{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T09:50:52Z","timestamp":1777542652997,"version":"3.51.4"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":5250,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1999,9]]},"abstract":"<jats:p>In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levels<jats:italic>U<jats:sub>\u03b1<\/jats:sub><\/jats:italic><jats:italic>V<jats:sub>\u03b1<\/jats:sub><\/jats:italic>. The recursive definition of the<jats:italic>V<jats:sub>\u03b1<\/jats:sub>'<\/jats:italic>s is:<\/jats:p><jats:p><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600006892_inline1\"\/><\/jats:p><jats:p>Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory (<jats:bold>ZF<\/jats:bold>) shows that<jats:italic>V\u03c9<\/jats:italic>, the first transfinite level of the hierarchy, is a model of all the axioms of<jats:bold>ZF<\/jats:bold>with the exception of the axiom of infinity. And, in general, one finds that if<jats:italic>\u03ba<\/jats:italic>is a strongly inaccessible ordinal, then<jats:italic>V\u03ba<\/jats:italic>is a model of all of the axioms of<jats:bold>ZF<\/jats:bold>. (For all these models, we take<jats:italic>\u2208<\/jats:italic>to be the standard element-set relation restricted to the members of the domain.) Doubtless, when cast as a first-order theory,<jats:bold>ZF<\/jats:bold>does not characterize the structures \u3008<jats:italic>V<jats:sub>\u03ba<\/jats:sub>,\u2208\u2229<\/jats:italic>(<jats:italic>V<jats:sub>\u03ba<\/jats:sub>\u00d7V<jats:sub>\u03ba<\/jats:sub><\/jats:italic>)\u3009 for<jats:italic>\u03ba<\/jats:italic>a strongly inaccessible ordinal, by the L\u00f6wenheim-Skolem theorem. Still, one of the main achievements of [12] consisted in establishing that a characterization of these models can be attained when one ventures into second-order logic. For let second-order<jats:bold>ZF<\/jats:bold>be, as usual, the theory that results from<jats:bold>ZF<\/jats:bold>when the axiom schema of replacement is replaced by its second-order universal closure. Then, it is a remarkable result due to Zermelo that second-order<jats:bold>ZF<\/jats:bold>can only be satisfied in models of the form \u3008<jats:italic>V<jats:sub>\u03ba<\/jats:sub>,\u2208\u2229<\/jats:italic>(<jats:italic>V<jats:sub>\u03ba<\/jats:sub>\u00d7V<jats:sub>\u03ba<\/jats:sub><\/jats:italic>)\u3009 for<jats:italic>\u03ba<\/jats:italic>a strongly inaccessible ordinal.<\/jats:p>","DOI":"10.2307\/421182","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:12:56Z","timestamp":1146985976000},"page":"289-302","source":"Crossref","is-referenced-by-count":19,"title":["Models of Second-Order Zermelo Set Theory"],"prefix":"10.1017","volume":"5","author":[{"given":"Gabriel","family":"Uzquiano","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600006892_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.2\/0392570"},{"key":"S1079898600006892_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0061160"},{"key":"S1079898600006892_ref003","volume-title":"Mathematics and mind","author":"Boolos","year":"1994"},{"key":"S1079898600006892_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4153-7"},{"key":"S1079898600006892_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/2268864"},{"key":"S1079898600006892_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02308-2"},{"key":"S1079898600006892_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/2267328"},{"key":"S1079898600006892_ref006","volume-title":"Foundations of set theory","author":"Fraenkel","year":"1973"},{"key":"S1079898600006892_ref012","doi-asserted-by":"crossref","first-page":"29","DOI":"10.4064\/fm-16-1-29-47","article-title":"\u00dcber Grenzzahlen und Mengenbereiche: Neue Untersuchungen \u00fcber die Grundlagen der Mengenlehre","volume":"16","author":"Zermelo","year":"1930","journal-title":"Fundamenta Mathematicae"},{"key":"S1079898600006892_ref004","volume-title":"Set theory: An introduction to large cardinals","author":"Drake","year":"1974"},{"key":"S1079898600006892_ref010","doi-asserted-by":"crossref","first-page":"323","DOI":"10.21136\/CMJ.1957.100254","article-title":"A contribution to G\u00f6del's axiomatic set theory","volume":"7","author":"Rieger","year":"1957","journal-title":"Czechoslovak Mathematical Journal"},{"key":"S1079898600006892_ref008","first-page":"131","volume-title":"Formal systems and recursive functions","author":"Montague","year":"1967"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600006892","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T20:56:10Z","timestamp":1627332970000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600006892\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,9]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1999,9]]}},"alternative-id":["S1079898600006892"],"URL":"https:\/\/doi.org\/10.2307\/421182","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,9]]}}}