{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,16]],"date-time":"2024-02-16T10:41:28Z","timestamp":1708080088067},"reference-count":12,"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":4667,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2001,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Working in <jats:italic>Z<\/jats:italic> + <jats:italic>KP<\/jats:italic>, we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula \u03a6(<jats:italic>\u03bb, a<\/jats:italic>) such that for any sequence \u27e8<jats:italic>A<jats:sub>\u03bb<\/jats:sub><\/jats:italic> \u2223 <jats:italic>\u03bb<\/jats:italic> a limit ordinal\u27e9 where for each <jats:italic>\u03bb<\/jats:italic>. <jats:italic>A<jats:sub>\u03bb<\/jats:sub><\/jats:italic> \u2286 <jats:sup><jats:italic>\u03bb<\/jats:italic><\/jats:sup>2, there is a supertransitive inner model of Zermelo containing all ordinals in which for every <jats:italic>\u03bb<\/jats:italic><jats:italic>A<jats:sub>\u03bb<\/jats:sub><\/jats:italic> = {<jats:italic>a<\/jats:italic> \u2223 \u03a6(<jats:italic>\u03bb, a<\/jats:italic>)}.<\/jats:p>","DOI":"10.2307\/2695026","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:05:52Z","timestamp":1146938752000},"page":"487-496","source":"Crossref","is-referenced-by-count":20,"title":["Slim models of Zermelo set theory"],"prefix":"10.1017","volume":"66","author":[{"given":"A. R. D.","family":"Mathias","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200010823_ref010","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700028755"},{"key":"S0022481200010823_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4153-7"},{"key":"S0022481200010823_ref008","unstructured":"Mathias A. R. D. , Notes on Mac Lane Set Theory, CRM Preprint n\u00fam. 348, gener 1997."},{"key":"S0022481200010823_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF01969698"},{"key":"S0022481200010823_ref009","volume-title":"Annals of Pure and Applied Logic","author":"Mathias"},{"key":"S0022481200010823_ref006","volume-title":"Elements of Set Theory","author":"Enderton","year":"1977"},{"key":"S0022481200010823_ref002","first-page":"113","article-title":"Axiome et schema de fondement dans le syst\u00e8me de Zermelo","volume":"17","author":"Boffa","year":"1969","journal-title":"Bulletin Acad. Polon. Sc. s\u00e9r. sc. math."},{"key":"S0022481200010823_ref004","first-page":"242","article-title":"Axiom and Scheme of Foundation","volume":"22","author":"Boffa","year":"1970","journal-title":"Bulletin de la Soci\u00e9t\u00e9 Math\u00e9matique de Belgique"},{"key":"S0022481200010823_ref005","first-page":"61","article-title":"Eine neue Theorie der Convergenz und Divergenz von Reihen mit positiven Gliedern","volume":"76","author":"du Bois-Reymond","year":"1873","journal-title":"Journal f\u00fcr Mathematik"},{"key":"S0022481200010823_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01594179"},{"key":"S0022481200010823_ref003","first-page":"205","article-title":"Une infinit\u00e9 d\u2019axiomes de l\u2019infini mutuellement ind\u00e9pendants dans le syst\u00e8me de Zermelo","volume":"268","author":"Boffa","year":"1969","journal-title":"Comptes Rendus de l\u2019 Acad\u00e9mie des Sciences, Paris"},{"key":"S0022481200010823_ref012","first-page":"217","volume-title":"Matematikerkongressen i Helsingfors den 4\u20137 Juli 1922, Den femte skandinaviska matematikerkongressen, Redog\u00f6relse","author":"Skolem","year":"1923"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200010823","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,8]],"date-time":"2019-05-08T00:20:34Z","timestamp":1557274834000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200010823\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,6]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2001,6]]}},"alternative-id":["S0022481200010823"],"URL":"https:\/\/doi.org\/10.2307\/2695026","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,6]]}}}