{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T08:57:54Z","timestamp":1760345874729},"reference-count":11,"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":11515,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,9]]},"abstract":"<jats:p>Let \u2112 be the first order language of field theory with an additional one place predicate symbol. In [B2] it was shown that the elementary theory <jats:italic>T<\/jats:italic> of the class <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044030_inline01\" \/> of all pairs of real closed fields, i.e., \u2112-structures \u2039K, L\u203a, <jats:italic>K<\/jats:italic> a real closed field, <jats:italic>L<\/jats:italic> a real closed subfield of <jats:italic>K<\/jats:italic>, is undecidable.<\/jats:p><jats:p>The aim of this paper is to show that the elementary theory <jats:italic>T<jats:sub>s<\/jats:sub><\/jats:italic> of a nontrivial subclass of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044030_inline01\" \/> containing many naturally occurring pairs of real closed fields is decidable (Theorem 3, \u00a75). This result was announced in [B2]. An explicit axiom system for <jats:italic>T<jats:sub>s<\/jats:sub><\/jats:italic> will be given later. At this point let us just mention that any model of <jats:italic>T<jats:sub>s<\/jats:sub><\/jats:italic>, is elementarily equivalent to a pair of power series fields \u2039<jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub>((<jats:italic>T<jats:sup>A<\/jats:sup><\/jats:italic>)), <jats:italic>R<\/jats:italic><jats:sub>1<\/jats:sub>((<jats:italic>T<jats:sup>B<\/jats:sup><\/jats:italic>))\u203a where <jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub> is the field of real numbers, <jats:italic>R<\/jats:italic><jats:sub>1<\/jats:sub> = <jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub> or the field of real algebraic numbers, and <jats:italic>B<\/jats:italic> \u2286 <jats:italic>A<\/jats:italic> are ordered divisible abelian groups. Conversely, all these pairs of power series fields are models of <jats:italic>T<\/jats:italic><jats:sub>s<\/jats:sub>.<\/jats:p><jats:p>Theorem 3 together with the undecidability result in [B2] answers some of the questions asked in Macintyre [M]. The proof of Theorem 3 uses the model theoretic techniques for valued fields introduced by Ax and Kochen [A-K] and Ershov [E] (see also [C-K]). The two main ingredients are<\/jats:p><jats:p>(i) the completeness of the elementary theory of real closed fields with a distinguished dense proper real closed subfield (due to Robinson [R]),<\/jats:p><jats:p>(ii) the decidability of the elementary theory of pairs of ordered divisible abelian groups (proved in \u00a7\u00a71-4).<\/jats:p><jats:p>I would like to thank Angus Macintyre for fruitful discussions concerning the subject. The valuation theoretic method of classifying theories of pairs of real closed fields is taken from [M].<\/jats:p>","DOI":"10.2307\/2273596","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:00:58Z","timestamp":1146938458000},"page":"669-679","source":"Crossref","is-referenced-by-count":3,"title":["On the elementary theory of pairs of real closed fields. II"],"prefix":"10.1017","volume":"47","author":[{"given":"Walter","family":"Baur","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044030_ref011","doi-asserted-by":"publisher","DOI":"10.4064\/fm-47-2-179-204"},{"key":"S0022481200044030_ref010","volume-title":"Lectures on formally real fields","author":"Prestel","year":"1975"},{"key":"S0022481200044030_ref007","volume-title":"Partially ordered algebraic systems","author":"Fuchs","year":"1963"},{"key":"S0022481200044030_ref006","first-page":"1390","article-title":"On the elementary theory of maximal normed fields","volume":"165","author":"Ershov","year":"1965","journal-title":"Doklady Akademie Nauk SSSR"},{"key":"S0022481200044030_ref005","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200044030_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/2373066"},{"key":"S0022481200044030_ref001","first-page":"161","volume-title":"Proceedings of Symposia in Pure Mathematics","volume":"20","author":"Ax","year":"1971"},{"key":"S0022481200044030_ref008","doi-asserted-by":"publisher","DOI":"10.4064\/fm-59-1-109-116"},{"key":"S0022481200044030_ref009","unstructured":"[M] Macintyre A. , Classifying pairs of real closed fields, Ph. D. Dissertation, Stanford University, 1968."},{"key":"S0022481200044030_ref004","unstructured":"[B2] Baur W. , Die Theorie der Paare reell abgeschlossener K\u00f3rper, l'Enseignement Math\u00e9matique (to appear)."},{"key":"S0022481200044030_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1976-0416890-9"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044030","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T16:47:33Z","timestamp":1558716453000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044030\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,9]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1982,9]]}},"alternative-id":["S0022481200044030"],"URL":"https:\/\/doi.org\/10.2307\/2273596","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,9]]}}}