{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T07:13:02Z","timestamp":1679469182823},"reference-count":10,"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":16301,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb.\n                        log."],"published-print":{"date-parts":[[1969,7,25]]},"abstract":"<jats:p>An <jats:italic>\u03c9-set<\/jats:italic> is a subset of the recursive ordinals whose\n                        complement with respect to the recursive ordinals is unbounded and has order\n                        type \u03c9. This concept has proved fruitful in the study of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200094433_inline01\" \/> sets in relation to metarecursion theory. We prove\n                        that the metadegrees of the <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200094433_inline02\" \/> sets\n                        coincide with those of the meta-r.e. \u03c9-sets. We then show that, given any\n                           <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200094433_inline02\" \/> set, a\n                        metacomplete <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200094433_inline01\" \/> set can be\n                        found which is weakly metarecursive in it. It then follows that weak\n                        relative metarecursiveness is not a transitive relation on the\n                           <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200094433_inline01\" \/> sets,\n                        extending a result of G. Driscoll [2, Theorem 3.1]. Coincidentally, we\n                        discuss the notions of total and complete regularity. Finally, we solve\n                        Post's problem for the transitive closure of weak relative\n                        metarecursiveness. We recommend the reader look at pp. 324\u2013328 of the\n                        fundamental article [6] of Kreisel and Sacks before proceeding. He will find\n                        there a proof of the following very basic fact: a subset of the integers is\n                           <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200094433_inline03\" \/> iff it is\n                        metarecursively enumerable (metafinite).<\/jats:p>","DOI":"10.2307\/2271094","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T20:53:51Z","timestamp":1146948831000},"page":"194-204","source":"Crossref","is-referenced-by-count":4,"title":["\u03a0<sub>1<\/sub><sup>1<\/sup> Sets,\n                        \u03c9-Sets, and metacompleteness"],"prefix":"10.1017","volume":"34","author":[{"suffix":"Jr.","given":"James C.","family":"Owings","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200094433_ref008","volume-title":"Sets, models, and recursion theory","author":"Sacks","year":"1965"},{"key":"S0022481200094433_ref007","first-page":"1","article-title":"Post's problem","volume":"124","author":"Sacks","year":"1966","journal-title":"admissible ordinals, and regularity, Transactions of the American Mathematical Society"},{"key":"S0022481200094433_ref004","first-page":"150","article-title":"On notations for ordinal numbers","volume":"3","author":"Kleene","year":"1938","journal-title":"this Journal"},{"key":"S0022481200094433_ref002","unstructured":"Driscoix G. C. Jr. , Contributions to metarecursion theory, Ph.D. thesis, Cornell University, Ithaca, New York, 1965."},{"key":"S0022481200094433_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1954-0063995-6"},{"key":"S0022481200094433_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1966-11416-7"},{"key":"S0022481200094433_ref006","first-page":"318","article-title":"Metarecursive sets","volume":"30","author":"Kreisel","year":"1965","journal-title":"this Journal"},{"key":"S0022481200094433_ref005","doi-asserted-by":"publisher","DOI":"10.2307\/2372632"},{"key":"S0022481200094433_ref010","first-page":"151","article-title":"Recursive well-orderings","volume":"20","author":"Spector","year":"1955","journal-title":"this Journal"},{"key":"S0022481200094433_ref003","first-page":"389","article-title":"Metarecursively enumerable sets and their metadegrees","volume":"33","author":"Driscoll","year":"1968","journal-title":"this Journal"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200094433","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T06:35:51Z","timestamp":1679466951000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200094433\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1969,7,25]]},"references-count":10,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1969,7,25]]}},"alternative-id":["S0022481200094433"],"URL":"https:\/\/doi.org\/10.2307\/2271094","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1969,7,25]]}}}