{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T17:23:35Z","timestamp":1709832215030},"reference-count":5,"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":8685,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,6]]},"abstract":"<jats:p>Let <jats:italic>\u03c9<\/jats:italic> be the set of natural numbers, i.e. {0,1,2,\u2026}. A set <jats:italic>A<\/jats:italic> (\u2264<jats:italic>\u03c9<\/jats:italic>) is called <jats:italic>n-generic<\/jats:italic> if it is Cohen-generic for <jats:italic>n<\/jats:italic>-quantifier arithmetic. As characterized by Jockusch [4], this is equivalent to saying that for every <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026116_inline1\" \/> set of strings <jats:italic>S<\/jats:italic>, there is a <jats:italic>\u03c3<\/jats:italic> &lt; <jats:italic>A<\/jats:italic> such that \u03c3 \u2208 <jats:italic>S<\/jats:italic> or \u2200\u03bd \u2265 \u03c3(\u03bd \u2209 <jats:italic>S<\/jats:italic>). When we say degree, we mean Turing degree (of unsolvability). We call a degree <jats:italic>n-generic<\/jats:italic> if it has an <jats:italic>n<\/jats:italic>-generic representative. A nonrecursive degree a is called <jats:italic>minimal<\/jats:italic> if there is no nonrecursive degree <jats:bold>b<\/jats:bold> with <jats:bold>b<\/jats:bold> &lt; <jats:bold>a<\/jats:bold>. Jockusch [4] exhibited various properties of generic degrees, and he showed that any 2-generic degree bounds no minimal degree. Chong and Jockusch [1] showed that any 1-generic degree below <jats:bold>0<\/jats:bold>\u2032 bounds no minimal degree. Haught [3] refuted one of the conjectures in [1] and showed that if a is a 1-generic degree and <jats:bold>0<\/jats:bold> &lt; <jats:bold>b<\/jats:bold> &lt; <jats:bold>a<\/jats:bold> &lt; <jats:bold>0<\/jats:bold>\u2032 then <jats:bold>b<\/jats:bold> is also 1-generic. We show here that there is a 1-generic degree which bounds a minimal degree. This gives an affirmative answer to questions in [1] and [4], As any 1-generic degree below <jats:bold>0<\/jats:bold>\u2032 bounds no minimal degree, we see that our 1-generic degree which bounds a minimal degree is not below <jats:bold>0<\/jats:bold>\u2032, but can be constructed recursively in <jats:bold>0<\/jats:bold>\u2033. Furthermore we see that the initial segments below 1-generic degrees are not order isomorphic.<\/jats:p>","DOI":"10.2307\/2274661","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:35:29Z","timestamp":1146940529000},"page":"733-743","source":"Crossref","is-referenced-by-count":9,"title":["A 1-generic degree which bounds a minimal degree"],"prefix":"10.1017","volume":"55","author":[{"given":"Masahiro","family":"Kumabe","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026116_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21755-9"},{"key":"S0022481200026116_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0067135"},{"key":"S0022481200026116_ref003","unstructured":"Haught C. , Turing and truth table degrees of 1-generic and recursively enumerable sets, Ph.D. thesis, Cornell University, Ithaca, New York, 1985."},{"key":"S0022481200026116_ref004","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511629181.004"},{"key":"S0022481200026116_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0099479"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026116","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T17:13:36Z","timestamp":1558199616000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026116\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,6]]},"references-count":5,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1990,6]]}},"alternative-id":["S0022481200026116"],"URL":"https:\/\/doi.org\/10.2307\/2274661","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,6]]}}}