{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:38:55Z","timestamp":1775839135919,"version":"3.50.1"},"reference-count":12,"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":9050,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1989,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Work in the setting of the recursively enumerable sets and their Turing degrees. A set <jats:italic>X<\/jats:italic> is <jats:italic>low<\/jats:italic> if <jats:italic>X<\/jats:italic>\u2032, its Turing jump, is recursive in \u2205\u2032 and <jats:italic>high<\/jats:italic> if <jats:italic>X<\/jats:italic>\u2032 computes \u2205\u2033. Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. <jats:italic>W<\/jats:italic> is <jats:italic>deep<\/jats:italic>, if for each recursively enumerable set <jats:italic>A<\/jats:italic>, the jump of <jats:italic>A<\/jats:italic> \u2295 <jats:italic>W<\/jats:italic> is recursive in the jump of <jats:italic>A<\/jats:italic>. We prove that there are no deep degrees other than the recursive one.<\/jats:p><jats:p>Given a set <jats:italic>W<\/jats:italic>, we enumerate a set <jats:italic>A<\/jats:italic> and approximate its jump. The construction of <jats:italic>A<\/jats:italic> is governed by strategies, indexed by the Turing functionals <jats:italic>\u03a6<\/jats:italic>. Simplifying the situation, a typical strategy converts a failure to recursively compute <jats:italic>W<\/jats:italic> into a constraint on the enumeration of <jats:italic>A<\/jats:italic>, so that (<jats:italic>W<\/jats:italic> \u2295 <jats:italic>A<\/jats:italic>)\u2032 is forced to disagree with <jats:italic>\u03a6<\/jats:italic>(\u2212;<jats:italic>A<\/jats:italic>\u2032). The conversion has some ambiguity; in particular, <jats:italic>A<\/jats:italic> cannot be found uniformly from <jats:italic>W<\/jats:italic>.<\/jats:p><jats:p>We also show that there is a \u201cmoderately\u201d deep degree: There is a low nonzero degree whose join with any other low degree is not high.<\/jats:p>","DOI":"10.2307\/2274854","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:30:21Z","timestamp":1146940221000},"page":"376-395","source":"Crossref","is-referenced-by-count":5,"title":["A limit on relative genericity in the recursively enumerable sets"],"prefix":"10.1017","volume":"54","author":[{"given":"Steffen","family":"Lempp","sequence":"first","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":"S0022481200027031_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0076222"},{"key":"S0022481200027031_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"},{"key":"S0022481200027031_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/042\/791052"},{"key":"S0022481200027031_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1963-0155747-3"},{"key":"S0022481200027031_ref005","first-page":"809","volume":"47","author":"Maass","year":"1982","journal-title":"Recursively enumerable generic sets"},{"key":"S0022481200027031_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19660120125"},{"key":"S0022481200027031_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/1970214"},{"key":"S0022481200027031_ref003","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-16.1.537"},{"key":"S0022481200027031_ref008","doi-asserted-by":"publisher","DOI":"10.2307\/1970393"},{"key":"S0022481200027031_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71892-5"},{"key":"S0022481200027031_ref009","volume-title":"Annals of Pure and Applied Logic","author":"Shore"},{"key":"S0022481200027031_ref001","unstructured":"Bickford M. and Mills C. F. , Lowness properties of r.e. sets (to appear)."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200027031","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,19]],"date-time":"2019-05-19T16:26:50Z","timestamp":1558283210000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200027031\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,6]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1989,6]]}},"alternative-id":["S0022481200027031"],"URL":"https:\/\/doi.org\/10.2307\/2274854","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,6]]}}}