{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T20:07:53Z","timestamp":1648584473021},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":20190,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1958,12]]},"abstract":"<jats:p>In this note we shall prove a certain relative recursiveness lemma concerning countable models of set theory (Lemma 5). From this lemma will follow two results about special types of such models.<\/jats:p><jats:p>Kreisel [5] and Mostowski [6] have shown that certain (finitely axiomatized) systems of set theory, formulated by means of the \u03f5 relation and certain additional non-logical constants, do not possess recursive models. Their purpose in doing this was to construct consistent sentences without recursive models. As a first corollary of Lemma 5, we obtain a very simple proof, not involving any formal constructions within the system of the notions of truth and satisfiability, of an extension of the Kreisel-Mostowski theorems. Namely, set theory with the single non-logical constant \u03f5 does not possess any recursively enumerable model. Thus we get, as a side product, an easy example of a consistent sentence containing a single binary relation which does not possess any recursively enumerable model; this sentence being the conjunction of the (finitely many) axioms of set theory.<\/jats:p>","DOI":"10.2307\/2964015","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T19:59:54Z","timestamp":1146945594000},"page":"408-416","source":"Crossref","is-referenced-by-count":11,"title":["On recursively enumerable and arithmetic models of set theory"],"prefix":"10.1017","volume":"23","author":[{"given":"Michael O.","family":"Rabin","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200057558_ref006","doi-asserted-by":"publisher","DOI":"10.4064\/fm-40-1-56-61"},{"key":"S0022481200057558_ref005","first-page":"39","article-title":"Note on arithmetic models for consistent formulae of the predicate calculus II","volume":"14","author":"Kreisel","journal-title":"Proceedings of the XIth International Congress of Philosophy, Bruxelles, August 20-26, 1953"},{"key":"S0022481200057558_ref002","volume-title":"The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory","author":"G\u00f6del","year":"1951"},{"key":"S0022481200057558_ref001","first-page":"79","article-title":"\u00dcber die Gleichheitsbeziehung in der Mengenlehre","volume":"157","author":"Fraenkel","year":"1927","journal-title":"Journal f\u00fcr Mathematik"},{"key":"S0022481200057558_ref004","first-page":"800","article-title":"A symmetric form of G\u00f6del's theorem","volume":"53","author":"Kleene","year":"1950","journal-title":"Koninklijke Nederlandsche Akademie van Wetenschappen, Proceedings of the section of sciences"},{"key":"S0022481200057558_ref003","volume-title":"Introduction to metamathematics","author":"Kleene","year":"1952"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200057558","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,5]],"date-time":"2019-06-05T20:25:05Z","timestamp":1559766305000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200057558\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1958,12]]},"references-count":6,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1958,12]]}},"alternative-id":["S0022481200057558"],"URL":"https:\/\/doi.org\/10.2307\/2964015","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1958,12]]}}}