{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,18]],"date-time":"2025-05-18T14:45:40Z","timestamp":1747579540186},"reference-count":7,"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":9781,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1987,6]]},"abstract":"<jats:p>In this paper we give a closer analysis of the elementary properties of the Banach spaces <jats:italic>C<\/jats:italic>(<jats:italic>K<\/jats:italic>), where <jats:italic>K<\/jats:italic> is a totally disconnected, compact Hausdorff space, in terms of the Boolean algebra <jats:italic>B<\/jats:italic>(<jats:italic>K<\/jats:italic>) of clopen subsets of <jats:italic>K<\/jats:italic>. In particular we sharpen a result in [4] by showing that if <jats:italic>B<\/jats:italic>(<jats:italic>K<\/jats:italic><jats:sub>1<\/jats:sub>) and <jats:italic>B<\/jats:italic>(<jats:italic>K<\/jats:italic><jats:sub>2<\/jats:sub>) satisfy the same sentences with \u2264 <jats:italic>n<\/jats:italic> alternations of quantifiers, then the same is true of <jats:italic>C<\/jats:italic>(<jats:italic>K<\/jats:italic><jats:sub>1<\/jats:sub>) and <jats:italic>C<\/jats:italic>(<jats:italic>K<\/jats:italic><jats:sub>2<\/jats:sub>). As a consequence we show that for each <jats:italic>n<\/jats:italic> there exist <jats:italic>C<\/jats:italic>(<jats:italic>K<\/jats:italic>) spaces which are elementarily equivalent for sentences with \u2264 <jats:italic>n<\/jats:italic> quantifier alternations, but which are not elementary equivalent in the full sense. Thus the elementary properties of Banach spaces cannot be determined by looking at sentences with a bounded number of quantifier alternations.<\/jats:p><jats:p>The notion of elementary equivalence for Banach spaces which is studied here was introduced by the second author [4] and is expressed using the language of positive bounded formulas in a first-order language for Banach spaces. As was shown in [4], two Banach spaces are elementarily equivalent in this sense if and only if they have isometrically isomorphic Banach space ultrapowers (or, equivalently, isometrically isomorphic nonstandard hulls.)<\/jats:p><jats:p>We consider Banach spaces over the field of real numbers. If <jats:italic>X<\/jats:italic> is such a space, <jats:italic>B<\/jats:italic><jats:sub><jats:italic>x<\/jats:italic><\/jats:sub> will denote the closed unit ball of <jats:italic>X<\/jats:italic>, <jats:italic>B<\/jats:italic><jats:sub><jats:italic>x<\/jats:italic><\/jats:sub> = {<jats:italic>x<\/jats:italic> \u03f5 <jats:italic>X<\/jats:italic>\u2223 \u2223\u2223<jats:italic>x<\/jats:italic>\u2223\u2223 \u2264 1}. Given a compact Hausdorff space <jats:italic>K<\/jats:italic>, we let <jats:italic>C(K)<\/jats:italic> denote the Banach space of all continuous real-valued functions on <jats:italic>K<\/jats:italic>, under the supremum norm. We will especially be concerned with such spaces when <jats:italic>K<\/jats:italic> is a totally disconnected compact Hausdorff space. In that case <jats:italic>B(K)<\/jats:italic> will denote the Boolean algebra of all clopen subsets of <jats:italic>K<\/jats:italic>. We adopt the standard notation from model theory and Banach space theory.<\/jats:p>","DOI":"10.2307\/2274386","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:21:40Z","timestamp":1146939700000},"page":"368-373","source":"Crossref","is-referenced-by-count":5,"title":["A note on elementary equivalence of <i>C(K)<\/i> spaces"],"prefix":"10.1017","volume":"52","author":[{"given":"S.","family":"Heinrich","sequence":"first","affiliation":[]},{"given":"C. Ward","family":"Henson","sequence":"additional","affiliation":[]},{"suffix":"Jr.","given":"L. C.","family":"Moore","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200034320_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01166070"},{"key":"S0022481200034320_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF02756565"},{"key":"S0022481200034320_ref001","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200034320_ref002","first-page":"17","article-title":"Decidability of the elementary theory of relatively complemented distributive lattices and the theory of filters","volume":"3","author":"Er\u0161ov","year":"1964","journal-title":"Algebra i Logika"},{"key":"S0022481200034320_ref003","first-page":"79","volume-title":"Banach space theory and its applications, proceedings, Bucharest, 1981","volume":"991","author":"Heinrich","year":"1983"},{"key":"S0022481200034320_ref005","unstructured":"Henson C. W. , Banach space model theory. I (to appear)."},{"key":"S0022481200034320_ref007","first-page":"64","article-title":"Arithmetical classes and types of Boolean algebras","volume":"55","author":"Tarski","year":"1949","journal-title":"Bulletin of the American Mathematical Society"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200034320","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,21]],"date-time":"2019-05-21T15:48:20Z","timestamp":1558453700000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200034320\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1987,6]]}},"alternative-id":["S0022481200034320"],"URL":"https:\/\/doi.org\/10.2307\/2274386","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,6]]}}}