{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T04:43:50Z","timestamp":1648701830538},"reference-count":5,"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":5580,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1998,12]]},"abstract":"<jats:p>We denote by KP_ the fragment of set-theory containing the axioms of extensionality, pairing, union and foundation as well as the schemas of \u2206<jats:sub>0<\/jats:sub>-comprehension and \u2206<jats:sub>0<\/jats:sub>-collection, that is: Kripke-Platek set-theory (KP) with the axiom of foundation in place of the \u2208-induction schema. The theory KP is obtained by adding to KP_ the schema of \u2208-induction<\/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=\"S0022481200014328_Uequ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>Using \u2208-induction it is possible to prove the existence of the transi tive closure without appealing to the axiom of infinity (see, e.g., [1]). Vice versa, when a theory proves the existence of the transitive closure, some induction is immediately ensured (by foundation and comprehension). This is not true in general: e.g., the whole of Zermelo-Fraenkel set-theory without the axiom of infinity does not prove \u2208-induction (in fact, it does not prove the existence of the transitive closure; see, e.g., [3]). Open-induction is the schema of \u2208-induction restricted to open formulas. We prove the following theorem.<\/jats:p><jats:p>KP_ <jats:italic>proves open-induction<\/jats:italic>.<\/jats:p><jats:p>We reason in a fixed but arbitrary model of KP_ whom we refer to as <jats:italic>the<\/jats:italic> model. The language is extended with a name for every set in the model. We call this constants <jats:italic>parameters<\/jats:italic>. Let <jats:italic>\u03c6<\/jats:italic><jats:italic>(x)<\/jats:italic> be a satisfiable open-formula possibly depending on parameters and with no free variable but <jats:italic>x<\/jats:italic>. We show that <jats:italic>\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>) is satisfied by an \u2208-minimal set, that is, a set <jats:italic>a<\/jats:italic> such that <jats:italic>\u03c6<\/jats:italic>(<jats:italic>a<\/jats:italic>) and (\u2200<jats:italic>x<\/jats:italic> \u2208 <jats:italic>a<\/jats:italic>) \u00ac<jats:italic>\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>). We assume that no ordinal satisfies <jats:italic>\u03c6<\/jats:italic>(<jats:italic>x<\/jats:italic>), otherwise the existence of a \u2208-minimal set follows from foundation and comprehension.<\/jats:p>","DOI":"10.2307\/2586657","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:01:33Z","timestamp":1146938493000},"page":"1399-1403","source":"Crossref","is-referenced-by-count":1,"title":["Foundation versus induction in Kripke-Platek set theory"],"prefix":"10.1017","volume":"63","author":[{"given":"Domenico","family":"Zambella","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200014328_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"},{"key":"S0022481200014328_ref003","unstructured":"Mancini A. , A note on recursive models of set theories, forthcoming."},{"key":"S0022481200014328_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19960420105"},{"key":"S0022481200014328_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1988-0938682-2"},{"key":"S0022481200014328_ref002","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19780241902"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200014328","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,10]],"date-time":"2019-05-10T19:52:00Z","timestamp":1557517920000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200014328\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,12]]},"references-count":5,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1998,12]]}},"alternative-id":["S0022481200014328"],"URL":"https:\/\/doi.org\/10.2307\/2586657","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,12]]}}}