{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T21:22:48Z","timestamp":1775856168797,"version":"3.50.1"},"reference-count":18,"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":14894,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1973,6]]},"abstract":"<jats:p>The jump <jats:bold>a<\/jats:bold>\u2032 of a degree <jats:bold>a<\/jats:bold> is defined to be the largest degree recursively enumerable in <jats:bold>a<\/jats:bold> in the upper semilattice of degrees of unsolvability. We examine below some of the ways in which the jump operation is related to the partial ordering of the degrees. Fried berg [3] showed that the equation <jats:bold>a = x\u2032<\/jats:bold> is solvable if and only if <jats:italic>a \u2265 0\u2032<\/jats:italic>. Sacks [13] showed that we can find a solution of <jats:bold>a = x\u2032<\/jats:bold> which is <jats:bold>\u2264 0\u2032<\/jats:bold> (and in fact is r.e.) if and only if <jats:bold>a \u2265 0\u2032<\/jats:bold> and is r.e. in <jats:bold>0\u2032<\/jats:bold>. Spector [16] constructed a minimal degree and Sacks [13] constructed one <jats:bold>\u2264 0\u2032<\/jats:bold>. So far the only result concerning the relationship between minimal degrees and the jump operator is one due to Yates [17] who showed that there is a minimal predecessor for each non-recursive r.e. degree, and hence that there is a minimal degree with jump <jats:bold>0\u2032<\/jats:bold>. In \u00a71, we obtain an analogue of Friedberg's theorem by constructing a minimal degree solution for <jats:bold>a = x\u2032<\/jats:bold> whenever <jats:bold>a \u2265 0\u2032<\/jats:bold>. We incorporate Friedberg5s original number-theoretic device with a complicated sequence of approximations to the nest of trees necessary for the construction of a minimal degree. The proof of Theorem 1 is a revision of an earlier, shorter presentation, and incorporates many additions and modifications suggested by R. Epstein. In \u00a72, we show that any hope for a result analogous to that of Sacks on the jumps of r.e. degrees cannot be fulfilled since <jats:bold>0\u2033<\/jats:bold> is not the jump of any minimal degree below <jats:bold>0\u2032<\/jats:bold>. We use a characterization of the degrees below <jats:bold>0\u2032<\/jats:bold> with jump <jats:bold>0\u2033<\/jats:bold> similar to that found for r.e. degrees with jump <jats:bold>0\u2032<\/jats:bold> by R. W. Robinson [12]. Finally, in \u00a73, we give a proof that every degree <jats:bold>a \u2264 0\u2032<\/jats:bold> with <jats:bold>a\u2032 = 0\u2033<\/jats:bold> has a minimal predecessor. Yates [17] has already shown that every nonzero r.e. degree has a minimal predecessor, but that there is a nonzero degree <jats:bold>\u2264 0\u2032<\/jats:bold> with no minimal predecessor (see [18]; or for the original unrelativized result see [10] or [4]).<\/jats:p>","DOI":"10.2307\/2272061","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:22:43Z","timestamp":1146950563000},"page":"249-271","source":"Crossref","is-referenced-by-count":36,"title":["Minimal degrees and the jump operator"],"prefix":"10.1017","volume":"38","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":"S0022481200077859_ref005","unstructured":"Jockusch Carl G. Jr. , Upward closure and cohesive degrees (to appear)."},{"key":"S0022481200077859_ref015","first-page":"539","volume":"31","author":"Schoenfield","year":"1966","journal-title":"A theorem on minimal degrees"},{"key":"S0022481200077859_ref011","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19680140704"},{"key":"S0022481200077859_ref013","volume-title":"Annals of Mathematics Studies","author":"Sacks","year":"1963"},{"key":"S0022481200077859_ref006","first-page":"489","volume":"34","author":"Jockusch","year":"1969","journal-title":"The degrees of hyperhyperimmune sets"},{"key":"S0022481200077859_ref002","first-page":"A","article-title":"Sets recursively enumerable in high degrees","volume":"19","author":"Cooper","year":"1972","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200077859_ref001","first-page":"598","volume":"37","author":"Cooper","year":"1972","journal-title":"Jump equivalence of the \u039420 hyperhyperimmune sets"},{"key":"S0022481200077859_ref003","first-page":"159","volume":"22","author":"Friedberg","year":"1957","journal-title":"A criterion for completeness of degrees of unsohability"},{"key":"S0022481200077859_ref004","first-page":"1","article-title":"Initial segments of Turing degrees","volume":"19","author":"Hugill","year":"1968","journal-title":"Proceedings of the London Mathematical Society"},{"key":"S0022481200077859_ref008","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-16.1.537"},{"key":"S0022481200077859_ref012","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19680142105"},{"key":"S0022481200077859_ref017","first-page":"243","volume":"35","author":"Yates","year":"1970","journal-title":"Initial segments of the degrees of unsohability, II. Minimal degrees"},{"key":"S0022481200077859_ref014","first-page":"383","volume":"35","author":"Sasso","year":"1970","journal-title":"A cornucopia of minimal degrees"},{"key":"S0022481200077859_ref016","doi-asserted-by":"publisher","DOI":"10.2307\/1969604"},{"key":"S0022481200077859_ref018","unstructured":"Yates C. E. M. , Initial segments and implications for the structure of degrees (to appear)."},{"key":"S0022481200077859_ref010","unstructured":"Martin Donald A. , Measure, category and degrees of unsohability (unpublished)."},{"key":"S0022481200077859_ref009","first-page":"273","volume":"28","author":"Martin","year":"1963","journal-title":"A theorem on hyperhypersimple sets"},{"key":"S0022481200077859_ref007","volume-title":"Introduction to metamathematics","author":"Kleene","year":"1952"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200077859","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T19:54:59Z","timestamp":1559246099000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200077859\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1973,6]]},"references-count":18,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1973,6]]}},"alternative-id":["S0022481200077859"],"URL":"https:\/\/doi.org\/10.2307\/2272061","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1973,6]]}}}