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The Theory of Countable Analytical Sets Author(S): Alexander S The Theory of Countable Analytical Sets Author(s): Alexander S. Kechris Source: Transactions of the American Mathematical Society, Vol. 202 (Feb., 1975), pp. 259-297 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1997311 . Accessed: 22/05/2013 14:36 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society. http://www.jstor.org This content downloaded from 131.215.71.79 on Wed, 22 May 2013 14:36:53 PM All use subject to JSTOR Terms and Conditions TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 202, 1975 THE THEORYOF COUNTABLEANALYTICAL SETS(1) BY ALEXANDER S. KECHRIS ABSTRACT. The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable n set of reals, en (this is also true for n even, replacing 1n by En and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). The internal structure of the sets Con is then investigated in detail, the point of departure being the fact that each CO is a set of A1-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, w-models of analysis, higher- level analogs of the constructible universe, inductive definability, etc. It is a classicaltheorem of effectivedescriptive set theorythat a S thin (i.e. containingno perfectset) subsetof the continuumis countableand in fact containsonly Al reals. As a consequence,among the countable S1 sets of realsthere is no largestone. Solovay[41] showedthat everythin 2 set con- tainsonly constructiblereals and therefore(in sharpcontrast with the previous case),assuming a measurablecardinal exists, thereis a largestcountable T4 set of reals,namely the set of constructibleones. In [19] Moschovakisand the author extendedSolovay's theorem to all evenlevels of the analyticalhierarchy. It was provedthere that, assumingprojective determinacy (PD), thereis a largestcount- able 4,n set for all n > 1, whichwe denoteby C2n. The firstmain result we provein this paper(see ?1) is the existenceof a largestcountable H?,+ 1 set (whichwe denoteby C2n + ) for all n > 0, assumingPD again. For n = 0 our proofshows, in ZF + DC only, the existenceof a largestthin Hl set of realsC1, a fact whichwas also independentlydiscovered by D. Guaspari[12] andG. E. Sacks[38]. (To completethe picture,we remarkhere that it was shown in [17], using PD, that no largest countable 2An-+1 or I21n sets exist.) Once the existenceof the largestcountable sets Cn is establishedthe rest of ?1 is devotedto the study of theirinternal structure. We show here that Cn is actuallya set of An-degreeswhich is lwellordered under the usualordering of Al-degrees. Let us denoteby {d-}<pn the increasinghierarchy of the Al- Received by the editors September 13, 1973. AMS (MOS) subject classifications (1970). Primary 04A15, 02K30, 28A05, 54H05; Secondary 02F35, 02K05, 02K25, 02K35, 04A30. ( ) Research partially supported by NSF grant GP 27964. Copyriglht ? 1975. AmiiericaniMatlhcmiiatical Society 259 This content downloaded from 131.215.71.79 on Wed, 22 May 2013 14:36:53 PM All use subject to JSTOR Terms and Conditions 260 A. S. KECHRIS degreesof (elementsof) Cn. As we (hope to) illustratein this paperthe hierarchy {dn},<p playsa significantrole in the generalstructure theory of the analytical sets of the nth level.It provides a commonframe of referenceand a startingpoint for investigationsin a varietyof diversetopics such as specificationof bases, co- modelsof analysis,higher-level analogs of L, inductivedefinability, etc. After the generaltheory is discussed,we examinein moredetail in ?2 the structureof C1, the largestthin HII set (partof the resultshere have been in- dependentlydiscovered by D. Guaspari[12] andG. E. Sacks[38]). Weshow, amongother things, that {df}l<Pl=tj coincideswith the hierarchyof the hy- perdegreesof the Boolos-Putnamcomplete sets of integers.A naturalquasi- hierarchyof hyperdegreesis then defined,which extends (through all the con- structiblereals) the naturalhierarchy of hyperdegreesand it is shownto be ex- actly {d1},<,q. ? 3 accountsfor the structuraldifferences between C21n. with n > 0 and C1. The basicreason for themis the fact that the notion of "wellordering on co" is 'An 1 if n > 0, but not Al. Martinand Solovay[29] madea remarkableuse of this fact to producea counterexampleto the well-knowncon- jecturethat the Kleenebasis theorem for 11 generalizes,under PD, in a straight- forwardfashion to all I',,+2n+1 1 with n > 0. The Martin-Solovaydiscovery re- vealedfor the first time an importantstructural difference between the first and the higherodd levelsof the analyticalhierarchy. In the firstpart of ? 3 we deal with reflectingpointclasses and we unearthmore suchstructural differences. This eventuallyleads us to the subjectof Q-theory(see ?3B), whichwas developed by Martinand Solovayand independentlyby the author.And we end ? 3 with a briefsummary of the connectionbetween countable analytical sets andhigher level analogsof L. Finally,in ? 4, questionsconcerning countable analytical sets in innermodels like L are discussedand various consistency and independenceresults are obtained. ACKNOWLEDGEMENTS.Part of the resultsin this paperis containedin the author'sPh.D. thesis (UCLA 1972). Wewish to thankour thesissupervisor ProfessorY. N. Moschovakisfor his guidanceand encouragementand most of all for creatingour interestin descriptiveset theoryand for directingour attention to the problemof countablesets. Weare also gratefulto Dr. L. Harringtonand ProfessorsD. A. Martin,G. E. Sacksand R. Solovayfor numeroushelpful dis- cussionson this subject. 0. Preliminaries.OA. Let co = {O, 1, 2, . .} be the set of natural numbersand R = 'co the set of all functionsfrom X to X or (for sim- plicity) reals. Letters i, j, k, 1, m, . denote elements of co and a, j, 7, 6, . reals. Subsetsof co aremany times below identified with theircharac- teristicfunctions. We studysubsets of the productspaces X = Xl X X2 X... X Xk, This content downloaded from 131.215.71.79 on Wed, 22 May 2013 14:36:53 PM All use subject to JSTOR Terms and Conditions THE THEORY OF COUNTABLE ANALYTICAL SETS 261 x&hereX, is X or R. We call such subsetspointsets. Sometimeswe thinkof themas relationsand we writeinterchangeably x EA * A(x). A pointclassis a classof pointsets,usually in all productspaces. Weshall be concernedprimarily in this paperwith the analyticalpointclasses S1, ni, Al and theircorresponding projective pointclasses El, f1, Al. For information aboutthem we referto [37], [39] and the introductionof [30] fromwhich we also drawsome of our nonstandardnotation. If r is a pointclasswe put F={X-A:AEr,A C X} = the dual of r, and A = r n r, while for each a E R, 1(a) = {A: For some B E1r, A(x) * B(x, a)} and A(a) = r(a) n r(a). Wealso let r =U 1r(a) and A =frn r. aER If A C X X co, put 3wA = {x: 3nA(x, a)}, V 'A = {x: VnA(x, n)} andif A C X X R, put 3RA= {x: 3aA(x, a)}, VRA= {x: VaA(x, a)}. If 4) is any operationon pointsets(like e.g. 3W above), let (Dr={(DA: A Er} andsay that r is closedunder 4) if (Drc r. Let r be a pointclass,X a productspace (X = co or R is enoughfor this definition).We say that r is X-parametrizedif for any productspace y there isa GEr, GC YXX sothat f Gx = {y:(y,x)EG}then {Gx:xEX}= {A CY: Aer}. In this casewe say that G is X-universalfor r subsetsof Y. If A = Gx, x is calleda code of A. A pointclassr is adequateif it containsall the recursivesets (in all prod- uct spaces) and is closedunder conjunction, disjunction, number quantification of both kindsand substitution by total recursivefunctions. It is clearthat all analyticalpointclasses are adequateand all 11, HI are -parametrizedwhile all projectivepointclasses are adequate and all Z1, II' are R-parametrized.In generalif r is adequateand c-parametrized,r is adequateand R-parametrized Moreoverthe R-universalsets for r can be takento be in r. Wheneverwe workwith a parametrizedpointclass which has also some closureproperties we shallalways assume that universalsets for this classare chosenso that the pointclassis uniformlyclosed (e.g. if r is assumedclosed under A n B, a fixed recursivefunction will computea code for A n B from codesof A, B). OB. Let A be a pointset. A norm on A is a map 0: A -*+X from A This content downloaded from 131.215.71.79 on Wed, 22 May 2013 14:36:53 PM All use subject to JSTOR Terms and Conditions 262 A. S. KECHRIS onto an ordinal X, the lengthof 0 (notation: length(q) = I?1 =X). With eachsuch norm we associatethe prewellordering(i.e. the reflexive,transitive, connected,wellfounded relation) v on A, definedby x < *y (x) <?(y). Put also x <0y *(x)<4OM. Converselyeach prewellordering< with field A givesrise to a uniquenorm 4: A -X such that < we thencall X the lengthof ;. If r is a pointclassand 0 is a normon A, we say that ? is a r-normif thereexist relations?r, 4 in r, r respectivelyso that y E A =V x {[x E A & @(x)S y)] *x <,? y * x? y}.
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