{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T07:46:57Z","timestamp":1648799217346},"reference-count":8,"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":15533,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1971,9]]},"abstract":"<jats:p>As was first mentioned in [3, \u00a75], if <jats:italic>A<\/jats:italic> is any <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200082220_inline2\" \/> \u2013 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200082220_inline3\" \/> set, <jats:italic>A<\/jats:italic> is the union of two disjoint <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200082220_inline2\" \/> \u2013 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200082220_inline3\" \/> sets <jats:italic>B<\/jats:italic>(0), <jats:italic>B<\/jats:italic>(1). In metarecursion theory this is proven as follows. Let \u0192 be a one-to-one metarecursive function whose range is <jats:italic>A<\/jats:italic>, let <jats:italic>R<\/jats:italic> be an unbounded metarecursive set whose complement is also unbounded, and set <jats:italic>B(0)<\/jats:italic> = <jats:italic>f(R), B(1)<\/jats:italic> = <jats:italic>f<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200082220_inline1\" \/>). The corresponding fact of ordinary recursion theory, namely that any r.e. but not recursive set can be split into two other such sets, was proved by Friedberg [2, Theorem 1], using a clever priority argument. Sacks [7, Corollary 2] then showed that any r.e. but not recursive set is the union of two disjoint r.e. sets neither of which was recursive in the other, a much stronger result. In this paper we attempt to prove the analogous result for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200082220_inline2\" \/> \u2013 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200082220_inline3\" \/> sets A, but succeed only in the case <jats:italic><jats:bold>A<\/jats:bold><\/jats:italic> is <jats:italic>simple<\/jats:italic>; i.e., the complement of <jats:italic><jats:bold>A<\/jats:bold><\/jats:italic> contains no infinite <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200082220_inline2\" \/> subset. As a corollary we show the <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200082220_inline2\" \/> metadegrees are dense, a fact already announced by Sacks [8, Corollary 1], but only proven by him for nonzero <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200082220_inline2\" \/> metadegrees.<\/jats:p>","DOI":"10.2307\/2269950","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:09:50Z","timestamp":1146949790000},"page":"433-438","source":"Crossref","is-referenced-by-count":3,"title":["A splitting theorem for simple \u03a0<sub>1<\/sub><sup>1<\/sup> Sets"],"prefix":"10.1017","volume":"36","author":[{"suffix":"Jr","given":"James C.","family":"Owings","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200082220_ref005","first-page":"194","volume":"34","author":"Owings","year":"1969","journal-title":"\u03a011 sets, \u03c9-sets, and metacompleteness"},{"key":"S0022481200082220_ref002","first-page":"309","volume":"23","author":"Friedberg","year":"1958","journal-title":"Three theorems on recursive enumeration"},{"key":"S0022481200082220_ref001","first-page":"389","volume":"33","author":"Driscoll","year":"1968","journal-title":"Metarecursively enumerable sets and their metadegrees"},{"key":"S0022481200082220_ref003","first-page":"318","volume":"30","author":"Kreisel","year":"1965","journal-title":"Metarecursive sets"},{"key":"S0022481200082220_ref007","doi-asserted-by":"publisher","DOI":"10.2307\/1970214"},{"key":"S0022481200082220_ref004","unstructured":"Owings J. C. Jr. , Topics in metacursion theory, Ph.D. thesis, Cornell University, 1966."},{"key":"S0022481200082220_ref006","first-page":"223","volume":"35","author":"Owings","year":"1970","journal-title":"The metarecursively enumerable sets but not the \u03a011 sets, can be enumerated without duplication"},{"key":"S0022481200082220_ref008","first-page":"243","volume-title":"Sets, models, and recursion theory","author":"Sacks","year":"1965"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200082220","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T19:53:50Z","timestamp":1559332430000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200082220\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1971,9]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1971,9]]}},"alternative-id":["S0022481200082220"],"URL":"https:\/\/doi.org\/10.2307\/2269950","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1971,9]]}}}