{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T09:28:30Z","timestamp":1773221310278,"version":"3.50.1"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12885,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,12]]},"abstract":"<jats:p>This paper concerns certain relationships between the ordering of degrees of unsolvability and the jump operation. It is shown that every minimal degree <jats:bold><jats:italic>a<\/jats:italic> &lt; 0\u2032<\/jats:bold> satisfies <jats:bold><jats:italic>a<\/jats:italic>\u2033<\/jats:bold> = <jats:bold>0\u2033<\/jats:bold>. To restate this result in more suggestive language and compare it with related results, we shall use notation based on the now standard terminology of \u201chigh\u201d and \u201clow\u201d degrees. Let <jats:italic>H<\/jats:italic><jats:sub>n<\/jats:sub> be the class of degrees <jats:bold><jats:italic>a<\/jats:italic> &lt; 0\u2032<\/jats:bold> such that <jats:italic><jats:bold>a<\/jats:bold><jats:sup>(n)<\/jats:sup><\/jats:italic> =  <jats:bold>0<\/jats:bold><jats:sup>(n+1)<\/jats:sup>, and let <jats:italic>L<jats:sub>n<\/jats:sub><\/jats:italic> be the class of degrees <jats:italic><jats:bold>a<\/jats:bold><\/jats:italic> \u2264 <jats:bold>0\u2032<\/jats:bold> such that <jats:italic><jats:bold>a<\/jats:bold><\/jats:italic><jats:sup>n<\/jats:sup> = <jats:bold>0<\/jats:bold><jats:sup>n<\/jats:sup>. (Observe that <jats:italic>H<jats:sub>i<\/jats:sub><\/jats:italic> \u2286 <jats:italic>L<jats:sub>j<\/jats:sub><\/jats:italic> and <jats:italic>L<\/jats:italic><jats:sub>i<\/jats:sub> \u2286 <jats:italic>L<\/jats:italic><jats:sub>j<\/jats:sub>, whenever <jats:italic>i<\/jats:italic> \u2264 <jats:italic>j<\/jats:italic>, and <jats:italic>H<\/jats:italic><jats:sub>i<\/jats:sub> \u2229 <jats:italic>L<\/jats:italic><jats:sub>j<\/jats:sub> = \u2205 for all <jats:italic>i<\/jats:italic> and<jats:italic>j.<\/jats:italic>) The result mentioned above may now be restated in the form that every minimal degree <jats:italic><jats:bold>a<\/jats:bold><\/jats:italic> \u2264 0\u2032 is in <jats:italic>L<\/jats:italic><jats:sub>2<\/jats:sub>. This extends an earlier result of S. B. Cooper ([1], see also [4]) that no minimal degree <jats:italic><jats:bold>a<\/jats:bold> &lt; 0\u2032 is in <jats:italic>H<\/jats:italic><jats:sub>1<\/jats:sub>. In the other direction, Sasso, Epstein, and Cooper ([10], [15]) have shown that there is a minimal degree <jats:bold><jats:italic>a<\/jats:italic> &lt;  0\u2032<\/jats:bold> which is not in <jats:italic>L<\/jats:italic><jats:sub>1<\/jats:sub>. Also, C.E.M. Yates [14, Corollary 11.14], showed the existence of a minimal degree <jats:bold><jats:italic>a<\/jats:italic> &lt;  0\u2032<\/jats:bold><\/jats:italic> in <jats:italic>L<\/jats:italic><jats:sub>1<\/jats:sub>. Thus each minimal degree <jats:bold><jats:italic>a &lt; 0\u2032<\/jats:italic><\/jats:bold> lies in exactly one of the two classes <jats:italic>L<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>L<\/jats:italic><jats:sub>2<\/jats:sub> \u2013 <jats:italic>L<\/jats:italic><jats:sub>1<\/jats:sub> and each of the classes contains minimal degrees.<\/jats:p><jats:p>Our results are not restricted to the degrees below <jats:bold>0\u2032<\/jats:bold>. We show in fact that every minimal degree <jats:italic><jats:bold>a<\/jats:bold><\/jats:italic> satisfies <jats:italic>a\u2033<\/jats:italic> = (<jats:bold>a<\/jats:bold> \u222a <jats:bold>0\u2032<\/jats:bold>)\u2032. To restate this result and discuss extensions of it, we extend the \u201chigh-low\u201d classification of degrees from the degrees below <jats:bold>0\u2032<\/jats:bold> to degrees in general. There are a number of fairly plausible ways of doing this, but we choose the one way we know of doing so which leads to interesting results. Let <jats:italic>GH<jats:sub>n<\/jats:sub><\/jats:italic> be the class of degrees <jats:italic>a<\/jats:italic> such that <jats:italic>a<jats:sup>(n)<\/jats:sup> = (a \u222a 0\u2032)<jats:sup>(n)<\/jats:sup><\/jats:italic>, and let <jats:italic>GL<jats:sub>n<\/jats:sub><\/jats:italic> be the class of degrees <jats:italic>a<\/jats:italic> such that <jats:italic>a<jats:sup>(n)<\/jats:sup><\/jats:italic> = (a \u222a 0\u2032)<jats:sup>(n\u22121)<\/jats:sup>.<\/jats:p>","DOI":"10.2307\/2273510","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:48:57Z","timestamp":1146952137000},"page":"715-724","source":"Crossref","is-referenced-by-count":36,"title":["Double jumps of minimal degrees"],"prefix":"10.1017","volume":"43","author":[{"suffix":"Jr","given":"Carl G.","family":"Jockusch","sequence":"first","affiliation":[]},{"given":"David B.","family":"Posner","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049227_ref015","doi-asserted-by":"publisher","DOI":"10.4064\/fm-82-3-217-237"},{"key":"S0022481200049227_ref013","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0392534-3"},{"key":"S0022481200049227_ref012","volume-title":"Degrees of unsolvability","author":"Shoenfield","year":"1971"},{"key":"S0022481200049227_ref011","doi-asserted-by":"publisher","DOI":"10.2307\/1970028"},{"key":"S0022481200049227_ref016","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100052221"},{"key":"S0022481200049227_ref003","first-page":"A","article-title":"Degrees of generic sets","volume":"22","author":"Jockusch","year":"1975","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200049227_ref007","unstructured":"Posner D. , High degrees, Doctoral dissertation, University of California, Berkeley, 1977."},{"key":"S0022481200049227_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90011-3"},{"key":"S0022481200049227_ref004","article-title":"Simple proofs of some theorems on high degrees","author":"Jockusch","journal-title":"Canadian Journal of mathematics"},{"key":"S0022481200049227_ref009","volume-title":"Annals of Mathematical Studies","author":"Sacks","year":"1963"},{"key":"S0022481200049227_ref014","first-page":"243","volume":"35","author":"Yates","year":"1970","journal-title":"Initial segments of the degrees of unsolvability, Part II: Minimal degrees"},{"key":"S0022481200049227_ref010","first-page":"571","volume":"39","author":"Sasso","year":"1974","journal-title":"A minimal degree not realizing least possible jump"},{"key":"S0022481200049227_ref001","first-page":"249","volume":"38","author":"Cooper","year":"1973","journal-title":"Minimal degrees and the jump operator"},{"key":"S0022481200049227_ref005","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1969.30.67"},{"key":"S0022481200049227_ref006","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19660120125"},{"key":"S0022481200049227_ref008","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049227","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:07:58Z","timestamp":1558984078000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049227\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,12]]},"references-count":16,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1978,12]]}},"alternative-id":["S0022481200049227"],"URL":"https:\/\/doi.org\/10.2307\/2273510","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,12]]}}}