{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T18:17:17Z","timestamp":1769019437563,"version":"3.49.0"},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6493,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The Sacks Density Theorem [7] states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of <jats:italic>P<\/jats:italic><jats:sup>\u2212<\/jats:sup> + <jats:italic>B<\/jats:italic>\u03a3<jats:sub>2<\/jats:sub>. The proof has two components: a lemma that in any model of <jats:italic>P<\/jats:italic><jats:sup>\u2212<\/jats:sup> + <jats:italic>B<\/jats:italic>\u03a3<jats:sub>2<\/jats:sub>, if <jats:italic>B<\/jats:italic> is recursively enumerable and incomplete then <jats:italic>I<\/jats:italic>\u03a3<jats:sub>1<\/jats:sub> holds relative to <jats:italic>B<\/jats:italic> and an adaptation of Shore's [9] blocking technique in <jats:italic>\u03b1<\/jats:italic>-recursion theory to models of arithmetic.<\/jats:p>","DOI":"10.2307\/2275670","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:58:29Z","timestamp":1146956309000},"page":"450-467","source":"Crossref","is-referenced-by-count":8,"title":["The Sacks density theorem and \u03a3<sub>2<\/sub>-bounding"],"prefix":"10.1017","volume":"61","author":[{"given":"Marcia J.","family":"Groszek","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael E.","family":"Mytilinaios","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Theodore A.","family":"Slaman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200017370_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90007-3"},{"key":"S0022481200017370_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/1970214"},{"key":"S0022481200017370_ref004","first-page":"38","volume":"54","author":"Mytilinaios","year":"1989","journal-title":"Finite injury and \u03a32-induction"},{"key":"S0022481200017370_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90004-6"},{"key":"S0022481200017370_ref005","first-page":"212","volume":"53","author":"Mytilinaios","year":"1988","journal-title":"\u03a32-collection and the infinite injury priority method"},{"key":"S0022481200017370_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0083670"},{"key":"S0022481200017370_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"},{"key":"S0022481200017370_ref001","first-page":"349","article-title":"\u03a3n definability without \u03a3n induction","volume":"334","author":"Chong","year":"1992","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200017370_ref003","unstructured":"Groszek M. J. and Slaman T. A. , On Turing reducibility, preprint, 1994."},{"key":"S0022481200017370_ref002","first-page":"205","volume-title":"Recursion theory week, Oberwolfach 1989","volume":"1432","author":"Groszek","year":"1990"},{"key":"S0022481200017370_ref007","doi-asserted-by":"publisher","DOI":"10.2307\/1970393"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200017370","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,12]],"date-time":"2019-05-12T21:29:14Z","timestamp":1557696554000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200017370\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1996,6]]}},"alternative-id":["S0022481200017370"],"URL":"https:\/\/doi.org\/10.2307\/2275670","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,6]]}}}