{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T07:59:04Z","timestamp":1771574344266,"version":"3.50.1"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14256,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,3]]},"abstract":"<jats:p>In a recent paper [3] H. Gaifman investigated some model theoretic consequences of Matijasevi\u010d's theorem [5], and posed some further problems which naturally arise. We provide here partial answers to two of these problems, the results having been previously announced in the postscript of [3].<\/jats:p><jats:p>Firstly, it is shown in [3] that if <jats:italic>M<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>M<\/jats:italic><jats:sub>2<\/jats:sub> are models of the Peano axioms <jats:italic>P<\/jats:italic> and <jats:italic>M<\/jats:italic><jats:sub>1<\/jats:sub> \u2286 <jats:italic>M<\/jats:italic><jats:sub>2<\/jats:sub>, then <jats:italic>M<\/jats:italic><jats:sub>1<\/jats:sub> is closed under the recursive functions of <jats:italic>M<\/jats:italic><jats:sub>2<\/jats:sub>. The converse of this statement is false. Moreover, Gaifman asks: Is every initial segment of a model <jats:italic>M<\/jats:italic> of <jats:italic>P<\/jats:italic> which is closed under the recursive functions of <jats:italic>M<\/jats:italic> (or the \u2211<jats:sub>n<\/jats:sub>-definable functions) also a model of <jats:italic>P<\/jats:italic>? We show that this is false and our method gives, en route, another proof of a theorem of Rabin [7] stating the <jats:italic>P<\/jats:italic> is not implied by any consistent set of \u2211<jats:sub>n<\/jats:sub> sentences for any <jats:italic>n<\/jats:italic>.<\/jats:p><jats:p>Secondly, we partially answer a question posed on p. 129 of [3] by proving (some-what more than) every countable nonstandard model of <jats:italic>P<\/jats:italic> has an end extension in which a diophantine equation, not solvable in the original model, has a solution. We can, in fact, take the new model to be isomorphic to the original one. This generalises (apart from the countability restriction) a theorem of Rabin [6].<\/jats:p>","DOI":"10.2307\/2272268","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:33:55Z","timestamp":1146936835000},"page":"41-47","source":"Crossref","is-referenced-by-count":7,"title":["On models of arithmetic\u2014Answers to two problems raised by H. Gaifman"],"prefix":"10.1017","volume":"40","author":[{"given":"Alex","family":"Wilkie","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200054244_ref007","first-page":"287","volume-title":"Essays on foundations of mathematics","author":"Rabin","year":"1966"},{"key":"S0022481200054244_ref006","first-page":"151","volume-title":"Logic, methodology and philosophy of science: Proceedings of the 1960 International Congress","author":"Rabin","year":"1962"},{"key":"S0022481200054244_ref003","volume-title":"A note on models and submodels of arithmetic","author":"Gaifman","year":"1970"},{"key":"S0022481200054244_ref002","first-page":"105","volume-title":"Mathematical logic and foundations of set theory","author":"Gaifman","year":"1968"},{"key":"S0022481200054244_ref008","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200054244_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066789"},{"key":"S0022481200054244_ref004","volume-title":"Introduction to metamathematics","author":"Kleene","year":"1967"},{"key":"S0022481200054244_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70846-2"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200054244","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T16:05:52Z","timestamp":1559145952000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200054244\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,3]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1975,3]]}},"alternative-id":["S0022481200054244"],"URL":"https:\/\/doi.org\/10.2307\/2272268","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,3]]}}}