{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T22:24:23Z","timestamp":1649111063905},"reference-count":19,"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":3388,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,12]]},"abstract":"<jats:title>Abstract.<\/jats:title><jats:p>Working in the theory \u201dZF + There is a nontrivial elementary embedding <jats:italic>j<\/jats:italic> : <jats:italic>V<\/jats:italic> \u2192 <jats:italic>V<\/jats:italic>\u201c, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every cardinal \u03bc \u2265 \u2135<jats:sub>2<\/jats:sub> satisfies the square bracket infinite exponent partition relation <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007519_inline1\" \/>. We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding <jats:italic>j<\/jats:italic>: <jats:italic>V<\/jats:italic> \u2192 <jats:italic>V<\/jats:italic>. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice.<\/jats:p>","DOI":"10.2178\/jsl\/1102022223","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:52:57Z","timestamp":1109800377000},"page":"1267-1281","source":"Crossref","is-referenced-by-count":1,"title":["Jonsson-like partition relations and <i>j: V \u2192 V<\/i>"],"prefix":"10.1017","volume":"69","author":[{"given":"Arthur W.","family":"Apter","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Grigor","family":"Sargsyan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007519_ref019","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-96-03281-9"},{"key":"S0022481200007519_ref018","first-page":"1591\u20131594","volume":"64","author":"Suzuki","year":"1999","journal-title":"NO elementary embedding from V into V is definable from parameters"},{"key":"S0022481200007519_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BF01352930"},{"key":"S0022481200007519_ref015","volume-title":"Set theory: An introduction to independence proofs","volume":"102","author":"Kunen","year":"1980"},{"key":"S0022481200007519_ref010","first-page":"41\u201368","article-title":"Regular cardinals in models of ZF","volume":"290","author":"Gitik","year":"1985","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200007519_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(99)00052-4"},{"key":"S0022481200007519_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF01269948"},{"key":"S0022481200007519_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF02774081"},{"key":"S0022481200007519_ref016","first-page":"189\u2013205","volume-title":"Axiomatic set theory","volume":"13","author":"Reinhardt","year":"1974"},{"key":"S0022481200007519_ref004","first-page":"591\u2013603","volume":"51","author":"Apter","year":"1986","journal-title":"Large cardinal structures below \u2135\u03c9"},{"key":"S0022481200007519_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760939"},{"key":"S0022481200007519_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BF02485843"},{"key":"S0022481200007519_ref011","volume-title":"The axiom of choice","author":"Jech","year":"1973"},{"key":"S0022481200007519_ref012","volume-title":"The higher infinite","author":"Kanamori","year":"1994"},{"key":"S0022481200007519_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0069385"},{"key":"S0022481200007519_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(83)90051-9"},{"key":"S0022481200007519_ref014","first-page":"407\u2013413","volume":"36","author":"Kunen","year":"1971","journal-title":"Elementary embeddings and infinitary combinatorics"},{"key":"S0022481200007519_ref007","first-page":"19\u201323","article-title":"On a problem of B. J\u00f3nsson","volume":"14","author":"Erd\u0151s","year":"1966","journal-title":"Bulletin de l' Acad\u00e9mie Polonaise des Sciences S\u00e9rie des Sciences Math\u00e9matiques Astronomiques, et Physiques"},{"key":"S0022481200007519_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-99-02554-4"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200007519","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T19:39:20Z","timestamp":1557171560000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200007519\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,12]]},"references-count":19,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2004,12]]}},"alternative-id":["S0022481200007519"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1102022223","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,12]]}}}