{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T16:38:21Z","timestamp":1648658301945},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12976,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,9]]},"abstract":"<jats:p>Martin [10] has computed the degrees of several classes of <jats:italic>\u03c9<\/jats:italic>-r.e. sets. Lachlan [3] and Shoenfield [14] have obtained some additional results. For most of the examples, a class of r.e. sets is given by a property that is first-order definable over the lattice of <jats:italic>\u03c9<\/jats:italic>-r.e. sets. Then the set of degrees of sets in this class is computed. The only sets of nonzero degrees which arise from the known examples are the following: \u03d5, {<jats:italic><jats:bold>a<\/jats:bold><\/jats:italic>\u2223<jats:italic><jats:bold>a<\/jats:bold><\/jats:italic> is <jats:italic>\u03c9<\/jats:italic>-r.e. and <jats:italic><jats:bold>a<\/jats:bold><\/jats:italic> \u2260 0}, {<jats:italic><jats:bold>a<\/jats:bold><\/jats:italic>\u2223<jats:italic><jats:bold>a<\/jats:bold><\/jats:italic> is <jats:italic>\u03c9<\/jats:italic>-r.e. and <jats:italic><jats:bold>a<\/jats:bold><\/jats:italic>\u2032 = 0\u2033}, and {<jats:italic><jats:bold>a<\/jats:bold><\/jats:italic>\u2223<jats:italic><jats:bold>a<\/jats:bold><\/jats:italic> is <jats:italic>\u03c9<\/jats:italic>-r.e. and <jats:italic><jats:bold>a<\/jats:bold><\/jats:italic>\u2033 &gt; 0\u2033} (see [14]).<\/jats:p><jats:p>There have been some results in this direction for <jats:italic>\u03b1<\/jats:italic> an arbitrary admissible ordinal. Sacks [13] has shown that every nonzero <jats:italic>\u03b1<\/jats:italic>-r.e. <jats:italic>\u03b1<\/jats:italic>-degree contains a regular <jats:italic>\u03b1<\/jats:italic>-r.e. set. Thus regularity does not separate the <jats:italic>\u03b1<\/jats:italic>-r.e. <jats:italic>\u03b1<\/jats:italic>-degrees. Simpson [18] has several theorems which give information about the kinds of <jats:italic>\u03b1<\/jats:italic>-r.e. sets a given <jats:italic>\u03b1<\/jats:italic>-degree can contain. The classes of <jats:italic>\u03b1<\/jats:italic>-r.e. sets considered do separate the <jats:italic>\u03b1<\/jats:italic>-r.e. <jats:italic>\u03b1<\/jats:italic>-degrees into two nonempty pieces for some <jats:italic>\u03b1<\/jats:italic>'s, but they are not necessarily given by properties which are definable over the lattice of <jats:italic>\u03b1<\/jats:italic>-r.e. sets. Using some definability results of Lerman [6], we shall make some comments about the definability of these classes of sets.<\/jats:p>","DOI":"10.2307\/2273521","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:48:21Z","timestamp":1146952101000},"page":"456-474","source":"Crossref","is-referenced-by-count":1,"title":["<i>\u03b1<\/i>-Degrees of maximal <i>\u03b1<\/i>-r.e. sets"],"prefix":"10.1017","volume":"43","author":[{"given":"Anne","family":"Leggett","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049331_ref021","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-65-03247-3"},{"key":"S0022481200049331_ref003","first-page":"431","volume":"33","author":"Lachlan","year":"1968","journal-title":"Degrees of recursively enumerable sets which have no maximal supersets"},{"key":"S0022481200049331_ref016","first-page":"351","article-title":"On the jump of an \u03b1-recursively enumerable set","volume":"217","author":"Shore","year":"1976","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200049331_ref012","volume-title":"Annals of Mathematics Studies","author":"Sacks","year":"1966"},{"key":"S0022481200049331_ref010","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19660120125"},{"key":"S0022481200049331_ref007","first-page":"341","article-title":"Maximal \u03b1-r.e. sets","volume":"188","author":"Lerman","year":"1974","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200049331_ref018","unstructured":"Simpson S. G. , Admissible ordinals and recursion theory, Ph.D. dissertation, Massachusetts Institute of Technology, 1971."},{"key":"S0022481200049331_ref020","volume-title":"Perspectives in Mathematical Logic","author":"Simpson"},{"key":"S0022481200049331_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90005-X"},{"key":"S0022481200049331_ref002","first-page":"318","volume":"30","author":"Kreisel","year":"1965","journal-title":"Metarecursive sets"},{"key":"S0022481200049331_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(74)90004-7"},{"key":"S0022481200049331_ref006","first-page":"405","volume":"41","author":"Lerman","year":"1976","journal-title":"Congruence relations, filters, ideals, and definability in lattices of \u03b1-recursively enumerable sets"},{"key":"S0022481200049331_ref008","unstructured":"Lerman M. , On elementary theories of some lattices of \u03b1-recursively enumerable sets (to appear)."},{"key":"S0022481200049331_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF02764882"},{"key":"S0022481200049331_ref013","first-page":"1","article-title":"Post's problem, admissible ordinals, and regularity","volume":"124","author":"Sacks","year":"1966","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200049331_ref014","first-page":"695","volume":"41","author":"Shoenfield","year":"1976","journal-title":"Degrees of classes of RE sets"},{"key":"S0022481200049331_ref015","volume-title":"Degrees of unsolvability","author":"Shoenfield","year":"1971"},{"key":"S0022481200049331_ref017","unstructured":"Shore R.A. , Some more minimal pairs of \u03b1-recursively enumerable degrees (to appear)."},{"key":"S0022481200049331_ref019","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70586-X"},{"key":"S0022481200049331_ref011","first-page":"173","volume":"32","author":"Owings","year":"1967","journal-title":"Recursion, metarecursion, and inclusion"},{"key":"S0022481200049331_ref005","first-page":"681","volume":"41","author":"Leggett","year":"1976","journal-title":"Types of simple \u03b1-recursively enumerable sets"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049331","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:33:34Z","timestamp":1558985614000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049331\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,9]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1978,9]]}},"alternative-id":["S0022481200049331"],"URL":"https:\/\/doi.org\/10.2307\/2273521","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,9]]}}}