{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T13:46:44Z","timestamp":1772372804800,"version":"3.50.1"},"reference-count":8,"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":16358,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1969,5,29]]},"abstract":"<jats:p>The problem of finding an infinite set of natural numbers which contains no subsets of higher (Turing) degree was first posed by W. Miller [3] and was brought to our attention by C. G. Jockusch, Jr., who proved that such a set, if it existed, could not be hyperarithmetic.<jats:sup>2<\/jats:sup> In this paper we construct an infinite set which is not recursive in any of its coinfinite subsets, and thus contains no subset of higher degree. Our original proof made use of the result (attributed to Ehrenfeucht) that every subset of <jats:italic>2<jats:sup>\u03c9<\/jats:sup><\/jats:italic> which is open (in the standard topology) is \u201cRamsey\u201d.<\/jats:p>","DOI":"10.2307\/2270981","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T16:52:03Z","timestamp":1146934323000},"page":"53-56","source":"Crossref","is-referenced-by-count":20,"title":["Sets with no subset of higher degree"],"prefix":"10.1017","volume":"34","author":[{"given":"Robert I.","family":"Soare","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200095074_ref008","first-page":"644","article-title":"Sets with no subsets of higher arithmetic degree","volume":"15","author":"Soare","year":"1968","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200095074_ref007","first-page":"513","article-title":"Sets with no subset of higher degree","volume":"15","author":"Soare","year":"1968","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200095074_ref006","volume-title":"Annals of Mathematics Studies","author":"Sacks","year":"1963"},{"key":"S0022481200095074_ref003","unstructured":"Miller W. , Sets of integers and degrees of unsolvability, Master's thesis, University of Washington."},{"key":"S0022481200095074_ref002","first-page":"521","article-title":"Uniformly introreducible sets","volume":"33","author":"Jockuschi","year":"1968","journal-title":"this Journal"},{"key":"S0022481200095074_ref004","unstructured":"Prikry K. and Galvin F. , A combinatorial theorem (to appear)."},{"key":"S0022481200095074_ref005","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200095074_ref001","doi-asserted-by":"publisher","DOI":"10.4064\/fm-61-2-215-223"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200095074","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T16:13:20Z","timestamp":1559405600000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200095074\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1969,5,29]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1969,5,29]]}},"alternative-id":["S0022481200095074"],"URL":"https:\/\/doi.org\/10.2307\/2270981","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1969,5,29]]}}}