{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:42:00Z","timestamp":1775839320034,"version":"3.50.1"},"reference-count":3,"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":17358,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1966,9,2]]},"abstract":"<jats:p>J. R. Shoenfield conjectured in a talk at the Berkeley Model Theory Symposium (1963) that, if <jats:bold>b<\/jats:bold> and <jats:bold>d<\/jats:bold> are non-zero recursively enumerable (r.e.) degrees such that <jats:italic>b<\/jats:italic> &lt; <jats:italic>d<\/jats:italic> then there exists an r.e. degree c such that <jats:bold>c<\/jats:bold> &lt; <jats:bold>d<\/jats:bold> and <jats:bold>b U c = d.<\/jats:bold> G. E. Sacks echoed this conjecture at the end of [3]. In this paper the conjecture is disproved. We construct r.e. degrees <jats:bold>b, d<\/jats:bold> such that <jats:bold>0 &lt; b &lt; d<\/jats:bold> and such that for no r.e. degree c is it true that <jats:bold>c<\/jats:bold> &lt; <jats:bold>d<\/jats:bold> and <jats:bold>b U c = d.<\/jats:bold> We are grateful to G. E. Sacks for suggesting this problem.<\/jats:p>","DOI":"10.2307\/2270459","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T20:34:36Z","timestamp":1146947676000},"page":"434-454","source":"Crossref","is-referenced-by-count":11,"title":["The impossibility of finding relative complements for recursively enumerable degrees"],"prefix":"10.1017","volume":"31","author":[{"given":"A. H.","family":"Lachlan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200066834_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1944-08111-1"},{"key":"S0022481200066834_ref001","first-page":"550","volume-title":"Introduction to Metamathematics","author":"Kleene","year":"1952"},{"key":"S0022481200066834_ref003","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\/S0022481200066834","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,2]],"date-time":"2019-06-02T20:45:24Z","timestamp":1559508324000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200066834\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1966,9,2]]},"references-count":3,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1966,9,2]]}},"alternative-id":["S0022481200066834"],"URL":"https:\/\/doi.org\/10.2307\/2270459","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1966,9,2]]}}}