{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T20:57:05Z","timestamp":1757624225491,"version":"3.44.0"},"reference-count":24,"publisher":"SAGE Publications","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["COM"],"published-print":{"date-parts":[[2023,12,21]]},"abstract":"<jats:p>We prove various sufficient conditions for the effective infinity of classes of computable numberings. Then we apply them to show that for every computable family of left-c.e. reals without the greatest element the class of its Friedberg computable numberings is effectively infinite. In particular, this result covers the families of all left-c.e. and all Martin-L\u00f6f random left-c.e. reals whose Friedberg computable numberings have been constructed by Broadhead and Kjos-Hanssen in their paper (In Mathematical Theory and Computational Practice, CiE 2009 (2009) 49\u201358 Springer). In addition, for every infinite computable family of left-c.e. reals we prove that the classes of all its computable, positive and minimal numberings are effectively infinite.<\/jats:p>","DOI":"10.3233\/com-230461","type":"journal-article","created":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T11:06:42Z","timestamp":1693307202000},"page":"339-350","source":"Crossref","is-referenced-by-count":4,"title":["Effectively infinite classes of numberings and computable families of reals"],"prefix":"10.1177","volume":"12","author":[{"given":"Marat","family":"Faizrahmanov","sequence":"first","affiliation":[{"name":"Volga Region Scientific-Educational Centre of Mathematics, Kazan (Volga Region) Federal University, 35, Kremlevskaya Str., Kazan, Russia"}]},{"given":"Zlata","family":"Shchedrikova","sequence":"additional","affiliation":[{"name":"Faculty of computer science and engineering, Innopolis University, 1, Universitetskaya Str., Innopolis, Russia"}]}],"member":"179","reference":[{"key":"10.3233\/COM-230461_ref1","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1007\/BF00967026","article-title":"Positive enumerations","volume":"18","author":"Badaev","year":"1977","journal-title":"Sib. Math. J."},{"key":"10.3233\/COM-230461_ref2","doi-asserted-by":"publisher","DOI":"10.1515\/9783110813418-005"},{"key":"10.3233\/COM-230461_ref3","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/257\/04025"},{"key":"10.3233\/COM-230461_ref4","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1023\/A:1012516217265","article-title":"Rogers semilattices of families of arithmetic sets","volume":"40","author":"Badaev","year":"2001","journal-title":"Algebra and Logic"},{"key":"10.3233\/COM-230461_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-03073-4_6"},{"key":"10.3233\/COM-230461_ref6","doi-asserted-by":"crossref","unstructured":"R.G.\u00a0Downey and D.R.\u00a0Hirschfeld, Algorithmic Randomness and Complexity, Springer-Verlag, New York, 2010.","DOI":"10.1007\/978-0-387-68441-3_8"},{"key":"10.3233\/COM-230461_ref7","first-page":"19","article-title":"Theorie der numerierungen I","volume":"19","author":"Ershov","year":"1973","journal-title":"Z. Math. Logik und Grundlag. Math."},{"key":"10.3233\/COM-230461_ref8","unstructured":"Y.L.\u00a0Ershov, Theory of Numberings, Nauka, Moscow, 1977, In Russian."},{"key":"10.3233\/COM-230461_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(99)80030-5"},{"key":"10.3233\/COM-230461_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-80049-9_17"},{"key":"10.3233\/COM-230461_ref11","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1215\/00294527-10670074","article-title":"A local version of the Slaman\u2013Wehner theorem and families closed under finite differences","volume":"64","author":"Faizrahmanov","year":"2023","journal-title":"Notre Dame J. Form. Log."},{"key":"10.3233\/COM-230461_ref12","doi-asserted-by":"publisher","first-page":"96","DOI":"10.26907\/0021-3446-2023-5-96-100","article-title":"Effectively infinite classes of numberings of computable families of reals","volume":"67","author":"Faizrahmanov","year":"2023","journal-title":"Izv. Vyssh. Uchebn. Zaved. Mat."},{"key":"10.3233\/COM-230461_ref13","doi-asserted-by":"publisher","first-page":"224","DOI":"10.1002\/malq.201400015","article-title":"Limitwise monotonic sets of reals","volume":"61","author":"Faizrahmanov","year":"2015","journal-title":"Math. Log. Quart."},{"key":"10.3233\/COM-230461_ref14","doi-asserted-by":"publisher","first-page":"309","DOI":"10.2307\/2964290","article-title":"Three theorems on recursive enumeration. I. Decomposition. II. Maximal set. III. Enumeration without duplication","volume":"23","author":"Friedberg","year":"1958","journal-title":"J. Symb. Logic"},{"key":"10.3233\/COM-230461_ref15","doi-asserted-by":"publisher","first-page":"621","DOI":"10.1007\/BF02671553","article-title":"Generalized computable numerations and nontrivial Rogers semilattices","volume":"36","author":"Goncharov","year":"1997","journal-title":"Algebra and Logic"},{"key":"10.3233\/COM-230461_ref16","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1016\/0168-0072(93)90076-P","article-title":"Some effectively infinite classes of enumerations","volume":"60","author":"Goncharov","year":"1993","journal-title":"Ann. and Pure Appl. Log."},{"key":"10.3233\/COM-230461_ref17","doi-asserted-by":"publisher","first-page":"1097","DOI":"10.2178\/jsl\/1230396908","article-title":"\u03b7-representation of sets and degrees","volume":"73","author":"Harris","year":"2008","journal-title":"J. Symb. Logic"},{"key":"10.3233\/COM-230461_ref18","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1016\/0304-3975(90)90141-4","article-title":"An easy priority-free proof of a theorem of Friedberg","volume":"74","author":"Kummer","year":"1990","journal-title":"Theoret. Comput. Sci."},{"key":"10.3233\/COM-230461_ref19","first-page":"278","article-title":"Positive and negative enumerations","volume":"160","author":"Mal\u2019tsev","year":"1965","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"10.3233\/COM-230461_ref20","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1002\/malq.19550010205","article-title":"Creative sets","volume":"1","author":"Myhill","year":"1955","journal-title":"Z. Math. Logik und Grundlag. Math."},{"key":"10.3233\/COM-230461_ref21","doi-asserted-by":"crossref","unstructured":"A.\u00a0Nies, Computability and Randomness, Oxford University Press, New York, 2009.","DOI":"10.1093\/acprof:oso\/9780199230761.001.0001"},{"key":"10.3233\/COM-230461_ref22","doi-asserted-by":"crossref","unstructured":"R.I.\u00a0Soare, Recursively Enumerable Sets and Degrees. A Study of Computable Funtions and Computably Generated Sets, Perspect. Math. Log., Springer-Verlag, Berlin, 1987.","DOI":"10.1007\/978-3-662-02460-7"},{"key":"10.3233\/COM-230461_ref23","doi-asserted-by":"crossref","unstructured":"R.I.\u00a0Soare, Turing Computability: Theory and Applications, Springer, Berlin, 2016.","DOI":"10.1007\/978-3-642-31933-4"},{"key":"10.3233\/COM-230461_ref24","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1002\/malq.19570031202","article-title":"Some notes on recursively enumerable sets","volume":"3","author":"Uspenskii","year":"1957","journal-title":"Z. Math. Log. und Grundl. Math."}],"container-title":["Computability"],"original-title":[],"link":[{"URL":"https:\/\/content.iospress.com\/download?id=10.3233\/COM-230461","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T12:22:17Z","timestamp":1757420537000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/COM-230461"}},"subtitle":[],"editor":[{"given":"Nikolay","family":"Bazhenov","sequence":"additional","affiliation":[]},{"given":"Takayuki","family":"Kihara","sequence":"additional","affiliation":[]},{"given":"Svetlana","family":"Selivanova","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2023,12,21]]},"references-count":24,"journal-issue":{"issue":"4"},"URL":"https:\/\/doi.org\/10.3233\/com-230461","relation":{},"ISSN":["2211-3576","2211-3568"],"issn-type":[{"type":"electronic","value":"2211-3576"},{"type":"print","value":"2211-3568"}],"subject":[],"published":{"date-parts":[[2023,12,21]]}}}