{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,7]],"date-time":"2023-09-07T22:45:33Z","timestamp":1694126733227},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14437,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1974,9]]},"abstract":"<jats:p>The Hanf number for sentences of a language <jats:italic>L<\/jats:italic> is defined to be the least cardinal <jats:italic>\u03ba<\/jats:italic> with the property that for any sentence <jats:italic>\u03c6<\/jats:italic> of <jats:italic>L<\/jats:italic>, if <jats:italic>\u03c6<\/jats:italic> has a model of power \u2265 <jats:italic>\u03ba<\/jats:italic> then <jats:italic>\u03c6<\/jats:italic> has models of arbitrarily large cardinality. We shall be interested in the language <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1,<jats:italic>\u03c9<\/jats:italic><\/jats:sub> (see [3]), which is obtained by adding to the formation rules for first-order logic the rule that the conjunction of countably many formulas is also a formula.<\/jats:p><jats:p>Lopez-Escobar proved [4] that the Hanf number for sentences of <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1,<jats:italic>\u03c9<\/jats:italic><\/jats:sub> is \u2290<jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>, where the cardinals \u2290<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> are defined recursively by \u2290<jats:sub>0<\/jats:sub> = \u2135<jats:sub>0<\/jats:sub> and \u2290<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> = \u03a3{2<jats:sup>\u2290<\/jats:sup><jats:sub><jats:italic>\u03b2<\/jats:italic><\/jats:sub>: <jats:italic>\u03b2<\/jats:italic> &lt; <jats:italic>\u03b1<\/jats:italic>} for all cardinals <jats:italic>\u03b1<\/jats:italic> &gt; 0. Here <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> denotes the least uncountable ordinal.<\/jats:p><jats:p>A sentence of <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1,<jats:italic>\u03c9<\/jats:italic><\/jats:sub> is <jats:italic>complete<\/jats:italic> if all its models satisfy the same <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1,<jats:italic>\u03c9<\/jats:italic><\/jats:sub>-sentences. In [5], Malitz proved that the Hanf number for complete sentences of <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1,<jats:italic>\u03c9<\/jats:italic><\/jats:sub> is also \u2290<jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>, but his proof required the generalized continuum hypothesis (GCH). The purpose of this paper is to give a proof that does not require GCH.<\/jats:p><jats:p>More precisely, we will prove the following:<\/jats:p><jats:p>Theorem 1. <jats:italic>For any countable ordinal <jats:italic>\u03b1<\/jats:italic>, there is a complete <jats:italic>L<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1,<jats:italic>\u03c9<\/jats:italic><\/jats:sub>-sentence<\/jats:italic> \u03c3<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub><jats:italic>which has models of power \u2290<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> but no models of higher cardinality<\/jats:italic>.<\/jats:p><jats:p>Our basic approach is identical with Malitz's. We simply use a different combinatorial fact at the crucial point.<\/jats:p>","DOI":"10.2307\/2272900","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:31:33Z","timestamp":1146936693000},"page":"575-578","source":"Crossref","is-referenced-by-count":8,"title":["The Hanf number for complete <i>L<\/i><sub><i>\u03c9<\/i><sub>1<\/sub>,<i>\u03c9<\/i><\/sub>-sentences (without GCH)"],"prefix":"10.1017","volume":"39","author":[{"given":"James E.","family":"Baumgartner","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200076738_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0079689"},{"key":"S0022481200076738_ref003","volume-title":"Languages with expressions of infinite length","author":"Karp","year":"1964"},{"key":"S0022481200076738_ref001","unstructured":"Baumgartner J. , Almost-disjoint sets, the dense-set problem, and the partition calculus (to appear)."},{"key":"S0022481200076738_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90017-4"},{"key":"S0022481200076738_ref007","first-page":"329\u2013341","volume-title":"The theory of models","author":"Scott","year":"1965"},{"key":"S0022481200076738_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/sm-6-1-18-19"},{"key":"S0022481200076738_ref004","first-page":"13\u201321","article-title":"On defining well-orderings","volume":"59","author":"Lopez-Escobar","year":"1966","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\/S0022481200076738","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T17:03:09Z","timestamp":1559149389000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200076738\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1974,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1974,9]]}},"alternative-id":["S0022481200076738"],"URL":"https:\/\/doi.org\/10.2307\/2272900","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1974,9]]}}}