{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T20:14:32Z","timestamp":1773260072339,"version":"3.50.1"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12885,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,12]]},"abstract":"<jats:p>Martin [12] has shown that the determinacy of analytic games is a consequence of the existence of sharps. Our main result is the converse of this:<\/jats:p><jats:p>Theorem. <jats:italic>If analytic games are determined, then x<jats:sup>2<\/jats:sup> exists for all reals x<\/jats:italic>.<\/jats:p><jats:p>This theorem answers question 80 of Friedman [5]. We actually obtain a somewhat sharper result; see Theorem 4.1. Martin had previously deduced the existence of sharps from 3 \u2212 \u03a0<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">1<\/jats:sup>-determinacy (where \u03b1 \u2212 \u03a0<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">1<\/jats:sup> is the \u03b1th level of the difference hierarchy based on \u2212 \u03a0<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">1<\/jats:sup> see [1]). Martin has also shown that the existence of sharps implies &lt; \u03c9<jats:sup>2<\/jats:sup> \u2212 \u03a0<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">1<\/jats:sup>-determinacy.<\/jats:p><jats:p>Our method also produces the following:<\/jats:p><jats:p>Theorem. <jats:italic>If all analytic, non-Borel sets of reals are Borel isomorphic, then x* exists for all reals x<\/jats:italic>.<\/jats:p><jats:p>The converse to this theorem had been previously proven by Steel [7], [18].<\/jats:p><jats:p>We owe a debt of gratitude to Ramez Sami and John Steel, some of whose ideas form basic components in the proofs of our results.<\/jats:p><jats:p>For the various notation, definitions and theorems which we will assume throughout this paper, the reader should consult [3, \u00a7\u00a75, 17], [8], [13] and [14, Chapter 16].<\/jats:p><jats:p>Throughout this paper we will concern ourselves only with methods for obtaining 0<jats:sup>#<\/jats:sup> (rather than x<jats:sup>#<\/jats:sup> for all reals <jats:italic>x<\/jats:italic>). By relativizing our arguments to each real <jats:italic>x<\/jats:italic>, one can produce <jats:italic>x<\/jats:italic><jats:sup>2<\/jats:sup>.<\/jats:p>","DOI":"10.2307\/2273508","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:48:57Z","timestamp":1146937737000},"page":"685-693","source":"Crossref","is-referenced-by-count":82,"title":["Analytic determinacy and 0<sup>#<\/sup>"],"prefix":"10.1017","volume":"43","author":[{"given":"Leo","family":"Harrington","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049203_ref008","volume-title":"Lecture Notes in Mathematics","author":"Jech","year":"1971"},{"key":"S0022481200049203_ref016","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0392534-3"},{"key":"S0022481200049203_ref005","first-page":"113","volume":"40","author":"Friedman","year":"1975","journal-title":"One hundred and two problems in mathematical logic"},{"key":"S0022481200049203_ref007","first-page":"A","article-title":"Analytic sets and Borel isomorphism","volume":"23","author":"Harrington","year":"1976","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200049203_ref013","volume-title":"Handbook for logic","author":"Martin","year":"1976"},{"key":"S0022481200049203_ref001","first-page":"26","volume-title":"Proceedings of the 1960 International Congress of Logic, Methodology and Philosophy of Science","author":"Addison","year":"1960"},{"key":"S0022481200049203_ref002","first-page":"481","volume":"41","author":"Baumgartner","year":"1976","journal-title":"Adding a closed unbounded set"},{"key":"S0022481200049203_ref010","volume-title":"Model theory for infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200049203_ref003","volume-title":"Lecture Notes in Mathematics","author":"Devlin","year":"1973"},{"key":"S0022481200049203_ref004","doi-asserted-by":"publisher","DOI":"10.4064\/fm-72-1-79-95"},{"key":"S0022481200049203_ref006","unstructured":"Harrington L.A. , \u03a31 1 sets and hyperdegrees (in preparation)."},{"key":"S0022481200049203_ref009","volume-title":"Handbook for logic","author":"Kechris","year":"1976"},{"key":"S0022481200049203_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1968-11995-0"},{"key":"S0022481200049203_ref012","doi-asserted-by":"crossref","first-page":"287","DOI":"10.4064\/fm-66-3-287-291","article-title":"Measurable cardinals and analytic games","volume":"66","author":"Martin","year":"1970","journal-title":"Fundamenta Mathematicae"},{"key":"S0022481200049203_ref014","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200049203_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90013-6"},{"key":"S0022481200049203_ref017","unstructured":"Steel J. , Ph.D. Thesis, University of California, Berkeley, 1976."},{"key":"S0022481200049203_ref018","unstructured":"Steel J. , Analytic sets and Borel isomorphisms (to appear)."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049203","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T15:07:45Z","timestamp":1558969665000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049203\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,12]]},"references-count":18,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1978,12]]}},"alternative-id":["S0022481200049203"],"URL":"https:\/\/doi.org\/10.2307\/2273508","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,12]]}}}