{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,31]],"date-time":"2023-08-31T17:07:36Z","timestamp":1693501656246},"reference-count":7,"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":14164,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,6]]},"abstract":"<jats:p>Our results concern the natural models of Ackermann-type set theories, but they can also be viewed as results about the definability of ordinals in certain sets.<\/jats:p><jats:p>Ackermann's set theory <jats:italic>A<\/jats:italic> was introduced in [1] and it is now formulated in the first order predicate calculus with identity, using \u2208 for membership and an individual constant <jats:italic>V<\/jats:italic> for the class of all sets. We use the letters <jats:italic>\u03d5, \u03c7, \u03b8, and \u03c7<\/jats:italic> to stand for formulae which do not contain <jats:italic>V<\/jats:italic> and capital Greek letters to stand for any formulae. Then, the axioms of <jats:italic>A<\/jats:italic>* are the universal closures of<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200053706_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>where all the free variables are shown in A4 and <jats:italic>z<\/jats:italic> does not occur in the \u0398 of A2. <jats:italic>A<\/jats:italic> is the theory <jats:italic>A<\/jats:italic>* \u2212 A5.<\/jats:p><jats:p>Most of our notation is standard (for instance, <jats:italic>\u03b1, \u03b2, \u03b3, \u03b4, \u03ba, \u03bb, \u03be<\/jats:italic> are variables ranging over ordinals) and, in general, we follow the notation of [7]. When <jats:italic>x<\/jats:italic> \u2286 <jats:italic>R<\/jats:italic>\u03b1, we use Df(<jats:italic>R\u03b1, x<\/jats:italic>) for the set of those elements of <jats:italic>R\u03b1<\/jats:italic> which are definable in \u3008<jats:italic>R\u03b1<\/jats:italic>, \u2208\u3009, using a first order \u2208-formula and parameters from <jats:italic>x<\/jats:italic>.<\/jats:p><jats:p>We refer the reader to [7] for an outline of the results which are known about <jats:italic>A<\/jats:italic>, but we shall summarise those facts which are frequently used in this paper.<\/jats:p>","DOI":"10.2307\/2271895","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:35:13Z","timestamp":1146951313000},"page":"151-158","source":"Crossref","is-referenced-by-count":2,"title":["Natural models and Ackermann-type set theories"],"prefix":"10.1017","volume":"40","author":[{"given":"John","family":"Lake","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200053706_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90011-2"},{"key":"S0022481200053706_ref004","unstructured":"Lake J. , Some topics in set theory, Ph.D. Thesis, University of London, 1973."},{"key":"S0022481200053706_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01350103"},{"key":"S0022481200053706_ref006","doi-asserted-by":"publisher","DOI":"10.4064\/fm-37-1-77-83"},{"key":"S0022481200053706_ref002","first-page":"481","volume":"34","author":"Grewe","year":"1969","journal-title":"Natural models of Ackermann's set theory"},{"key":"S0022481200053706_ref003","first-page":"410","volume":"38","author":"Lake","year":"1973","journal-title":"On an Ackermann-type set theory"},{"key":"S0022481200053706_ref005","volume-title":"Constructible sets with applications","author":"Mostowski","year":"1969"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200053706","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T19:40:36Z","timestamp":1559158836000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200053706\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1975,6]]}},"alternative-id":["S0022481200053706"],"URL":"https:\/\/doi.org\/10.2307\/2271895","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,6]]}}}