{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T21:22:17Z","timestamp":1648761737957},"reference-count":26,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":2420,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[2007,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>There are \u03a0<jats:sub>5<\/jats:sub> formulas in the language of the Turing degrees, <jats:italic>D<\/jats:italic>, with \u2264, \u22c1 and \u22c0, that define the relations <jats:bold>x<\/jats:bold>\u2033 \u2264 <jats:bold>y<\/jats:bold>\u2033, <jats:bold>x<\/jats:bold>\u2033 = <jats:bold>y<\/jats:bold>\u2033 and so <jats:bold>x<\/jats:bold> \u2208 <jats:bold>L<\/jats:bold><jats:sub>2<\/jats:sub>(<jats:bold>y<\/jats:bold>) = {<jats:bold>x<\/jats:bold> \u2265 <jats:bold>y<\/jats:bold> \u2223 <jats:bold>x<\/jats:bold>\u2033 = <jats:bold>y<\/jats:bold>\u2033} in any jump ideal containing <jats:bold>0<\/jats:bold><jats:sup>(\u03c9)<\/jats:sup>. There are also \u03a3<jats:sub>6<\/jats:sub> &amp; \u03a0<jats:sub>6<\/jats:sub> and \u03a0<jats:sub>8<\/jats:sub> formulas that define the relations <jats:bold>w<\/jats:bold> = <jats:bold>x<\/jats:bold>\u2033 and <jats:bold>w<\/jats:bold> = <jats:bold>x<\/jats:bold>\u2032, respectively, in any such ideal <jats:italic>I<\/jats:italic>. In the language with just \u2264 the quantifier complexity of each of these definitions increases by one. On the other hand, no \u03a0<jats:sub>2<\/jats:sub> or \u03a3<jats:sub>2<\/jats:sub> formula in the language with just \u2264 defines <jats:bold>L<\/jats:bold><jats:sub>2<\/jats:sub> or <jats:bold>x<\/jats:bold> \u2208 <jats:bold>L<\/jats:bold><jats:sub>2<\/jats:sub>(<jats:bold>y<\/jats:bold>). Our arguments and constructions are purely degree theoretic without any appeals to absoluteness considerations, set theoretic methods or coding of models of arithmetic. As a corollary, we see that every automorphism of <jats:italic>I<\/jats:italic> is fixed on every degree above <jats:bold>0<\/jats:bold>\u2033 and every relation on <jats:italic>I<\/jats:italic> that is invariant under double jump or joining with <jats:bold>0<\/jats:bold>\u2033 is definable over <jats:italic>I<\/jats:italic> if and only if it is definable in second order arithmetic with set quantification ranging over sets whose degrees are in <jats:italic>I<\/jats:italic>. Similar direct coding arguments show that every hyperjump ideal <jats:italic>I<\/jats:italic> is rigid and biinterpretable with second order arithmetic with set quantification ranging over sets with hyperdegrees in <jats:italic>I<\/jats:italic>. Analogous results hold for various coarser degree structures.<\/jats:p>","DOI":"10.2178\/bsl\/1185803806","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T14:55:58Z","timestamp":1197557758000},"page":"226-239","source":"Crossref","is-referenced-by-count":1,"title":["Local Definitions in Degree Structures: The Turing Jump, Hyperdegrees and Beyond"],"prefix":"10.1017","volume":"13","author":[{"given":"Richard A.","family":"Shore","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600002225_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12013-2"},{"key":"S1079898600002225_ref023","doi-asserted-by":"publisher","DOI":"10.2307\/1971028"},{"key":"S1079898600002225_ref001","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-53.2.193"},{"key":"S1079898600002225_ref021","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-01-06015-4"},{"key":"S1079898600002225_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90023-1"},{"key":"S1079898600002225_ref026","article-title":"Definability in degree structures","author":"Slaman","year":"2007","journal-title":"Journal of Mathematical Logic"},{"key":"S1079898600002225_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0086116"},{"key":"S1079898600002225_ref006","doi-asserted-by":"crossref","first-page":"325","DOI":"10.4064\/fm-56-3-325-345","article-title":"Some applications of the notion of forcing and generic sets","volume":"56","author":"Feferman","year":"1965","journal-title":"Fundamenta Mathematicae"},{"key":"S1079898600002225_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3870(200101)47:1<3::AID-MALQ3>3.0.CO;2-A"},{"key":"S1079898600002225_ref003","volume-title":"On a conjecture of Kleene and Post","author":"Cooper","year":"1993"},{"key":"S1079898600002225_ref009","unstructured":"Jockusch C. 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