{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T08:30:10Z","timestamp":1775464210958,"version":"3.50.1"},"reference-count":3,"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":15990,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1970,6]]},"abstract":"<jats:p>In this paper we investigate the possibility of extending Friedberg's enumeration of the recursively enumerable (r.e.) sets without duplication [1, p. 312] to meta-recursion theory. It turns out that all of our proposed extensions are impossible save one: the metarecursively enumerable (meta-r.e.) sets can be enumerated without duplication, but only if all the recursive ordinals are used as indices (Theorems 1 and 2). The <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200090885_inline1\"\/> sets cannot be so enumerated, even if the index set is all recursive ordinals (Theorems 3 and 4). As a corollary, one proves there is no <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200090885_inline1\"\/> predicate <jats:italic>P<\/jats:italic>(<jats:italic>n, x<\/jats:italic>) with the property that for each <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200090885_inline1\"\/> set <jats:italic>A<\/jats:italic> there is exactly one integer <jats:italic>n<\/jats:italic> for which <jats:italic>A<\/jats:italic> = {<jats:italic>x<\/jats:italic> \u2223 <jats:italic>P<\/jats:italic>(<jats:italic>n, x<\/jats:italic>)}. We also discuss enumerations of nonempty, infinite, and coinfinite <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200090885_inline1\"\/> and meta-r.e. sets.<\/jats:p>","DOI":"10.2307\/2270513","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:00:41Z","timestamp":1146949241000},"page":"223-229","source":"Crossref","is-referenced-by-count":8,"title":["The Meta-R.E. sets, but not the \u03a0<sub>1<\/sub><sup>1<\/sup> sets, can be enumerated without repetition"],"prefix":"10.1017","volume":"35","author":[{"suffix":"Jr.","given":"James C.","family":"Owings","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200090885_ref002","first-page":"318","article-title":"Metarecursive sets","volume":"30","author":"Kreisel","year":"1965","journal-title":"this Journal"},{"key":"S0022481200090885_ref003","first-page":"194","article-title":"sets, \u03c9-sets, and metacompleteness","volume":"34","author":"Owings","year":"1969","journal-title":"this Journal"},{"key":"S0022481200090885_ref001","first-page":"309","article-title":"Three theorems on recursive enumeration","volume":"23","author":"Friedberg","year":"1958","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\/S0022481200090885","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T19:12:15Z","timestamp":1559416335000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200090885\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1970,6]]},"references-count":3,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1970,6]]}},"alternative-id":["S0022481200090885"],"URL":"https:\/\/doi.org\/10.2307\/2270513","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1970,6]]}}}