{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T00:14:50Z","timestamp":1648858490932},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10603,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1985,3]]},"abstract":"<jats:p>Probably the two most famous examples of elementary embeddings between inner models of set theory are the embeddings of the universe into an inner model given by a measurable cardinal and the embeddings of the constructible universe<jats:italic>L<\/jats:italic>into itself given by 0<jats:sup>#<\/jats:sup>. In both of these examples, the \u201ctarget model\u201d is a subclass of the \u201cground model\u201d (and in the latter case they are equal). It is not hard to find examples of embeddings in which the target model is not a subclass of the ground model: if<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200033181_inline1\" \/>is a generic ultrafilter arising from forcing with a precipitous ideal on a successor cardinal<jats:italic>\u03ba<\/jats:italic>, then the ultraproduct of the ground model via<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200033181_inline1\" \/>collapses<jats:italic>\u03ba<\/jats:italic>. Such considerations suggest a classification of how close the target model comes to \u201cfitting inside\u201d the ground model.<\/jats:p><jats:p>Definition 1.1. Let<jats:italic>M<\/jats:italic>and<jats:italic>N<\/jats:italic>be inner models (transitive, proper class models) of ZFC, and let<jats:italic>j<\/jats:italic>:<jats:italic>M<\/jats:italic>\u2192<jats:italic>N<\/jats:italic>be an elementary embedding. The<jats:italic>co-critical point<\/jats:italic>of<jats:italic>j<\/jats:italic>is the least ordinal<jats:italic>\u03bb<\/jats:italic>, if any exist, such that there is<jats:italic>X<\/jats:italic>\u2286<jats:italic>\u03bb, X<\/jats:italic>\u2208<jats:italic>N<\/jats:italic>but<jats:italic>X<\/jats:italic>\u2209<jats:italic>M<\/jats:italic>. Such an<jats:italic>X<\/jats:italic>is called a<jats:italic>new subset<\/jats:italic>of<jats:italic>\u03bb<\/jats:italic>.<\/jats:p><jats:p>It is easy to see that the co-critical point of<jats:italic>j<\/jats:italic>:<jats:italic>M<\/jats:italic>\u2192<jats:italic>N<\/jats:italic>is a cardinal in<jats:italic>N<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273800","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:12:14Z","timestamp":1146953534000},"page":"220-226","source":"Crossref","is-referenced-by-count":0,"title":["Co-critical points of elementary embeddings"],"prefix":"10.1017","volume":"50","author":[{"given":"Michael","family":"Sheard","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200033181_ref004","first-page":"65","volume":"43","author":"Kunen","year":"1978","journal-title":"Saturated ideals"},{"key":"S0022481200033181_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90024-3"},{"key":"S0022481200033181_ref006","first-page":"57","volume":"39","author":"Mitchell","year":"1974","journal-title":"Sets constructible from sequences of ultrafilters"},{"key":"S0022481200033181_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511600586"},{"key":"S0022481200033181_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0290961"},{"key":"S0022481200033181_ref002","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200033181_ref003","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BFb0103104","volume-title":"Higher set theory (Proceedings, Oberwolfach, 1977)","volume":"669","author":"Kanamori","year":"1978"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200033181","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,4,14]],"date-time":"2020-04-14T14:19:19Z","timestamp":1586873959000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200033181\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1985,3]]}},"alternative-id":["S0022481200033181"],"URL":"https:\/\/doi.org\/10.2307\/2273800","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,3]]}}}