{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T13:35:33Z","timestamp":1773149733974,"version":"3.50.1"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11699,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,3]]},"abstract":"Relativization\u2014the principle that says one can carry over proofs and theorems about partial recursive functions and Turing degrees to functions partial recursive in any given set A<\/jats:italic> and the Turing degrees of sets in which A<\/jats:italic> is recursive\u2014is a pervasive phenomenon in recursion theory. It led H. Rogers, Jr. [15] to ask if, for every degree d<\/jats:bold>, (\u2265 d<\/jats:bold>), the partial ordering of Turing degrees above d<\/jats:bold>, is isomorphic to all the degrees . We showed in Shore [17] that this homogeneity conjecture is false. More specifically we proved that if, for some n<\/jats:italic>, the degree of Kleene's (the complete set) is recursive in d<\/jats:bold>(n<\/jats:italic>)<\/jats:sup> then \u2247 (\u2264 d<\/jats:bold>). The key ingredient of the proof was a new version of a result from Nerode and Shore [13] (hereafter NS I) that any isomorphism \u03c6: \u2192 (\u2265 d<\/jats:bold>) must be the identity on some cone, i.e., there is an a<\/jats:bold> called the base of the cone such that b<\/jats:bold> \u2265 a<\/jats:bold> \u21d2 \u03c6(b<\/jats:bold>) = b<\/jats:bold>. This result was combined with information about minimal covers from Jockusch and Soare [8] and Harrington and Kechris [3] to derive a contradiction from the existence of such an isomorphism if deg() \u2264 d<\/jats:bold>(n<\/jats:italic>)<\/jats:sup>.<\/jats:p>","DOI":"10.2307\/2273376","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:58:32Z","timestamp":1146952712000},"page":"8-16","source":"Crossref","is-referenced-by-count":21,"title":["On homogeneity and definability in the first-order theory of the Turing degrees"],"prefix":"10.1017","volume":"47","author":[{"given":"Richard A.","family":"Shore","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044650_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90023-1"},{"key":"S0022481200044650_ref017","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.76.9.4218"},{"key":"S0022481200044650_ref005","article-title":"Review of Selman [16]","volume":"47","author":"Jockusch","year":"1974","journal-title":"Mathematical Reviews"},{"key":"S0022481200044650_ref004","first-page":"601","volume":"43","author":"Hodes","year":"1978","journal-title":"Uniform upper bounds on ideals of Turing degrees"},{"key":"S0022481200044650_ref011","first-page":"A","article-title":"Initial segments of the degrees below 0\u2032","volume":"25","author":"Lerman","year":"1978","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200044650_ref003","first-page":"445","article-title":"A basis result for \u03a330 sets of reals with an application to minimal covers","volume":"53","author":"Harrington","year":"1975","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481200044650_ref001","first-page":"A","article-title":"Analysis and degrees \u2264 0\u2032","volume":"25","author":"Epstein","year":"1978","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200044650_ref006","first-page":"715","volume":"43","author":"Jockusch","year":"1978","journal-title":"Double jumps of minimal degrees"},{"key":"S0022481200044650_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF03007659"},{"key":"S0022481200044650_ref016","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-25.4.586"},{"key":"S0022481200044650_ref012","volume-title":"The degrees of unsolvability","author":"Lerman","year":"1982"},{"key":"S0022481200044650_ref015","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200044650_ref013","volume-title":"Proceedings of the Kleene Symposium","author":"Nerode","year":"1979"},{"key":"S0022481200044650_ref010","first-page":"289","volume":"41","author":"Lachlan","year":"1976","journal-title":"Countable initial segments of the degrees of unsolvability"},{"key":"S0022481200044650_ref002","volume-title":"Lecture Notes in Mathematics","volume":"759","author":"Epstein","year":"1979"},{"key":"S0022481200044650_ref018","doi-asserted-by":"publisher","DOI":"10.2307\/1971028"},{"key":"S0022481200044650_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1970-0265154-X"},{"key":"S0022481200044650_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(80)90004-2"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044650","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T21:42:37Z","timestamp":1558734157000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044650\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,3]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1982,3]]}},"alternative-id":["S0022481200044650"],"URL":"https:\/\/doi.org\/10.2307\/2273376","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,3]]}}}