{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T22:49:48Z","timestamp":1649198988507},"reference-count":13,"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:title>Abstract<\/jats:title><jats:p>Let \u0393 be a collection of relations on the reals and let <jats:italic>M<\/jats:italic> be a set of reals. We call <jats:italic>M<\/jats:italic> a <jats:italic>perfect set basis<\/jats:italic> for \u0393 if every set in \u0393 with parameters from <jats:italic>M<\/jats:italic> which is not totally included in <jats:italic>M<\/jats:italic> contains a perfect subset with code in <jats:italic>M<\/jats:italic>. A simple elementary proof is given of the following result (assuming mild regularity conditions on \u0393 and <jats:italic>M<\/jats:italic>): <jats:italic>If M is a perfect set basis for \u0393, the field of every wellordering in \u0393 is contained in M<\/jats:italic>. An immediate corollary is Mansfield's Theorem that the existence of a \u03a3<jats:sub arrange=\"stack\">2<\/jats:sub><jats:sup arrange=\"stack\">1<\/jats:sup> wellordering of the reals implies that every real is constructible. Other applications and extensions of the main result are also given.<\/jats:p>","DOI":"10.2307\/2273501","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:48:57Z","timestamp":1146937737000},"page":"630-634","source":"Crossref","is-referenced-by-count":2,"title":["The perfect set theorem and definable wellorderings of the continuum"],"prefix":"10.1017","volume":"43","author":[{"given":"Alexander S.","family":"Kechris","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049136_ref001","first-page":"41","volume-title":"Proceedings of the Second Scandinavian Logic Symposium","author":"Fenstad"},{"key":"S0022481200049136_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/155"},{"key":"S0022481200049136_ref005","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1970.35.451"},{"key":"S0022481200049136_ref009","first-page":"24","volume-title":"Mathematical logic and foundations of set theory","author":"Moschovakis","year":"1970"},{"key":"S0022481200049136_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1971-12789-1"},{"key":"S0022481200049136_ref011","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200049136_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF02764630"},{"key":"S0022481200049136_ref013","first-page":"58","volume-title":"Foundations of mathematics, Symposium Commemorating the 60th Birthday of K. G\u00f6del","author":"Solovay","year":"1966"},{"key":"S0022481200049136_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0419235-7"},{"key":"S0022481200049136_ref006","doi-asserted-by":"publisher","DOI":"10.4064\/fm-86-3-279-282"},{"key":"S0022481200049136_ref012","volume-title":"Mathematical logic","author":"Shoenfield","year":"1967"},{"key":"S0022481200049136_ref002","first-page":"269","volume":"39","author":"Kechris","year":"1974","journal-title":"On projective ordinals"},{"key":"S0022481200049136_ref007","author":"Martin","journal-title":"Projective sets and cardinal numbers: some questions related to the continuum problem"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049136","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T15:07:51Z","timestamp":1558969671000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049136\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,12]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1978,12]]}},"alternative-id":["S0022481200049136"],"URL":"https:\/\/doi.org\/10.2307\/2273501","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,12]]}}}