{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T13:07:47Z","timestamp":1649077667305},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12976,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,9]]},"abstract":"<jats:p>It is proved here, assuming Projective Determinacy, that every ascending sequence of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049392_inline1\" \/>-degrees has a minimal strict upper bound but no least strict upper bound. This generalizes a result of Friedman for <jats:italic>n<\/jats:italic> = 1.<\/jats:p><jats:p>Our general notation and terminology will be that of [Ke1] and [Mo1]. Letters <jats:italic>i,j, k<\/jats:italic>,\u2026 denote members of <jats:italic>\u03c9<\/jats:italic> and <jats:italic>\u03b1<\/jats:italic>, <jats:italic>\u03b2<\/jats:italic>, \u03d2,\u2026 members of <jats:italic>\u03c9<\/jats:italic><jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup> i.e. <jats:italic>reals<\/jats:italic>. Projective Determinacy (PD) is the hypothesis that every projective set of reals in determined, while in general for a collection of sets of reals \u0393, Determinacy (\u0393) abbreviates the statement that every set in \u0393 is determined.<\/jats:p><jats:p><jats:bold>\u00a71. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049392_inline1\" \/>-degrees.<\/jats:bold> For each <jats:italic>m<\/jats:italic> \u2265 1 and <jats:italic>\u03b1<\/jats:italic>, <jats:italic>\u03b2<\/jats:italic> \u0404 <jats:italic>\u03c9<\/jats:italic><jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup> let <jats:italic>\u03b1<\/jats:italic> \u2264<jats:sub><jats:italic>m<\/jats:italic><\/jats:sub><jats:italic>\u03b2<\/jats:italic> \u21d4 <jats:italic>\u03b1<\/jats:italic> \u0404 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049392_inline2\" \/>(<jats:italic>\u03b2<\/jats:italic>), <jats:italic>\u03b1<\/jats:italic> &lt;<jats:sup><jats:italic>m<\/jats:italic><\/jats:sup><jats:italic>\u03b2<\/jats:italic> \u21d4 <jats:italic>\u03b1<\/jats:italic> \u2264<jats:sub><jats:italic>m<\/jats:italic><\/jats:sub><jats:italic>\u03b2<\/jats:italic> \u2227 <jats:italic>\u03b2<\/jats:italic> \u2270<jats:sub><jats:italic>m<\/jats:italic><\/jats:sub><jats:italic>\u03b1<\/jats:italic>, and <jats:italic>\u03b1<\/jats:italic> \u2261<jats:sub><jats:italic>m<\/jats:italic><\/jats:sub><jats:italic>\u03b2<\/jats:italic> \u21d4 <jats:italic>\u03b1<\/jats:italic> \u2264<jats:sub><jats:italic>m<\/jats:italic><\/jats:sub><jats:italic>\u03b2<\/jats:italic> \u2227 <jats:italic>\u03b2<\/jats:italic> \u2264<jats:sub><jats:italic>m<\/jats:italic><\/jats:sub><jats:italic>\u03b1<\/jats:italic>.Clearly \u2261<jats:sub><jats:italic>m<\/jats:italic><\/jats:sub> is an equivalence relation on <jats:italic>\u03c9<\/jats:italic><jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup>. The \u2261<jats:sub><jats:italic>m<\/jats:italic><\/jats:sub> -equivalence class of <jats:italic>\u03b1<\/jats:italic> \u0404 <jats:italic>\u03c9<\/jats:italic><jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup> is called its <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049392_inline2\" \/>-degree, in symbols<\/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=\"S0022481200049392_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>If d = [<jats:italic>\u03b1<\/jats:italic>]<jats:sub><jats:italic>m<\/jats:italic><\/jats:sub>, e = [<jats:italic>\u03b2<\/jats:italic>]<jats:sub><jats:italic>m<\/jats:italic><\/jats:sub> then we define<\/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=\"S0022481200049392_eqnU2\" \/><\/jats:disp-formula><\/jats:p>","DOI":"10.2307\/2273527","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:48:21Z","timestamp":1146952101000},"page":"502-507","source":"Crossref","is-referenced-by-count":1,"title":["Minimal upper bounds for sequences of -degrees"],"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":"S0022481200049392_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(76)90187-0"},{"key":"S0022481200049392_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1971-12789-1"},{"key":"S0022481200049392_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/155"},{"key":"S0022481200049392_ref005","unstructured":"Kechris A. S. , The ordinal of the Q-degrees. II, mimeographed notes, 12 1973."},{"key":"S0022481200049392_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(73)90012-0"},{"key":"S0022481200049392_ref001","doi-asserted-by":"publisher","DOI":"10.4064\/fm-81-3-183-192"},{"key":"S0022481200049392_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-61-2-215-223"},{"key":"S0022481200049392_ref008","first-page":"331","article-title":"Forcing with perfect closed sets","volume":"13","author":"Sacks","year":"1971","journal-title":"American Mathematical Society Symposium in Pure Mathematics"},{"key":"S0022481200049392_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0419235-7"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049392","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:33:24Z","timestamp":1558985604000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049392\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,9]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1978,9]]}},"alternative-id":["S0022481200049392"],"URL":"https:\/\/doi.org\/10.2307\/2273527","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,9]]}}}