{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,19]],"date-time":"2024-02-19T10:21:06Z","timestamp":1708338066276},"reference-count":4,"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":5215,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1999,12]]},"abstract":"In 1970, Kenneth Kunen showed that there is no non-trivial elementary embedding of the universe V into itself<\/jats:italic> [2] using the axiom of choice. Kunen remarked in his paper that the result can be formalized in Morse-Kelley theory of sets and classes. In this paper, we will work within ZF<\/jats:bold>, Zermelo-Fraenkel axioms, and deal with embeddings definable with a formula and a parameter.<\/jats:p>In ZF<\/jats:bold>, a \u201cclass\u201d is usually synonymous with \u201cproperty\u201d, that is a class definable with a parameter, C<\/jats:italic> = {x: \u03c6(x,p)}, where \u03c6 is a formula in the language [\u2208}. Using this convention, let j<\/jats:italic> be a class. Then \u201cj<\/jats:italic> is an elementary embedding of V into V\u201d is not a single statement but a schema of statements \u201cj<\/jats:italic> preserves \u03c8\u201d for each formula \u03c8. We prove that this schema is expressible in the language {\u2208} by a single formula:<\/jats:p>Lemma<\/jats:sc>. An embedding j: V \u2192 V is elementary iff j preserves<\/jats:italic>\u03c8.<\/jats:p>Here \u03c8(\u03b1, \u03c8, a) is the property \u201ca is an ordinal, \u03c6 is a formula and V\u03b1<\/jats:sub>.\u201d<\/jats:p>The lemma is of course a schema of lemmas, one for each formula denning j<\/jats:italic> and for each \u03c8 to be preserved.<\/jats:p>Using this we prove our theorem in ZF<\/jats:bold> (again, a schema of theorems.):<\/jats:p>Theorem<\/jats:sc> 1.1. There is no nontrivial definable elementary embedding j: V \u2192 V<\/jats:italic>.<\/jats:p>Many symbols and their definitions follow those used by Drake's book [1]. The formula Sat expresses the satisfaction relation . The formula Fmla(u<\/jats:italic>) expresses that u<\/jats:italic> is the G\u00f6del-set for a formula.<\/jats:p>","DOI":"10.2307\/2586799","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T14:03:57Z","timestamp":1146924237000},"page":"1591-1594","source":"Crossref","is-referenced-by-count":6,"title":["No elementary embedding from v into v is definable from parameters"],"prefix":"10.1017","volume":"64","author":[{"given":"Akira","family":"Suzuki","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200012779_ref004","unstructured":"Woodin H. , absoluteness and supercompact cardinals, 05 15 1985, hand-written note."},{"key":"S0022481200012779_ref001","volume-title":"Set Theory: An Introduction to Large Cardinals","author":"Drake","year":"1974"},{"key":"S0022481200012779_ref002","volume":"36","author":"Kunen","year":"1971","journal-title":"Elementary embeddings and infinitary combinatorics"},{"key":"S0022481200012779_ref003","unstructured":"Vickers J. and Welch P. D. , On Elementary Embeddings from an Inner Model to the Universe, to appear."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200012779","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T15:57:54Z","timestamp":1557417474000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200012779\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,12]]},"references-count":4,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1999,12]]}},"alternative-id":["S0022481200012779"],"URL":"https:\/\/doi.org\/10.2307\/2586799","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,12]]}}}