On Projective Ordinals Author(s): Alexander S. Kechris Source: The Journal of Symbolic Logic, Vol. 39, No. 2 (Jun., 1974), pp. 269-282 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2272639 . Accessed: 22/05/2013 17:19
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This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions THE JouRNAL OF SymOLIC LoGic Volume 39. Number2, June1974
ON PROJECTIVE ORDINALS'
ALEXANDER S. KECHRIS
We studyin this paper the projectiveordinals 81, where81 = sup{e: e is the lengthof a Al prewellorderingof the continuum}.These ordinalswere introduced by Moschovakis in [8] to serve as a measure of the "definablelength" of the continuum.We prove firstin ?2 that projectivedeterminacy implies 81 < 81 forall even n > 0 (the same resultfor odd n is due to Moschovakis).Next, in the contextof fulldeterminacy, we partlygeneralize (in ?3) theclassical fact that Sf1 Ml and theresult of Martin that 81 = EO + 1 byproving that 81n+ 1 = A++ 1, where A2n+1 is a cardinalof cofinalityw. Finallywe discussin ?4 the connectionbetween the projectiveordinals and Solovay's uniformindiscernibles. We prove among otherthings that Va (a# exists)implies that every 81 withn >: 3 is a fixedpoint of the increasingenumeration of the uniformindiscernibles.
?1. Preliminaries. 1A. Let w = {0, 1, 2,.. .} be theset of naturalnumbers and ? =_ cwthe set of all functionsfrom co into co or (forsimplicity) reals. Lettersi, j, k, 1,m, . . . will denote elementsof co and a, A, y,8, ... elementsof R. We studysubsets of the productspaces ... -T = X1 X X Xk, whereXi is c or Rq.We call such subsetspointsets. Sometimes we thinkof themas relationsand writeinterchangeably x e A -coA(x). A pointclassis a class of pointsets,usually in all the productspaces. Most of the timewe shall be workinghere with the analyticalpointclasses El, fl1,Al and their correspondingprojective pointclasses El, X,,, Al. We use F.'(a), H'(a), Al(a) for the relativized(to any a e R) analyticalpointclasses. Various determinacyhypotheses occur frequentlyas assumptionsin the state- mentsof theoremsin thispaper. Neverthelesswe nevermake directuse of them. We simplydraw conclusionsfrom some of theirknown consequences. The reader can consult[10], [8] or the recentsurvey article [2] forthe basic factsconcerning
Received August 5, 1972. 1 The resultsin this paper are included in the author's doctoral dissertationsubmitted to the University of California, Los Angeles, in June 1972. The author would like to express his sincerestthanks to his thesis advisor, Professor Yiannis N. Moschovakis, both forcreating his interestin descriptiveset theoryand for his guidance and encouragement.The preparation of the paper was partiallysupported by NSF grant GP-27964. e 1974,Association for Symbolic Logic 269
This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions 270 ALEXANDER S. KECHRIS games,determinacy, etc. In generalwe writeDeterminacy(r), where r is a point- class, to indicatethat every set of reals in r is determined.Furthermore we put
ProjectiveDeterminacy(PD) v everyprojective set of reals is determined, Full Determinacy(AD)v everyset of reals is determined.
lB. A prewellorderingon a set X is a relation< c X x X whichsatisfies the followingconditions: (a) x < x, Vx e X; (b) x < y & y < z = x < z; (c) x Determinacy(A1n)=> Prewellordering(Ifl,+1) & Prewellordering(n+? 2) (thusalso Prewellordering(f7+11) & Prewellordering(2E:2%2)). A scale on a pointsetA is a sequence{al}~z0 of normson A withthe following limitproperty: If XI E A, forall i, iflim. xi = x and if,for each n and all largeenough i, an(x) =-A, thenx e A and,for each n, a,,(x) < <. (FollowingSolovay, we call thecondition " a,(x) ? An,"the semicontinuity property of scales.) If r is a pointclassand {a}new<] is a scaleon A we saythat {an}n,, is a P-scaleif thereexist relations Sri Se in r, r respectivelyso that y e A * Vx{[xe A & an(x) < an(y)] - Sr(n,x, y) Sp(n,x, y)}. We writeScale(P) if everyset in r admitsa P-scale. The notion of a scale was formulatedby Moschovakisin [9], wherethe scale propertyis called propertyY. One of the basic resultsof [9] is that & Scale(X2n+ 2) Determinacy(Alj)> Scale(HII,,+1) (similarlyfor the boldfaceclasses). This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions ON PROJECTIVEORDINALS 271 Finallyif {an}nec,) is a scale on a pointsetA we call fanjnr.,a A-scale,where A is an ordinal,if everya,, has length< A(or equivalentlyif each an,maps A onto A). REMARK. It happensvery often in practicethat one definesa sequence{an}nec of maps froma pointsetA into the ordinals,that has all the propertiesof a scale exceptpossibly that some an is not a norm,i.e., it is not ontoan ordinal.Then one can associateto {an}n-ew,a unique scale {Fn}n,, so that It is convenientto abuse languagehere and referto {ann,,ec itselfas a scale,although whatwe have in mindis {'n}necw 1C. We will have to deal veryoften in thispaper withweilfounded relations and trees.If X is a set,a wellfoundedrelation on X is a relation< c X x X such that for no sequence x0,xl, . . . of elementsof X we have < x2 -< x1 -< xO. The set Field(-<) = {x: 3y(x -< y) or 3y(y -< x)} is called thefield of <. For x e Field(-<), we definethe lengthof x by the -<- induction XI.<= sup{IyI< + l: y < x}, wherewe assumesup(0) = 0. The lengthof < itselfis givenby 1< 1 = sup{IxI.< + 1: xE Field(-<)}. Notice herethe followingminimality property of the functionIxI.<: If f: Field(-<) -- ordinals and x -< y =>f(x) < f(y) thenfor every x E Field(-<) we have IxI.< ? f(x). Now let X be a set.A treeon X is a set T of finitesequences from X closedunder subsequences,i.e., (x,. . ., xn) E T & k :5 n => (X1, . .., xk) E T. The emptysequence ( ) is alwaysa memberof a nonemptytree. A branchof T is a sequencef E OX such that,for every n, f r n = (f(O), . . ., f(n-1)) e T. We denotethe set of branchesof T by [T], followingMansfield. A treeT is well- foundedif it has no branches(i.e., [T] = 0) or equivalentlyif < rnT x T is well- founded,where < is theusual (proper)extension relation between finite sequences (xli,--, Xn) -< (yi, . .., y.) ,- n > m & xi = yi fori < m. Thus if T is a wellfoundedtree we can put,for each u E T, IUIT = SUP{IVIT+ 1: Ve T, v < u} = SUP{IU'(X)IT + 1 : u(x) e T} (whereuzv denotes concatenation) and we can definethe length of T by IT =I( )ITV Finallyfor u E T, let T, = {v: uzvE T}. Then IUIT= ITI . We will be usuallyworking with trees of pairsof integersand ordinals,i.e., trees This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions 272 ALEXANDERS. KECHRIS on sets X= c x A, where A is an ordinal. They contain elementsof the form ((ko 60),. . . (kne,6)), whereki e co and 6i < A,for all i. A branchof such a tree is a sequenceg e O(w x A),but forconvenience it willbe representedby theunique pair (af) e 4oo x WA,such thatg(n) = (a(n),f(n)), for all n. For each a e 9 the treeT(a) on A is definedby T(a) = {(for.. 6n) :((a_(0), 60) . (a(n),6n)) e T}. Notice that (for.). . 6fn) e T(a) & &-(nJr 1) = ,B(n+ 1) -> (fr .... ine T(g). From thisit followsimmediately that the sets .~o .. on)= {a: (fe ... s 6n)c T(a)} are all clopen. I D. We work in this paper entirelyin Zermelo-Fraenkelset theorywith dependentchoices (ZF + DC) where (DC) Vue x3v(u,v) e r = 3fVn(f(n),f(n+ 1)) e r. We stateall otherhypotheses explicitly. ?2. Relationsbetween projective ordinals. 2A. The projectiveordinals S'j are definedby 6' = sup{{: e is thelength of a Al prewellorderingof e}. They have been introducedby Moschovakisin [8] and severalresults about them wereproved there. It is clear that 81 = 81 < 6l < < 6' < 61+1 ?<*- , but is it possible that, forsome n > 0 81 = 819? Moschovakisproved in [8] that Determinacy(Aln)=' 821n+ < 621n+2- This is a consequenceof the followingbasic fact. THEOREM (2A-1) (MOSCHOVAKIS [8]). Assume Determinacy(Aln).Let a be a fl2n+ -normon a completenln+, set. Thenthe length of a is preciselyS'2n+l. (A 2n + 1 set A is completeif forany B E II2n + 1 thereis a continuousf so that x E B -f(x) e A.) Neverthelessthe problemof the relationshipbetween 82n and 82n+ 1 (forn > 0) was leftopen. We provebelow that,for n > 0, k,2n< 62n+, (assumingPD). 2B. The observationthat lies behindthe proof of thisfact is thatone can work much betterwith wellfounded relations than directlywith prewellorderings. We thereforefind it convenientto introducehere another kind of projectiveordinals. Let -n = sup{{: 6 is thelength of a El: wellfoundedrelation on reals}. The followingcan be provedusing a simplevariation of theproof of Lemma 10 in [8]. THEORFM (2B-1) (MosCHovAKIs). For any n, Determinacy(A,&l)= 4n+, = 81 This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions ON PROJECTIVEORDINALS 273 Using (2B-1) we now show THEOREM(2B-2). Assumen > 1. ThenDeterminacy(An) => 8,, < 8k2+1- PROOF. Since 82n < 2n and + = ,n+l it is enoughto provethat 4n < at2n+l. The idea is to "put together"all Z21 wellfoundedrelations to create a longerZn+1 wellfoundedrelation. This can be done as follows: Let S c Q3 be a ,2nset whichis universalfor En subsetsof R2. This means thatevery El,, subsetof R2, A, has the form A = Sa =((3, y): (a, py) e S}, forsome real a, a code of A. Let W = {a: Sa is wellfounded}be theset of codes of 2n wellfoundedrelations. Then our "big" wellfoundedrelation is on 2 and is givenby (CC, (Y, ) = y e W & (8, 8) E Sa. It is trivialto verifythat < is wellfoundedand it is not harderto observethat, if R = Sa. is wellfounded,then R is isomorphicto the restrictionof < to pairs of the form(ao, P). Thus IRI ? I- REMARK 1. Let -a' = sup{C: t is the lengthof a IIWwellfounded relation on reals}. Then we have PROPOSITION. For each nil1 =- + 1. PROOF. It is enoughto show > a,',+ 1. Let < be a Elj + 1 wellfoundedrelation. We shall definefor each real a a wellfoundedtree Ta on 3 such thatTa e II' and a -< P3= ITal < ITCH1.This impliesthat forany a E Field(-<) we have IaIs< ? Tal and since any H' tree has length %<-a we get I <1 ' sup{lTaI + 1: a E. } < 7s. Thus a +1 < 7tn4- To define Ta let a >- P - 3y((a, p, y) E P), where P E Ill. Applying the " unfold- ing trick" put for each a, = Ta {((a, o,0YO), (B31,P1r, l * * (k- 1, Pk, Yk)) (a, PO,YO) E P & (PO, ,1, Yl) E P &... & (Pk-1, Pk,yJ )e P}. Clearly Ta e 1la and Ta is wellfounded. If a < p3,pick a yo such that (p3,a, yo) E P. Then (in the notation of IC) Ta = (T)((abyo)) which implies ITaI < IT06. E Notice also that the proof of (2B-2) establishes (in ZF + DC only) that a' < a+ 1 (n > 0). REMARK 2. Kunen and Martin have independentlyshown that Determinacy(Aln,) = gn+ = 81n+ 2 (see [7]). Thus assuming projective determinacywe have the following picture: (81 =) 71 - 8 - < = 1 = < = < = =* ?3. Projective ordinals in the completelyplayful universe. 3A. Assuming the (full) Axiom of Determinacy (AD) Moschovakis has shown in [8] that every 81 is a cardinal. It is classically known that 81 = S, and Martin This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions 274 ALEXANDER S. KECHRIS provedin 1968 thatAD > 81 = 92 (see [7]). For some timeit seemedlikely that, withAD, 81. = k, would hold forevery n 2 1. Thus it came as a surprisewhen Martin proved in 1970 that AD => = X.,+1 (see [7]). Our main resultin this sectionpartly generalizes the classical result81 = X1 and Martin'stheorem. We prove AD -> 8'n + = (A2n+1)+, whereA2n + l is a cardinalof cofinalityco. (Here A+ = leastcardinal bigger than A.) This givesalso lowerbounds for the 81's. 3B. Beforewe proceedto provethis fact we have to set up some of the mach- ineryconcerning K-Souslin and K-Borelpointsets. (Further details can be foundin [7] or [4]). It was Martinwho firstapplied in a nontrivialway these methods to the studyof the projectivesets beyondthe second level of the hierarchy.In factour proof below parallels the argumentsMartin used to prove 6i3=1 c+ but in addition uses the basic theoremof Moschovakis on the existenceof scales on projectivesets. DEFINITION (3B-1). A pointsetA c X is called K-Souslin,where K is a cardinal, if it can be writtenas A = U n Aftn, feWc?neco wherefor each sequence (fo,..., i,,) fromK, A(40,.4.) is a clopen pointset.We denote by A,, the pointclass of K-Souslin pointsets. It is easy to see that a set A c 9? is K-Souslin if and only if there exists a tree T on c X K so that a e A - T(a) is not wellfounded (=.3f((a,f) e [T])). Similarly for subsets of the product spaces. DEFINITION (3B-2). A pointset A c X is called K-Borel, where K is a cardinal, if it belongs to the smallest pointclass which contains all open pointsets and is closed under complements and unions of length < K. This pointclass is denoted by RX - It is a standard fact that 40 = El, and the classical Souslin theorem asserts that ( ) A = Al. This last result has been generalized to all odd levels usingAD. Martin proved firstthat AD => .3 = A1 (see [7]) and also THEOREM (3B-3) (MARTIN [7]). For anyn, AD => + C: A t1- Then Moschovakis [9] showed the other inclusion of (3B-3), i.e., AD 1 =>A cln2l ,2 (In fact he needs only PD here.) We shall see later how s,, = El generalizes. The next fact connects the notions of K-Souslin and K-Borel. It was proved by Sierpin'ski for K = w (see [5, p. 32]), but his proof works as well for any K. THEOREM(3B-4) (SIERPI1ISKI). If A c X is K-Souslin,where K is a cardinal,then A is the intersectionof K+ sets in Ski+. Thus A E ARK+ +. PROOF. Assume for simplicityA c: R and let T be a tree on co x K such that a E A -. T(a) is not wellfounded. For each 0 ? e < K+ and any finitesequence u from K put = {a IT(a)uI < e}, This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions ON PROJECTIVEORDINALS 275 where,for a treeJ, IJI< e abbreviatesboth that J is wellfoundedand thatIJI < e. We agreethat JuJ= -1 if u 0 J and -1 < t forany ordinale. It is now easy to checkthat for length(u) = n we have AO = {a: ((a(O), uO),..., (a(n - 1), uAn.)) J}, At+ 1 = AX u n Az^(,,), T = {((a(O), ao(a)),.. (a(n), a,(am))): a e A}, where{sa}j,,cv is a A-scaleon A. We checkthat a E A - T(a) is not wellfounded. If a e A, then55(a) = (ao(a), a1(a), . . ., an(a),. . .) is a branchof T(a). Conversely, if (e%, j, . *, en,...) is a branchof T(a), thereexist reals a0,1,a1, a2 ... aCcne ... all in A, such that,for every n, ((an(O),ao(aA)), . . X (a.(n), an(ccn)))= ((a(OM,eo), * * *, (ex(n), en)). Then an cacand am(acn)= emfor all n ? m. So a e A. Now T is a treeon w x Aand it can be easilyreplaced by an isomorphicone on co x JA,without changing its integerpart. Thus A e .A9. M We are now readyto prove THEOREM (3B-7). For anyn, AD .> 81n , 1 = (A2n+ 1) +, whereA2, + 1 is a cardinal of cofinalityw. PROOF. Let S c a be a setwhich is Z2n+1 butnot HIn + I Sayaec- S co 3Q(a, 0), whereQ 6 fn. Let {am)}mecvbe a IU, + 1-scaleon Q. Since Q e 21n,the prewell- orderings <5m are actually A',,+,; thus length(?^m) < 821n+ 1, forall m. It is easy to see that cofinality(82n+ 1) > to, so that {am}me,is a A-scalefor some A < 812n+ Put A2n+1 = Al. We proceed to show that ()t2.+=)+ 8+ 1 and cofinality(A2n+l) = This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions 276 ALEXANDERS. KECHRIS Since Q admitsa A-scaleit is A2n,+1-Souslinby (3B-6) and thus,as a simpleargu- mentshows, S is A2n,+1-Souslin. Then, by (3B-4), S E (\2%+1) + +* If (A2,+ 1) + was less than81n+ 1 (A2,,n 1) + + wouldbe at most8ln+ 1 (recallthat each 8kis a cardinal), thereforeS e d C Aln+1 (by(3B-3)), whichis a contradiction.Thus (A2n+ 1) + = 62nsl1 If cofinality(A2n+ l) > co, then(by (3B-5)) S C (2n + 1)=+ 6in+ 1 -2n+?1- again a contradiction.Thus cofinality(A2n+ 1) = cv. E COROLLARY(3B-8). For anyn, AD =::-2 +1 onlX 82n+2 2 ncon?+2 REMARK 1. One can easily check by examiningthe proof of (3B-7) that A2n +1 = smallestcardinal A such that Zln+1 $l C . If we put A2n = 62n-I (n > 1), thenas Martinalready observed, in [7],AD >n E = (n > 1), whichgeneralizes the factthat El = Yt9,.In factAn is the least such cardinal.Solovay (unpublished) provedthat AD &-Y =Y forA,K A is K-Souslin > A admits a K-scale. We prove firstthat this is true if cofinality(K)> co. PROPOsITION.Let Kbe a cardinaland assumecofinality(K) > co. Then,for any A c XI,A is K-Souslin=. A admitsa K-scale. PROOF. Let T be a tree on co x K and assume a e A - T(a) is not wellfounded. The firstattempt for defininga scale on A is to put, for a e A, c4(a) = hj,(O) = least 6 such that J(,) is not wellfounded, = h/(n + 1) least e such that JhjLr(n + l)-(,) is not wellfounded. We use <1, . ,n>, where ej < K, to denote the ordinal of the n-tuple (l, . . en) in the lexicographical wellordering of nK. One can easily check now that {a}}new is a scale. (Recall here the remark in lB.) In fact it is a K'0-scale(Ky0 denotes ordinal exponentiation), but not necessarily a K-scale. 2 The resultsin this remark (which extends to the end of 3B) will not be used in the rest of the paper. This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions ON PROJECTIVE ORDINALS 277 To avoid thisproblem we use the hypothesiscofinality(K) > Wto writea AE - 32 < K(T(ca) is not wellfounded),where of course TXis the restrictionof T to or- dinals < {. Then we put,for c- E A, o.l~a) = least t such thatTV(ca) is not wellfounded, an(cX)= AD > 82n + 2 = (821n+1)+ (see [7]). This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions 278 ALEXANDER S. KECHRIS (Thus 81 = (A,)+, forany n ? 1.) Thereforewe know AD 6 = +2 and we willknow all 81's,once we knowthe ones withodd n 2 5. Fromthe results K" already mentionedone is temptedto conjecturethat AD => 81 = Kw-n+l- Kunen (unpublished)disproved this by showingthat AD -> 8 > X* a+1 This resultand (3B-7) improvethe lowerbounds of (3B-8). Neverthelessthe problemof theexact computationof Si forn > 5 seemsvery difficult and remainsstill unsolved, although Kunen has made importantprogress. ?4. Projectiveordinals and uniformindiscernibles. 4A. The aim of thislast sectionis to establishconnections between the projec- tive ordinalsand Solovay's uniformindiscernibles. Inevitably we will have to use severalfacts from Silver's elaborate theory of indiscerniblesfor the modelsL[c], for czE A. We will also need the resultsof Solovay on "sharps" and uniform indiscernibles.We tryto summarizewhat we needin 4B below.One can finddetails in [11], [13] and [7]. 4B. Considerthe theory ZF+ V=L[e]+ eteR. abbreviatedZFL(d), in a language which besides e contains a constantac. Let V1, V2,FV3, .. . be thevariables of thislanguage. It is wellknown that in ZFL(a) one can definea formulax(d, Vl,v2) abbreviatedv1 => 92Fr(Xi,* ** , X.) -='(Y1i, - - *Yn)- (Superscript21 means as usual interpretation.)A set I of indiscerniblesgenerates 2t,if 21 is thesmallest elementary submodel of itselfcontaining I. This is equivalent to sayingthat every element of A can be writtenin the formt'(x1,..., x,), where x1, .. * *,Xn e I and xl {p(v1, v.): for some x1, . . ., xn e 1, with xl This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions ON PROJECTIVEORDINALS 279 charactercD,, which has the followingproperties (where Ord(v) abbreviates"v is an ordinal"): (a) "Ord(vj)" cE (b) "t0(v1,. . ., vn) (c) O0rd(t<,(vj, . . ., Vn, Vn+l, . . i Vn+k)) & tw(Vi, * . Vn, Vn+l, ** Vn+k) < Vn+1 =t0(v1,, v. I , Vn+1.+ . ., Vn,+k) = t0(V, .I Vn,Vn, k k *, Vn+2k)" (ax all p. (d) a(n) = m i"d(n) = mi" E Da, wheren is the nthinumeral. (e) For all I, PQ((Dc,6) is wellfounded. A charactersatisfying (a)-(e) is called remarkable(for a). If FiDis remarkable,then (by (e)), for each limit ordinal A, r(ID., A) is isomorphicto a unique L.-[a] and thereexists a unique subsetIxA s A*which is a generatingset of indiscerniblesfor L,,[a.] and has character(FD. Silver proved (1) I~ais cofinalin A*, (2) I~ais an initialsegment of Ill if A < FLand A* = Athelement of I, (3) LAx[a] is an elementary submodel of L,,-[a], if A < Fs, (4) Ka = K, if K is a cardinal, (5) ifIA = UAIa, thenI' is a closed unboundedclass of ordinalswhich contains all cardinalsand generatesL[a]. Call Ia theclass of Silverindiscernibles for L[a]. From (3) and (5) it followsthat (a = {cp(V1. , Vn)): LNj[a] k c(X1 . Xn)} so that a remarkable character for a is unique. It is customary after Solovay to write a# for the real coding D,, (i.e., a# : w -- 2 and a#(n) = 0 O n is the Godel number of a formula in 4(Da). It is important to state here that all the results about indiscernibles for L[a] can be deduced only from the assumption that the remarkable character for a exists, usually abbreviated "'a exists." In particular, if a#exists, the theory of indiscern- ibles for L[a] can be done in L[a#] (e.g., the class Pa is definable in L[a#]). Solovay proved that 8 = a# is a IH2 relation. Thus if Va3f(f = a#), i.e., if Va(a# exists), then a c-* a# is a A function from A into 9 which can be easily seen to have a recursive inverse on its range, i.e., for some recursivef: e .> we have f(a#) = a. Assuming Va(a# exists), Solovay defined the class of uniformindiscernibles by 91 = QnaP. Then clearly 96 is a closed unbounded class of ordinals containing all thecardinals. Let 91 = {u1, u2, . . Z, . .. .} in increasingorder. Then u1 = X1 = V1- We prove in the rest of this section that all the 61's are uniformindiscernibles and we study their position in the above enumeration. 4C. We begin with a result on subsets of X, constructible from a real. It provides a converse to the firsttheorem of [3] but gives also immediately that 12 >- U2 (which can be proved also directly; see Martin [7]). Let, for each a E .P, = {(m, n): a( This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions 280 ALEXANDER S. KECHRIS Put WO = {a: Code(A) = {a E WO: HalE A}. A set A c 9, is called r in thecodes (wherer is a pointclass)if Code(A) E r. It was provedin [3] thatCode(A) E El(a) > A E L[a]. Thus if L = Uae L[a], then Code(A) El =: A E L. We provehere a converseassuming Va(a# exists). THEOREM(4C-1). AssumeVa(a# exists). Then,for any A C Ml, A E L = A is la' in thecodes. PROOF. Let A ' RI and A e L (withoutloss of generalitysince the proof is "uniform").Then A c Lm +)L C Liu X where, for any transitivemodel X4 of ZF and any A E X4, (Ak) - least cardinal of f bigger than A and I = -11?= {tlo,tl ...., lt, . .} is the increasing enumeration of the Silver indiscerniblesfor L. Thus A = tap0 * *.i ik, 11 Ml+ml *...*I+mi), '7i < < - where *- 92 LA;[a] F e E tj(e'), forany limitA > I'. Thus takingA = e + cowe have - eA 3t '(e' e P & e < e' < (t + )?)*& L(g+ )] tz(e)) And going to the codes 8 e Code(A) 8 E WO & 323y[ 8 E Code(A) c> 8 E WO & 3, E-At(8, a#)3y E At(8, a)P(B, y, 8, a#), which shows that Code(A) E IIT(a#). 0 This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions ON PROJECTIVEORDINALS 281 COROLLARY(4C-2). AssumeVa(a# exists) and let A c M,. Then A E L c:-A is IIV in thecodes A is El in thecodes. COROLLARY(4C-3) (Proved also independentlyby Martin [7]). AssumeVa(a# exists).Then U2 < 81- PROOF. Let e < u2. By a theorem of Solovay (see [7]), U + 1 = Sup{(U )La]: a EJ}. Thus find a such that 6 < (SX)L((rJ. In L[a] there exists a mapf: - 8 , from X1 1-1 and onto {. Let 71 < 8 G>f(t) < f(8). Then < is a prewellorderingof length ton 91 and EL[a].c Thus Code(A) = {Qa, f) E W02: (jai, jal) E <} E IlI. But Code(A) is a IIl prewellorderingof WO of length6; therefore6 < S1. D Martin [7] proved that Va(a#exists) - 812 < U2. Thus Va(a#exists) 81 = U2 (Martin [7]). 4D. Since (assumingVa(a# exists)) ul = 81 and u2 = 81, one is confronted again witha temptingconjecture, i.e., ua = 81 forall n > 1. That this is not the case is obviousto the believerin some strongform of definabledeterminacy, e.g., determinacyof all games in L(R) = the smallestmodel of ZF containingall ordinalsand M?. This is equivalentto the assertionL(R) k AD. But fromwork of Moschovakis[8] and Kunen(unpublished) it is knownthat AD > all S1 are regular. On the otherhand Solovay provedthat Va(a#exists) => Cofinality(uz+ 1) = Cofinality(u2),; for all f > 1; see [7]. These two facts(and of course AD = Va(a#exists)) force 81 forn > 3 to be a fixedpoint of {UZ})Eord, in L(Q). But thisis absolutefrom L(S) to the world. Thus 81 = use, n > 3. This is what we prove below, using only Va(a#exists). The proof is motivatedby the following(unpublished) result of Solovay. THEOREM(4D-1) (SOLOVAY). AssumeVa(a# exists). Then, for every t > 1 and every77 < us+ 1,we canfind aformula ip, a real a and uniformindiscernibles uze < < upo < ue suchthat 77 = tLt~a(U,1,. .. uy). We are now readyto prove THEOREM(4D-2). AssumeVa(a# exists). Then, for any n > 3, 81 =us- . PROOF. Let n > 3. If 61 = uSA, then u, < 81 < uz,+1,for some t < 81. We R2 derivea contradictionby producinga map a: w x _u_ +, onto uz+1, whose correspondingprewellordering 77 = tLta](u~ilI IBr for some p and some a, A. Here -r= r(ri'), with red = G6del number of p and r: w -c co recursive.We assume also forconvenience that everyinteger is of the formrTp1. If F(c, v1) F(vl) is a functiondefinable in ZFL(c), whichmaps the This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions 282 ALEXANDER S. KECHRIS universeonto the ordinalskeeping them fixed and sendingeverything else to 0 we have also 77 = F (tLEta(U,(,6)..., ul(86>)), and the expression on the right always givesan ordinal.This suggestsdefining c~r, -C O = FL[aJ(tLt]I(UI(8)1j,.*.), Uj()7j)). That a maps co x 2 onto uz+1 is obviousby thepreceding remarks. That a(rXpl, a(,Y) a(rX', 8) F al[(t[aJ (ul(O)j,... .)) < pLY(tLt](6 )),, Lot, y] k 0(u I(,6)l, * ,ul(lI (fora b obtainedexplicitly from ap, X) - whereg: a)2 x -R . cv is A^ and REFERENCES [1] J. W. ADDISON and Y. N. MOSCHOVAKIS, Some consequencesof the axiom of definable determinateness,Proceedings of the National Academy of Sciences, U.S.A., vol. 59 (1968), pp. 708-712. [2] J. E. FENSTAD, The axiom of determinateness,Proceedings of the Second Scandinavian Logic Symposium(J. E. Fenstad, Editor), North-Holland, Amsterdam,1971, pp. 41-61. [3] A. S. KECHRIS and Y. N. MOSCHOVAKIS, Two theoremsabout projective sets, Israel Journalof Mathematics, vol. 12 (1972), pp. 391-399. [4] , Notes on the theoryof scales, multilithedcirculated manuscript. [5] K. KURATOWSKI,Topology, vol. 1, Academic Press, New York and London, 1966. [6] D. A. MARTIN, The axiom of determinatenessand reductionprinciples in the analytical hierarchy,Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687-689. [7] , Projective sets and cardinal numbers: some questions related to the continuum problem (to appear). [8] Y. N. MOSCHOVAKIs, Determinacyand prewellorderingsof the continuum,Mathematical logic and foundationsof set theory(Y. Bar-Hillel, Editor), North-Holland, Amsterdam and London, 1970, pp. 24-62. [9] , Uniformizationin a playful universe,Bulletin of the American Mathematical Society, vol. 77 (1970), pp. 731-736. [10] J. MYCIELSKI, On the axiom of determinateness,Fundamenta Mathematicae, vol. 53 (1964), pp. 205-224. [11] J. H. SILVER, Some applications of model theoryto set theory,Annals of Mathematical Logic, vol. 3 (1971), pp. 45-110. [12] J. R. SHOENFIELD,Mathematical logic, Addison-Wesley, Reading, Massachusetts, 1967. [131 R. M. SoLovAy, A nonconstructibleA3 set of integers,Transactions of the American Mathematical Society, vol. 127 (1967), pp. 50-75. UNIVERSITY OF CALIFORNIA LOS ANGELES, CALIFORNIA 90024 MASSACHUSETTS INSTITUTE OF TECHNOLOGY CAMBRIDGE, MASSACHUSETTS 02139 This content downloaded from 131.215.71.79 on Wed, 22 May 2013 17:19:05 PM All use subject to JSTOR Terms and Conditions