{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T15:13:54Z","timestamp":1649171634566},"reference-count":11,"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":12337,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1980,6]]},"abstract":"In this paper we prove that Beth's property does not hold in L<\u03c9<\/jats:sup><\/jats:italic>. This answers a question posed by Magidor and Malitz in [8]. Beth's property is a natural closure condition on a language which says that everything implicitly definable in the language is also explicitly definable in the language. That the first-order predicate calculus (L<\/jats:italic>0<\/jats:sub>) has the property was first shown by Beth [4]. Lopez-Escobar proved that also has Beth's property [7]. Malitz [9] showed the Beth's property fails in L\u03ba\u03bb<\/jats:sub><\/jats:italic> where \u03ba<\/jats:italic> \u2265 \u03bb<\/jats:italic> \u2265 \u03c9<\/jats:italic>1<\/jats:sub>. Friedman and Silver showed that the property fails in L\u03ba\u03bb<\/jats:sub><\/jats:italic> for \u03ba<\/jats:italic> > \u03c9<\/jats:italic>1<\/jats:sub>. Also Friedman [5] showed that extensions of elementary logic using cardinality quantifiers (L<\/jats:italic>)1<\/jats:sub>\u03ba<\/jats:sup> do not have Beth's property.<\/jats:p>The undefined notation used here is standard. If further clarification is needed, we refer the reader to [3]. \u03ba<\/jats:italic> and \u03bb<\/jats:italic> denote infinite cardinals. cX<\/jats:italic> and \u0223X<\/jats:italic>\u2223 denote the cardinality of X<\/jats:italic>. cf\u03bc will denote the cofinality of the order \u03bc. All languages discussed are assumed to have no function or constant symbols. All structures are relational. The type of a formula is the set of relation symbols appearing in the formula (excluding equality). The type of a set of formulas is the set of all relation symbols appearing in some formula of the set of formulas. For purpose of this paper we will assume that to each n<\/jats:italic>-ary relation in a structure there is associated an n<\/jats:italic>-ary relation symbol and that in any given structure this association is one-to-one. Using this convention we can define the type of a structure to be the set of those relation symbols to which there is associated a relation in the structure. Sometimes we use the term predicate instead of relation.<\/jats:p>","DOI":"10.2307\/2273188","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:52:48Z","timestamp":1146952368000},"page":"284-290","source":"Crossref","is-referenced-by-count":0,"title":["Beth's property fails in L<\/i><\u03c9<\/sup>"],"prefix":"10.1017","volume":"45","author":[{"given":"Lee","family":"Badger","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120004679X_ref001","first-page":"A","article-title":"Beth's theorem fails in L<\u03c9","volume":"21","author":"Badger","year":"1974","journal-title":"Notices of the American Mathematical Society"},{"key":"S002248120004679X_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(53)50042-3"},{"key":"S002248120004679X_ref010","unstructured":"Magidor M. , Makowsky J. and Stavi J. , Unpublished result that Craig's theorem fails in L <\u03c9."},{"key":"S002248120004679X_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90019-5"},{"key":"S002248120004679X_ref006","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1969.tb00356.x"},{"key":"S002248120004679X_ref011","first-page":"635","volume":"31","author":"Yasuhara","year":"1966","journal-title":"An axiomatic system for the first-order language with an equi-cardinality quantifier"},{"key":"S002248120004679X_ref009","first-page":"216","volume":"36","author":"Malitz","year":"1971","journal-title":"Infinitary analogs of theorems from first order model theory"},{"key":"S002248120004679X_ref002","unstructured":"Badger L. , The Malitz quantifier meets its Ehrenfeucht game, Ph.D. Thesis, University of Colorado, 1975."},{"key":"S002248120004679X_ref003","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1977.72.293"},{"key":"S002248120004679X_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02762675"},{"key":"S002248120004679X_ref007","doi-asserted-by":"crossref","first-page":"253","DOI":"10.4064\/fm-57-3-253-272","article-title":"An interpolation theorem for denumerably long formulas","volume":"57","author":"Lopez-Escobar","year":"1965","journal-title":"Fundamenta Mathematicae"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120004679X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T19:12:05Z","timestamp":1558897925000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120004679X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1980,6]]}},"alternative-id":["S002248120004679X"],"URL":"https:\/\/doi.org\/10.2307\/2273188","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,6]]}}}