{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T23:11:30Z","timestamp":1781651490063,"version":"3.54.5"},"reference-count":10,"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":13706,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1976,9]]},"abstract":"<jats:p>The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and <jats:italic>p<\/jats:italic>-adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy.<\/jats:p><jats:p>In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and <jats:italic>p<\/jats:italic>-adic fields. We shall outline a new treatment of the <jats:italic>p<\/jats:italic>-adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and <jats:italic>p<\/jats:italic>-adically closed fields.<\/jats:p><jats:p>We want to describe the definable subsets of <jats:italic>p<\/jats:italic>-adically closed fields. Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if <jats:italic>K<\/jats:italic> is a real-closed field and <jats:italic>X<\/jats:italic> is a subset of <jats:italic>K<\/jats:italic> first-order definable on <jats:italic>K<\/jats:italic> using parameters from <jats:italic>K<\/jats:italic> then <jats:italic>X<\/jats:italic> is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of <jats:italic>K<\/jats:italic>). In particular, if <jats:italic>X<\/jats:italic> is infinite, <jats:italic>X<\/jats:italic> has nonempty interior.<\/jats:p><jats:p>Now, there is an analogous question for <jats:italic>p<\/jats:italic>-adically closed fields. If <jats:italic>K<\/jats:italic> is <jats:italic>p<\/jats:italic>-adically closed, what are the definable subsets of <jats:italic>K<\/jats:italic>? To the best of our knowledge, this question has not been answered until now.<\/jats:p><jats:p>What is the difference between the two cases? Tarski's analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for <jats:italic>p<\/jats:italic>-adically closed fields has been achieved [3], but only when we take a cross-section <jats:italic>\u03c0<\/jats:italic> as part of our basic data. The problem is that in the presence of <jats:italic>\u03c0<\/jats:italic> it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.<\/jats:p>","DOI":"10.2307\/2272038","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:42:22Z","timestamp":1146951742000},"page":"605-610","source":"Crossref","is-referenced-by-count":151,"title":["On definable subsets of <i>p<\/i>-adic fields"],"prefix":"10.1017","volume":"41","author":[{"given":"Angus","family":"MacIntyre","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200051173_bib003","doi-asserted-by":"publisher","DOI":"10.2307\/1970476"},{"key":"S0022481200051173_bib002","doi-asserted-by":"publisher","DOI":"10.2307\/2373066"},{"key":"S0022481200051173_bib009","unstructured":"Tarski, A. , A decision procedure for elementary algebra and geometry, The Rand Corporation, Santa Monica, 1948."},{"key":"S0022481200051173_bib007","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/012\/0257030"},{"key":"S0022481200051173_bib001","doi-asserted-by":"publisher","DOI":"10.2307\/2373065"},{"key":"S0022481200051173_bib004","first-page":"79","volume":"36","author":"Baldwin","year":"1971","journal-title":"On strongly minimal sets"},{"key":"S0022481200051173_bib008","first-page":"173","article-title":"A theorem on quantifier elimination","volume":"5","author":"Shoenfield","year":"1971","journal-title":"Symposia Mathematica"},{"key":"S0022481200051173_bib010","unstructured":"Weispfenning, V. , On the elementary theory of Hensel fields, Thesis, Heidelberg, 1971."},{"key":"S0022481200051173_bib006","first-page":"1390","article-title":"On the elementary theory of maximal normed fields","volume":"5","author":"Ersov","year":"1965","journal-title":"Doklady Akademii Nauk SSSR"},{"key":"S0022481200051173_bib005","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160220202"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200051173","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T10:38:51Z","timestamp":1679481531000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200051173\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1976,9]]},"references-count":10,"aliases":["10.1017\/s0022481200051173","10.1017\/s0022481200051173"],"journal-issue":{"issue":"3","published-print":{"date-parts":[[1976,9]]}},"alternative-id":["S0022481200051173"],"URL":"https:\/\/doi.org\/10.2307\/2272038","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1976,9]]}}}