{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T13:01:41Z","timestamp":1773147701904,"version":"3.50.1"},"reference-count":13,"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:p>Friedberg [3] showed that every degree of unsolvability above <jats:bold>0<\/jats:bold>\u2032 is the jump of some degree, and Sacks [9] showed that the degrees above <jats:bold>0<\/jats:bold>\u2032 which are recursively enumerable (r.e.) in <jats:bold>0<\/jats:bold>\u2032 are the jumps of the r.e. degrees.<\/jats:p><jats:p>In this paper we examine the extent to which the Sacks jump theorem can be combined with the minimal pair theorem of Lachlan [4] and Yates [13]. We prove below that there is a degree <jats:bold>c<\/jats:bold> &gt; <jats:bold>0<\/jats:bold>\u2032 which is r.e. in <jats:bold>0<\/jats:bold>\u2032 but which is <jats:italic>not<\/jats:italic> the jump of half a minimal pair of r.e. degrees.<\/jats:p><jats:p>This extends Yates' result [13] proving the existence of <jats:italic>noncappable<\/jats:italic> degrees (that is, r.e. degrees <jats:bold>a<\/jats:bold> &lt; <jats:bold>0<\/jats:bold>\u2032 for which there is no corresponding r.e. <jats:bold>b<\/jats:bold> &gt; <jats:bold>0<\/jats:bold> with <jats:bold>a<\/jats:bold> \u2229 <jats:bold>b<\/jats:bold> = <jats:bold>0<\/jats:bold>).<\/jats:p><jats:p>It also throws more light on the class <jats:bold>PS<\/jats:bold> of <jats:italic>promptly simple degrees<\/jats:italic>. It was shown by Ambos-Spies, Jockusch, Shore and Soare [1] that <jats:bold>PS<\/jats:bold> coincides with the class <jats:bold>NC<\/jats:bold> of noncappable degrees, and with the class <jats:bold>LC<\/jats:bold> of all <jats:italic>low-cuppable<\/jats:italic> degrees, and (using earlier work of Maass, Shore and Stob [5]) that <jats:bold>PS<\/jats:bold> splits every class <jats:bold>H<\/jats:bold><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> or <jats:bold>L<\/jats:bold><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>, <jats:italic>n<\/jats:italic> \u2265 0, in the high-low hierarchy of r.e. degrees.<\/jats:p><jats:p>If <jats:bold>c<\/jats:bold> &gt; <jats:bold>0<\/jats:bold>\u2032, with <jats:bold>c<\/jats:bold> r.e. in <jats:bold>0<\/jats:bold>\u2032, let<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200027006_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>and call <jats:bold>c<\/jats:bold><jats:sup>\u22121<\/jats:sup> the <jats:italic>jump class<\/jats:italic> for <jats:bold>c<\/jats:bold>. It is easy to see that every jump class contains members of PS (= NC = LC). By Sacks [8] there exists a low <jats:bold>a<\/jats:bold> \u2208 <jats:bold>LC<\/jats:bold>, where of course [<jats:bold>a, 0<\/jats:bold>\u2032] (= {br.e. \u2223<jats:bold>a<\/jats:bold> \u2264 <jats:bold>b<\/jats:bold> \u2264 <jats:bold>0<\/jats:bold>\u2032}) \u2286 <jats:bold>LC = PS<\/jats:bold>. But by Robinson [7] [<jats:bold>a, 0<\/jats:bold>\u2032] intersects with every jump class.<\/jats:p>","DOI":"10.2307\/2274851","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:30:21Z","timestamp":1146954621000},"page":"324-353","source":"Crossref","is-referenced-by-count":6,"title":["A jump class of noncappable degrees"],"prefix":"10.1017","volume":"54","author":[{"given":"S. B.","family":"Cooper","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200027006_ref002","unstructured":"Bickford M. , The jump operator in strong reducibilities, Ph.D. thesis, University of Wisconsin, Madison, Wisconsin, 1983."},{"key":"S0022481200027006_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1963-0155747-3"},{"key":"S0022481200027006_ref013","first-page":"159","volume":"31","author":"Yates","year":"1966","journal-title":"A minimal pair of recursively enumerable degrees"},{"key":"S0022481200027006_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/042\/791052"},{"key":"S0022481200027006_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71892-5"},{"key":"S0022481200027006_ref007","first-page":"586","volume":"93","author":"Robinson","year":"1971","journal-title":"Jump restricted interpolation in the r.e. degrees"},{"key":"S0022481200027006_ref010","volume-title":"Annals of Pure and Applied Logic","author":"Shore"},{"key":"S0022481200027006_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090950"},{"key":"S0022481200027006_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1984-0719661-8"},{"key":"S0022481200027006_ref003","first-page":"159","volume":"22","author":"Friedberg","year":"1957","journal-title":"A criterion for completeness of degrees of unsolvability"},{"key":"S0022481200027006_ref004","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-16.1.537"},{"key":"S0022481200027006_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760850"},{"key":"S0022481200027006_ref008","volume-title":"Degrees of unsolvability","author":"Sacks","year":"1966"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200027006","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,19]],"date-time":"2019-05-19T20:26:06Z","timestamp":1558297566000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200027006\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,6]]},"references-count":13,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1989,6]]}},"alternative-id":["S0022481200027006"],"URL":"https:\/\/doi.org\/10.2307\/2274851","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,6]]}}}