{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T04:00:40Z","timestamp":1648958440693},"reference-count":7,"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":11607,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,6]]},"abstract":"<jats:p>Jockusch and Posner [4] showed that every minimal \u03c9-degree is GL<jats:sub>2<\/jats:sub>. This is achieved by exhibiting a function <jats:italic>f<\/jats:italic> recursive in <jats:bold>0<\/jats:bold>\u2032 which dominates every function of minimal \u03c9-degree. The function <jats:italic>f<\/jats:italic> has the peculiar property that for every <jats:italic>s<\/jats:italic>, <jats:italic>f<\/jats:italic>(<jats:italic>s<\/jats:italic>) is defined after a search (using <jats:bold>0<\/jats:bold>\u2032) over the power set of <jats:italic>L<jats:sub>s<\/jats:sub><\/jats:italic> (G\u00f6del's constructible hierarchy at the level <jats:italic>s<\/jats:italic>). It can be seen that a function defined in a similar manner over an infinite successor cardinal <jats:italic>k<\/jats:italic> will not be a total function, since for example if <jats:italic>k<\/jats:italic> = \u03c1<jats:sup>+<\/jats:sup>, then <jats:italic>f<\/jats:italic>(\u03c1) will not be defined until after all the subsets of \u03c1 have been examined, and this will take at least <jats:italic>k<\/jats:italic> steps. The following questions then naturally arise: (i) For successor cardinals <jats:italic>k<\/jats:italic>, is there a function dominating every set of minimal <jats:italic>k<\/jats:italic>-degree? (ii) For arbitrary cardinals <jats:italic>k<\/jats:italic>, is every minimal <jats:italic>k<\/jats:italic>-degree GL<jats:sub>2<\/jats:sub> (i.e. <jats:italic><jats:bold>b<\/jats:bold><\/jats:italic>\u2033 = (<jats:italic><jats:bold>b<\/jats:bold><\/jats:italic> \u2228 <jats:bold>0<\/jats:bold>\u2032) for <jats:italic><jats:bold>b<\/jats:bold><\/jats:italic> of minimal <jats:italic>k<\/jats:italic>-degree)? In this paper we answer (i) in the negative and provide a positive answer to (ii), assuming <jats:italic>V = L<\/jats:italic>. We show in fact that if <jats:italic>k<\/jats:italic> is a successor cardinal and <jats:italic>h<\/jats:italic> \u2264<jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> 0\u2032, then there is a function of minimal <jats:italic>k<\/jats:italic>-degree below 0\u2032 not dominated by <jats:italic>h<\/jats:italic> (Theorem 1). This implies that any refinement of the function <jats:italic>f<\/jats:italic> described above will not be able to remove the difficulties encountered. On the other hand, we introduce the notion of \u2018strong domination\u2019 to provide a positive answer to (ii) (Theorem 2 and Corollary 1). We end this paper by indicating that for limit cardinals <jats:italic>k<\/jats:italic>, there is a function below 0\u2032 dominating every function of minimal <jats:italic>k<\/jats:italic>-degree.<\/jats:p>","DOI":"10.2307\/2273144","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:59:42Z","timestamp":1146952782000},"page":"329-334","source":"Crossref","is-referenced-by-count":0,"title":["Double jumps of minimal degrees over cardinals"],"prefix":"10.1017","volume":"47","author":[{"given":"C. T.","family":"Chong","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044261_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90006-X"},{"key":"S0022481200044261_ref004","first-page":"715","volume":"43","author":"Jockusch","year":"1978","journal-title":"Double jumps of minimal degrees"},{"key":"S0022481200044261_ref006","unstructured":"Posner D. B. , High degrees, Ph.D Thesis, University of California, Berkeley, 1977."},{"key":"S0022481200044261_ref001","first-page":"157","article-title":"Generic sets and minimal \u03b1-degrees","volume":"254","author":"Chong","year":"1979","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200044261_ref003","doi-asserted-by":"publisher","DOI":"10.2307\/1971132"},{"key":"S0022481200044261_ref005","first-page":"456","volume":"43","author":"Leggett","year":"1978","journal-title":"\u03b1-degrees of maximal \u03b1-r.e. sets"},{"key":"S0022481200044261_ref002","first-page":"249","volume":"38","author":"Cooper","year":"1973","journal-title":"Minimal degrees and the jump operator"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044261","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T21:16:14Z","timestamp":1558732574000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044261\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1982,6]]}},"alternative-id":["S0022481200044261"],"URL":"https:\/\/doi.org\/10.2307\/2273144","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,6]]}}}