{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T23:50:33Z","timestamp":1649029833119},"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 [7] we proved that (I) if <jats:italic>T<\/jats:italic> is a countable \u2135<jats:sub>0<\/jats:sub>-categorical theory without finite models then <jats:italic>T<\/jats:italic> has a model companion; and several people have observed that (II) if <jats:italic>T<\/jats:italic> is a countable theory without finite models which is \u2135<jats:sub>1<\/jats:sub>-categorical and forcingcomplete for infinite forcing (i.e., <jats:italic>T<\/jats:italic>= <jats:italic>T<jats:sup>F<\/jats:sup><\/jats:italic>) then <jats:italic>T<\/jats:italic> is model-complete. It is natural to ask (1) whether in (I) we can replace \u2135<jats:sub>0<\/jats:sub> by \u2135<jats:sub>1<\/jats:sub>; (2) whether in (II) we can replace <jats:italic>T<jats:sup>F<\/jats:sup><\/jats:italic> by <jats:italic>T<jats:sup>f<\/jats:sup><\/jats:italic>; and (3) in connection with (II), whether the categoricity of the class of infinitely generic structures for a theory <jats:italic>K<\/jats:italic> in some or all infinite powers implies the existence of a model companion for <jats:italic>K<\/jats:italic>. The purpose of this note is to provide negative answers to (1), (2), and (3). Specifically, we will prove:<\/jats:p><jats:p>Theorem. <jats:italic>There exists a countable theory T such that<\/jats:italic><\/jats:p><jats:p>(i) <jats:italic>T has no finite models and is \u2135-categorical<\/jats:italic>;<\/jats:p><jats:p>(ii) <jats:italic>T is forcing-complete for finite forcing, i.e., T = T<jats:sup>f<\/jats:sup><\/jats:italic>;<\/jats:p><jats:p>(iii) <jats:italic>T has no model companion (i.e., in light of<\/jats:italic> (ii), <jats:italic>T is not model-complete<\/jats:italic>);<\/jats:p><jats:p>(iv) <jats:italic>the class of infinitely generic structures for T is categorical in every infinite power<\/jats:italic>;<\/jats:p><jats:p>(v) <jats:italic>every uncountable existentially complete structure for T is infinitely generic<\/jats:italic>;<\/jats:p><jats:p>(vi) <jats:italic>there is, up to isomorphism, precisely one countable existentially complete model of T<jats:sup>f<\/jats:sup>, and there are no uncountable e.c. models of T<jats:sup>f<\/jats:sup> (in particular, there is just one countable finitely generic structure and there are no uncountable ones<\/jats:italic>);<\/jats:p><jats:p>(vii) <jats:italic>there are precisely<\/jats:italic> \u2135<jats:sub>0<\/jats:sub><jats:italic>isomorphism types of countable existentially complete structures for T<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2272266","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:33:55Z","timestamp":1146951235000},"page":"31-34","source":"Crossref","is-referenced-by-count":1,"title":["A counterexample in the theory of model companions"],"prefix":"10.1017","volume":"40","author":[{"given":"D.","family":"Saracino","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200054220_ref006","first-page":"441","volume":"36","author":"Robinson","year":"1971","journal-title":"On the notion of algebraic closedness for noncommutative groups and fields"},{"key":"S0022481200054220_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90008-2"},{"key":"S0022481200054220_ref004","unstructured":"Mortimer M. , Model-completeness and categoricity, preprint."},{"key":"S0022481200054220_ref003","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10648"},{"key":"S0022481200054220_ref005","volume-title":"Proceedings of the Second Scandinavian Logic Symposium","author":"Robinson","year":"1971"},{"key":"S0022481200054220_ref008","first-page":"81","article-title":"Arithmetical extensions of relational systems","volume":"13","author":"Tarski","year":"1957","journal-title":"Compositio Mathematica"},{"key":"S0022481200054220_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1965-0175782-0"},{"key":"S0022481200054220_ref007","first-page":"591","article-title":"Model companions for \u21350-categorical theories","volume":"39","author":"Saracino","year":"1973","journal-title":"Proceedings 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\/S0022481200054220","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T20:05:47Z","timestamp":1559160347000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200054220\/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":["S0022481200054220"],"URL":"https:\/\/doi.org\/10.2307\/2272266","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,3]]}}}