor Suslin cardinal with the strong partition relation. Then K+ has the clopen strong partition relation.

From these results we immediately have:

Corollary 4.3. If K is a successor cardinal with the strong partition property, then A = S(K) f-l S(K) E I-:.

Corollary 4.4. There is no w2-sequence of distinct Ai sets.

This follows immediately as o1 has the strong partition property (a result of Martin-see [lo, Theorem 22.11). In fact, by recent results of the author, all the S:,,, have the strong partition relation. This full result, however, has not been published. The bulk of the analysis will appear in [4], and the complete analysis for S: will appear shortly in [5]. Granting this result, we have:

Corollary 4.5. There is no @,-sequence of distinct Ai, sets.

As we remarked in the introduction, for the case of ‘disjoint’, the result was already known-due to Kechris (see [B]).

94 S. Jackson

Proof of Theorem 4.1. We suppose the theorem fails, and fix a sequence

(A a : a < S) of distinct sets each in r We consider the partition 9 where we partition f E [ 61’ according to whether or not AfCO) $ nlrgrgCf) A,, where /3(f) is the least ordinal ~6 such that r)1~0+Cfj A, = fJIGBrvAB for all y 5 /3(f). Such an ordinal must exist from our hypothesis. We claim that on the homogeneous side of the partition, the property stated holds. We suppose not. Then by restricting our attention to a set homogeneous for the contrary side we may assume for all f E [a]” that AfCO) z nlSBGBCf)AB. We let B, = lJB3nAB. From our hypothesis it follows that there is an ordinal a0 < 6 such that B,, = B, for all CY~, (~~3 ao. Restricting our attention to ordinals ~=ab we may now assume that any x E lJnCGAU is in fact in cofinally many of the A,. We fix (Ye, cu2 < 6 and x E w” with x #A,, and x EA,,. Since x EA,*, x is in cofinally many of the A,. We choose f E [&I6 with f(0) = (Y~ and x E AfCaj for all (Y > 0. (Note that 6 is regular.) This, however, contradicts AfCo) 2 lJISBGBCf)AB. Restricting our attention now to sets homogeneous for the partition we may assume that AfCo) 2

n IsBrScfj A, for all f E [6] 4 In particular, A, 2 (-&A, for all (Y < 6. If we let

& = f-7,>, A, it now follows that the B, form a strictly increasing sequence, contradicting our hypothesis. Cl

Proof of Theorem 4.2. We fix a Suslin cardinal K = A+ having the strong partition property. Now, the same arguments that show that the odd ~3: are measurable also show that the w-c.u.b. filter on K is a normal measure (this also follows from the strong partition relation on K). Recall that from (A2) we have 3c is a Suslin cardinal, S(K) = ?$(A), and S(L) has the prewellordering property. This is enough to make Martin’s argument for showing that Ai is closed under <S: unions and intersections go through for A = S(n) fl S(n) and A. These results in turn suffice to carry out the Kunen tree construction. We get a tree T on o X K such that for any f : K + K there is an x E o o with T, well-founded and such that f(cu) < ITx rcxl for all LY in an w-c.u.b. set. In particular, the ultrapower of K by the w-cofinal normal measure on k-c, k(K), = K+.

We prove a lemma from which the theorem follows easily.

Lemma. For any increasing F : K+ + K+ with F(0) > K and 6 < K+, there is an increasing f : K +- K such that F(a) = jw (f) (K + a!) for all 0 S (Y G 6. Furthermore, if X E K is unbounded and F has range in jo(X), then f may be chosen to have range everywhere in X.

Proof. We fix X s K unbounded, F : K+- jo(X) increasing, and 6 < K+. We fix a well-ordering W of K with rank 1 W I> F(6). Notice that by normality the rank of W, I W 1, is represented by the function /3 + 1 W r/31. We fix cu, < K such that the function 6 + I W I/~(cIE~)~ represents 6, where 1 W r/3(cxo)I denotes the rank of a0 in W r/3. We define two functions Gr, G2 from 6 into K. We set G,(y) = that CY < K such that IW(cu)l = K + y. Hence, the function p+ IW ~P(cz)[ represents K + y.


We set G,(y) = that a < K such that 1 W(a)1 = F(y - K) (here y - K is the ordinal

y’ such that K + y’ = y, defined for y 2 K). We let C’r K be c.u.b. and closed

under the functions b,(a) = IW ~CY( and b,( (u) = the next element of X after a.

We let C s K be the set of closure points of C (i.e. (Y E C 3 a is the &h element

of C). We define now f(cu) for a < K by induction on a as follows:

Casel: aissuchthatforsome/IEC,p<a!, andy<p,a=IWlP(y)l. Bythe

choice of C, this 6 must be unique, and hence y is uniquely determined as well.

We let y’ = G,(IW(y)l).

Subcase (i): y’ < /I, I W r/?(y’)l E X, and IW r/3(y’)l~ SUP~,<~ (f(cu’) + 1).

We then set f(a) = IW r/3(y’)l.

Subcase (ii): Otherwise. We set f(a) = the next element of X >

sup&A@‘). Case 2: cy not of the above form. Proceed as in subcase (ii).

We claim that this f verifies the lemma for the given F and 6. Clearly f is

increasing and has range in X. So, fix a 6 < 6 to show jo(f) (K + 6) = F(6). Fix a

y< K such that K + 6 = IW(y)l. Hence, K + 6 is represented by the function

P+ IW VW- A s K + 6 3 K, for almost all p, IW r/?(r)1 Z= p. Hence, for almost

all /?, IW IP(r /3, 7, satisfy the hypothesis of Case 1. Let y’ = G2(lW(v)l). Note

that for almost all p, JW rP(y’)l> /3 as F h as range in the ordinals >K. We claim‘

that for almost all /3, (W rP(y’)l falls under subcase (i) above. Note that for

almost all /3, lW rP(y’)l EX as F has range in jw(X). If the claim fails, then for

almost all /3 < K there is a y&?) < /3 with y*(p) <_, 7, (W rP(y2)l 3 /3, and such

that if y;(P)= G2(lW(yz)l), then either y;(/3)aP or IWr/3(y;(/?))I $X. Now, for

almost all /I, y&3) is equal to a fixed y2< K. So, if y;= IW rP(y2)l, then for

almost all /3, IW r/3(y;)l $X. This, however, contradicts the fact that F(W(y,) -

K) ej,(X), and verifies the claim. Now, jo(f) (K + 6) is represented by the

function

V) =fW TPW = W’ rP(r’)l = IW ~PGWWl))l

= lw ~/%Gz<K + @)l .

However, the function /3+ IW ~/?(G*(K + 6))l represents F((K + 6) - K) = F(6).

Hence, j,(f)(~ + 6) = F(6), which completes the proof of the lemma. 0

We return now to the proof of Theorem 4.2. Let 9’ be a clopen partition of the

increasing functions from K+ to K+. We define the auxiliary partition ‘9” : [K]“+

(0, 1) by: S’(f) = 0 iff P&,(~)(K + a)) = 0 (by “(jo(f)(K + a))” we mean the

function F defined by F(a) =jm(f)(~ + a)). We let X G K be homogeneous for

P’, say P’(f) = 0 for all f E [Xl”. We claim that jW(X) - K is homogeneous for

9. Indeed, if F :K++~,(X) - K is increasing, we let 6 <K+ be such that

9(G) = P(F) for all G with G YS = F 18. By the lemma, there is an f : K-X

increasing such that jm(f)(~ + a) rb = F 16. Hence, s(f) = P’(jo(f)(~ + a)) =

0, and we are done. Cl

96 S. Jackson

5. Well-ordered sequences of sets of ordinals

A question related to the topics discussed concerns the lengths of sequences of subsets of a given cardinal. Specifically, we have the following conjecture, perhaps best attributed to the Cabal:

Conjecture. Assume AD + V = L(R). Let K < 0 be a cardinal (where 0 = the supremum of the lengths of the prewellorderings of the reals). Then any well-ordered sequence of subsets of K must have length <K+.

As we mentioned in the introduction, this question is related to the GCH in HOD. As we pointed out there, if &(K) = the supremum of the length of the sequences of distinct subsets of K, then d(~) = sup,,n [(2”)+ computed in HOD,]. In particular, if the conjecture fails, then the GCH fails in HOD, for a cone of x. We may view the conjecture as a sort of ‘bold-face’ version of the GCH conjecture for HOD.

If K is a Suslin cardinal, the result is known, and follows easily from the fact that K+ is measurable. For K a successor Suslin cardinal, the measurability of K+

follows as in the projective hierarchy. If K is a limit Suslin cardinal with cof(K) = w, then V'S(K) has the scale property with norms of length K+ from which the measurability of K+ again follows by standard arguments.

Finally, if K is a limit of uncountable cofinality, the measurability of K+ follows by combining the projective hierarchy arguments with an analysis similar to Lemma 4.10 of [13].

For K between Suslin cardinals, the problem becomes more difficult. Martin first showed that the conjecture is true for K = co, by an argument using the particular nature of wl. We prove a somewhat more general result which provides a more direct proof of this result as well.

Theorem 5.1. Suppose K is a cardinal with the strong partition property. Let K,,

denote the ultrapower of K by V, = the n-fold product of the o-cojinal normal measure on K. Suppose K, = IC+“. Then there is no (K,)+ sequence of distinct

subsets of K,,.

Corollary 5.2 (Martin). There is no w,+~ sequence of distinct subsets of CO,.

Likewise, using the strong partition relation for S:, and the fact that

in(#) = W,+(n+l), where in denotes the embedding from the ultrapower by the n-fold product of the w-cofinal normal measure on S; (which follows from the same argument as for K = wl), we get:

Corollary 5.3. There is no a~,_+,,+~ sequence of distinct subsets of 0,.

As S; = o,.+~ is measurable, the conclusion of Corollary 5.3 also holds for 0,. Hence, we have verified the above conjecture for all K < w,.~


As we mentioned earlier, recent results of the author establish the strong partition relation for all S:,,,. Also, by these results one computes

6l -x 2n+1- *“I 0.

. 1 2n - 1+ 1.

w

Using again the same argument for computing the ultrapowers of w1 by the products of its normal measures, one gets

Hence:

w

1 2n - 1+ (1 + n).

Corollary 5.4. The conjective is true for

K=X

Proof of Theorem 5.1. We fix K having the strong partition and n E o, and suppose to the contrary that (A, : a < K,+~ ) is a sequence of distinct subsets of K,. We may assume n is chosen minimal.

Definition. We let <” be the ordering on n-tuples of ordinals (ar, . . . , an)

with &r<(y;!<. a * < an < K defined by ((u,, . . . , a,J <” (PI, . . . , &) iff

(cu,, %-1, . . . 2 aI) -dex (/&, Pn-l, . . . , /3J, where <lex denotes lexicographic ordering. We say the function f : <“* K is of the correct type if f is order- preserving from <” (more precisely, the domain of <“) into K, is everywhere discontinuous, and f has uniform cofinality w; that is, there is g : <" x w+ K

order-preserving (with order (a,, . . . , (y,, 1) < (PI, . . . , &, m) iff ((u,, . . . ,

al, 4 dex (B”, . . . , PI, m)) such that f ((u,, . . . , LY,) = supm g(a,, . . . , cu,, m) for all (Y~ < LX* < . - a < Cu, < K.

The first lemma below is not needed for the proof of Theorem 5.1, but is of independent interest.

Lemma 1. The set of ordinals <K,, representable with respect to V, by a function F : CR ---, K of the correct type has order type K,, .

Proof. We let S E K, be the set of ordinals so representable. We define an order-preserving map n from K, into S. We fix ?j < K, and define n(v). Fix a function h : K”+ K representing q. Let a, < ~~ be 1eaSt such that if h(1): K+ K

represents cyl with respect to V, then for almost all (Y~, . . . , a,,, h(cuI, . . . , an) <

98 S. Jackson

h(l)(ar,). Clearly, al is well-defined. We fix h(1) : K- K representing czl. We may assume h(1) is monotonically increasing as otherwise we replace it with h(1) (a) = supSSah(l)(@, which agrees with h(1) almost everywhere. By chang- ing h on a set of measure 0 if necessary, we may also assume that for all

a17 * . . , %u,, h(4,. . *, an) < h(l)(a,). We define an auxiliary ordering <z on n + 1-tuples (oi, . . . , a,,, /3) with (Y~ < a2 < - * * <a, and OGPS~(~)(Q by

(ff1, * . * , an, B) <“* (Bl, . . . , A, y) iff (%7 %I-,, * . . 7 al, PI <‘yPn, . * * ,

B,1,;‘(; We define H : <: += K order-preserving by: H(a,, . . . , an, p) = the . . . ) a,, /3)lcn + 1)th ordinal greater than a,. Here I(a,, . . . , an, j3)Ict

denotes’ihe rank of (ki, . . . , a,, /?) in <” *. Notice that H: <“,+ K is of the correct type. Also using the normality of V, it is easy to see that H is well-defined in the following sense: if h(l), h(1) agree almost everywhere, then for almost all

a,, * * * , a,, with respect to V,, h(l)(~~) = h(l)(cu,) and for all OS /3 s h(l)(an), H(q, . . . , an, /3) = ii@,, . . . , an, /3), where H, fi refer to the functions constructed using h(l), h(1) respectively. We define now k(cr,, . . . , a,) for a,<au,<.. .<a,, < K by k(cu,, . . . , a,,) = H(q, . . . , an, h(a,, . . . , a,)). It follows readily that this is well-defined, that is, if h, h(1) agree with i, g(l) almost everywhere, and k, I? are constructed using h, h(1) and fi, i(l) respectively, then k and 1 agree almost everywhere. Also, k is of the correct type everywhere. We set n(q) = [k],. Thus, n is well-defined and maps into S. It also follows readily that n is order-preserving. This establishes the lemma. 0

We digress for a moment to recall some equivalent formulations of the strong partitiOn rehtiOn on K. First, suppose g is a partitiOn of the functions f : K --_, K of the correct type (i.e., increasing, everywhere discontinuous, and of uniform cofinality 0). An equivalent version of the strong partition relation asserts that there must be a c.u.b. C c K such that all f : K+ C of the correct type lie on the same side of the partition. Secondly, suppose <* is an ordering of length K. Then we have the partition property for functions f : dom(<*)+ K of the correct type (with obvious meaning), which follows by identifying dom(<*) with K in an order-preserving manner. A useful variation of this last version is the following: we partition finitely many functions fi, . . . , fn each order-preserving of the correct type from some ordering ci into K, and with the relative ordering of the values of the 6 specified (i.e. we have a fixed assignment for each pair o E dom ci, /3 E dom <j as to whether &(cx) <J(p), I =6(p), or J(a) >J(/3)). We then have a c.u.b. Cc K homogeneous for fi, . . . , fn into C of the correct type and correctly ordered. This again follows by identifying the disjoint union of the domains of the <i with a single ordering-which we are assuming to have order type K. It is this last version of the partition property which we shall make use of below.

We require a definition:

Definition. Let k : K~ * K be of the correct type. We define k(i) : K~+ K for

Partition properties and well-ordered sequences

lci<n by

99

For i = n, we set k(n) = k.

Notice that for i = 1 this agrees with our definition of h(1) for general h given previously.

Lemma 2. Suppose f : K -+ K of the correct type is given. Then the cardinality of

the set of ordinals <K,, which are representable by an f : <” +- K of the correct type

with [f(l)]” S r], is SK,,_~.

Proof. This follows immediately from the fact that if f, g are of the correct type and f (1) <g(l) almost everywhere, then f <g almost everywhere. 0

Lemma 3. There is a c. u. b. C G K such that for any f : <” + C of the correct type, the cardinal@ of the set of ordinals <K,,+~ which are representable by an f : Cn+l+ K of the correct type with f(n) =T for some fixed f : <” --;, K with 7 <f everywhere, is a~,,.

Proof. We partition the functions 7 : -?-+ K of the correct type according to whether or not the property stated in Lemma 3 holds. (Note: use of the partition property is not necessary for the proof of the lemma, but simplifies the presentation of the proof.)

We suppose towards a contradiction that C E K is c.u.b. and homogeneous for the contrary side. We let <$+l denote the ordering on tuples (LY~, . . . , crn+l, 6)

with ~yi < cu, <a - * < q,+l and OS j3 s an+r defined by: (LYE, . . . , an+,, p)

,:+l (PI, . . . , B,+1, Y) iff (a,+,, . . . , al, P) dex (Bn+l, . . . , PI, v). We let C’ = the set of closure points of C, and f-lx an E; : <“,+‘+ C’ of the correct type. Let

rl<Kn, and fix a function h : K”+ K representing q. Define F,, : <“+l-+ C by:

F,(R, . . . , %+d = F(R, . . . , G+~, h(a,, . . . , 4) if h(al, . . . , 4 < G+~, and = F(q, . . . , N~+~, 0) otherwise. It follows easily that [F,],+, depends only on q, that each Fq is of the correct type, that q+ [Fq15+, is order-preserving, and that for all q, F,(h) =y everywhere, for some fixed f : <” ---, K order-preserving and everywhere discontinuous. This establishes Lemma 3. Cl

We now consider the following partition: 9: we partition functions fi, f2 : < n+l+ K of the correct type together with

g : K + K of the correct type where fl, f2, g are ordered as follows:

(1) fl(%, . . * > %l+1)<.f2(~1, * * *, %+l)<fl(~l+L e2,. . .9 ct;l+d,

(2) fi(%. . . 9 K+~)<~(B) ifPacu,+l, and

MaI,-*., ~+d>g(P) forB<cu,+l.

100 S. Jackson

We partition such tuples (fr, f2, g) according to whether -- --L LL --- ’

ordinal ?I < K, with q(l) < [g]” (i.e. 77 is represented

[WIV s kid and

or nor mere is an by h:Kn*K with

rl E 4fd, * rl ~4LJ""~

It is easy to see that (l), (2) above completely specify the fi, f2, g, and determine a partition property for such tuples.

We claim that on the homogeneous side of the partition

relative ordering of

the property stated holds. We suppose not, and fix a c.u.b. C G K homogeneous for the contrary side. We let C’ = the set of closure points of C, and fix fi, f2: <‘,+I+ C’ of the correct type and ordered as in 9. Let q <K, be an ordinal for which Arfilvn, A,,,, disagree, and h : <” + K be of correct type with [h]“, > 7. Let g : K + C’ be of the” correct type with g(cu) > h(l)(a) and g(a) >fi(l)(cu) for all LY. We now claim that there are functions fr, &, g of the correct type and ordered as in 8, with

El, = k1v.Y El, = b-21> and [& = [g]“. We define fl, _&, g by induction as follows: Let D G C’ be c.u.b., closed under g, and such that for (Y E D the order

type of <” ]C(Sr, . . . , Pn): A < a) s a. Let D’ = the set of closure points of D. Case 1: an ED’. We set J;(q, . . . , qJ =fi(cu,, . . . , a,,), &(a,, . . . , LY,) =

JX% ’ * * , 4 and E(G) = g(a;l). Case 2: a;, $ D’. We set &(q, . . . , a;l) = the next oth element of D greater

than

max r SUP fi(Pl,. . . J&l), ,“,:t m&

(I% ,...I Bn)-3n1 ,...I 4 SUP

WI. . ..I f%W(~l,..., ad XV

n lY.A,},

Also, we set &(a,, . . . , an) = the next oth element of D greater than

J;i(%, * * * , 4, and g(qJ = the next wth element of D greater than

sul$& * * * , l-4-1, %I, PI < * * * < /%-I< %u,). Using the definition of D’ and the fact that fi(ar, . . . , LY,) > cu,, it follows

readily that fl, &, g are ordered as in 9 everywhere, and are of the correct type. It is also immediate that 5, 6, g have range in D E C, and agree with fi, fi, g almost everywhere. This contradicts the homogeneity of C for the contrary side of the partition.

Hence, on the homogeneous side of the partition, the property stated in 9 holds. We fix a c.u.b. C G K homogeneous for 9 and let C’ = the set of closure points of C. We may assume C satisfies Lemma 3 as well. From Lemma 3, there is an f : <” + C’ of the correct type and 7 : <” + C’ order-preserving and every discontinuous with 7 <f everywhere, and such that the set of ordinals 5 < K,, which are representable by a fE: C+’ + C of the correct type with fE(n) =f everywhere, has cardinality 2~~. Also, from the proof of Lemma 3, for

&<&<K,, fE,p f5, are of the correct type everywhere, ordered as in 9, and

fi(n) =.&(n) =_7 everywhere. We fix a g:~+ C of the correct type with 7, g ordered as in 9” (i.e. f( q, . . . , a;l) < g(B) for /I 3 a;, and f( (or, . . . , LX,,) > g(b)


otherwise). It follows then that fs,, fE2, g are ordered as in 9’ and have range in C. From the definition of 9 it now follows using Lemma 2 that we have a size K,, subset S of K,+~, such that if a # /3 are in S then A,, A, disgree on some ordinal from a fixed set of size <K,, (hence SK,_~). Th is violates the minimality of n, and

completes the proof of the theorem. Cl

References

[l] L.A. Harrington, On II: equivalence relations, to appear. [2] L. Harrington and A.S. Kechris, On the determinacy of games on ordinals, Ann. Math. Logic 20

(1981) 109-154. [3] L. Harrington and R. Sami, Equivalence relations, projective and beyond, in: M. Boffa, D. Van

Dalen and K. McAloon, eds., Logic Colloquium 78 (North-Holland, Amsterdam, 1979) 247-264.

[4] S. Jackson, AD and the projective ordinals, Cabal Seminar 81-85 (Springer, Berlin, to appear). [5] S. Jackson, A computation of S:, to appear. [6] S. Jackson and D.A. Martin, Pointclasses and well-ordered unions, Cabal Seminar 79-81,

Lecture Notes in Math. 1019 (Springer, Berlin, 1983) 56-66. [7] AS. Kechris, On transfinite sequences of projective sets with an application to Zi equivalence

relations, in: A. Macintyre, L. Pacholski and J. Paris, eds., Logic Colloquium ‘77 (North- Holland, Amsterdam, 1978) 155-160.

[8] A.S. Kechris, AD and infinite exponent partition relations, Unpublished manuscript, Dec. 1977. [9] A.S. Kechris, Souslin cardinals, K-Suslin sets, and the scale property in the hyperprojective

hierarchy, Cabal Seminar 77-79, Lecture Notes in Math. 839 (Springer, Berlin, 1981) 127-146. [lo] A.S. Kechris, AD and projective ordinals, Cabal Seminar 76-77, Lectures Notes in Math.

(Springer, Berlin, 1978) 91-132. [ll] A.S. Kechris and H. Woodin, Generic codes for uncountable ordinals, partition properties, and

elementary embeddings, Cabal Seminar 85-89 (Springer, Berlin, to appear). [12] Y.N. Moschovakis, Descriptive Set Theory (North-Holland, Amsterdam, 1980). [13] J.R. Steel, Scales in L(R), Cabal Seminar 79-81, Lecture Notes in Math. 1019 (Springer, Berlin,

1983) 107-156. [14] J.R. Steel, Closure properties of pointclasses, Cabal Seminar 77-79, Lecture Notes in Math. 839

(Springer, Berlin, 1981) 147-164.

partition properties and well-ordered sequences

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