{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T20:16:29Z","timestamp":1775506589791,"version":"3.50.1"},"reference-count":9,"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":12245,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1980,9]]},"abstract":"<jats:p>Let <jats:italic>T<\/jats:italic> extend the theory <jats:italic>P<\/jats:italic> of Peano arithmetic, and suppose <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046570_inline1\"\/><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046570_inline2\"\/>. Form from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046570_inline3\"\/> a model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046570_inline4\"\/>, in analogy to the way in which the ordered ring of integers is formed from the standard model of arithmetic. Let <jats:italic>P<\/jats:italic>\u2032 and <jats:italic>T<\/jats:italic>\u2032 be the corresponding analogues of <jats:italic>P<\/jats:italic> and <jats:italic>T<\/jats:italic> respectively. Now consider the group <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046570_inline5\"\/>. In [5] MacDowell and Specker set out to determine the structure of such groups. (The precise statement in [5] refers to the ring of integers rather than the ordered ring. However, as pointed out to us by J. Knight, since Lagrange's Theorem that a positive integer is the sum of four squares is provable in the analogue of <jats:italic>P<\/jats:italic>\u2032 for rings (see, for example, the proof in [7, p. 102]), the set of positive elements is definable in the ring, and consequently, so is the ordering. Thus, for the present purpose it makes no difference which of the two structures is used. Of course, one needs the ordering to discuss end extensions, as considered in [5]. On the other hand, one should be aware that in Pr\u2032 one cannot define an ordering, where the theory Pr\u2032 is the theory of the group of integers with distinguished element 1, \u3008Z, +, 1\u3009. The constant 1 is needed so that divisibility mod <jats:italic>n<\/jats:italic> can be expressed. We will return to this point later.) In \u00a71 we shall outline the results in this direction obtained in [5].<\/jats:p><jats:p>Lipshitz and Nadel, unaware that a similar question had been posed and investigated in [5] (though, of course aware that [5] contained the celebrated results on end extensions) set out to characterize those models \u3008<jats:italic>A<\/jats:italic>, +\u3009 of Pr = Presburger Arithmetic (the complete theory of \u3008<jats:italic>\u03c9<\/jats:italic>, +\u3009) which can be expanded to models \u3008<jats:italic>A<\/jats:italic>, +, \u00b7, 0, 1, \u2264\u3009 of <jats:italic>P<\/jats:italic>. They were able to give a complete characterization for countable models \u3008<jats:italic>A<\/jats:italic>, +\u3009 in [4], which we describe in \u00a72.<\/jats:p>","DOI":"10.2307\/2273426","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:53:41Z","timestamp":1146938021000},"page":"612-622","source":"Crossref","is-referenced-by-count":3,"title":["On a problem of MacDowell and Specker"],"prefix":"10.1017","volume":"45","author":[{"given":"Mark","family":"Nadel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200046570_ref009","doi-asserted-by":"publisher","DOI":"10.4064\/fm-41-2-203-271"},{"key":"S0022481200046570_ref007","volume-title":"An introduction to the theory of numbers","author":"Niven","year":"1960"},{"key":"S0022481200046570_ref005","first-page":"257","volume-title":"Infinitistic methods","author":"MacDowell","year":"1961"},{"key":"S0022481200046570_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1978-0491158-5"},{"key":"S0022481200046570_ref003","volume-title":"Infinite Abelian groups","author":"Kaplansky","year":"1954"},{"key":"S0022481200046570_ref002","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200046570_ref001","first-page":"531","volume":"41","author":"Barwise","year":"1976","journal-title":"An introduction to recursively saturated and resplendent models"},{"key":"S0022481200046570_ref006","volume-title":"Introduction to mathematical logic","author":"Mendelson","year":"1964"},{"key":"S0022481200046570_ref008","volume-title":"Saturated model theory","author":"Sacks","year":"1972"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200046570","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,25]],"date-time":"2019-05-25T17:39:43Z","timestamp":1558805983000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200046570\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,9]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1980,9]]}},"alternative-id":["S0022481200046570"],"URL":"https:\/\/doi.org\/10.2307\/2273426","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,9]]}}}