{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T17:30:35Z","timestamp":1767893435988,"version":"3.49.0"},"reference-count":7,"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":4119,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,12]]},"abstract":"<jats:p>We show that the \u201ceffective cardinality\u201d of the collection of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009117_inline1\"\/> sets is strictly bigger than the effective cardinality of the <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009117_inline2\"\/>. The phrase <jats:italic>effective cardinality<\/jats:italic> is vague but can be made exact in the usual ways. For instance:<\/jats:p><jats:p>Theorem 1.1. <jats:italic>Assume<\/jats:italic> AD<jats:sup><jats:italic>L<\/jats:italic>(\u211d)<\/jats:sup><jats:italic>Then in L<\/jats:italic>(\u211d) <jats:italic>there is no injection<\/jats:italic><\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200009117_Uequ1\"\/><\/jats:disp-formula>.<\/jats:p><jats:p>A few years ago Tony Martin showed a similar result, establishing the non-existence of an injection from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009117_inline3\"\/> to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009117_inline2\"\/> for <jats:italic>m<\/jats:italic> sufficiently larger than <jats:italic>n<\/jats:italic>. His method did not seem to work for <jats:italic>m<\/jats:italic> = <jats:italic>n<\/jats:italic> + 1.<\/jats:p><jats:p>This present paper gives level by level calculations for the projective hierarchy, but it too falls short of a complete analysis, in as much as it leaves the position of the effective cardinals in the Wadge degrees largely obscure. At the low levels it takes some time for any new cardinals to appear. Whenever \u0393<jats:sub>1<\/jats:sub>, \u0393<jats:sub>2<\/jats:sub> are non-trivial Wadge degrees strictly included in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009117_inline4\"\/> one has<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200009117_Uequ2\"\/><\/jats:disp-formula>.<\/jats:p>","DOI":"10.2178\/jsl\/1190150289","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:16:54Z","timestamp":1197573414000},"page":"1351-1372","source":"Crossref","is-referenced-by-count":3,"title":["Cardinalities in the projective hierarchy"],"prefix":"10.1017","volume":"67","author":[{"given":"Greg","family":"Hjorth","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009117_ref003","first-page":"663","volume":"63","author":"Hjorth","year":"1998","journal-title":"An absoluteness principle for Borel sets"},{"key":"S0022481200009117_ref001","volume-title":"Notes on descriptive set theory","author":"Andretta","year":"2000"},{"key":"S0022481200009117_ref004","volume-title":"American Mathematical Society Mathematical Surveys and Monographs Series","author":"Hjorth","year":"2000"},{"key":"S0022481200009117_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1982-15009-1"},{"key":"S0022481200009117_ref007","volume-title":"Descriptive Set Theory","author":"Moschovakis","year":"1980"},{"key":"S0022481200009117_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0071697"},{"key":"S0022481200009117_ref002","first-page":"1199","volume":"60","author":"Hjorth","year":"1995","journal-title":"A dichotomy for the definable universe"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009117","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T00:18:09Z","timestamp":1557188289000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009117\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,12]]},"references-count":7,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2002,12]]}},"alternative-id":["S0022481200009117"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150289","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,12]]}}}