{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,8,12]],"date-time":"2022-08-12T01:25:51Z","timestamp":1660267551988},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":3571,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,6]]},"abstract":"We will mainly be concerned with a result which refutes a stronger variant of a conjecture of Macpherson about finitely axiomatizable \u03c9<\/jats:italic>-categorical theories. Then we prove a result which implies that the \u03c9<\/jats:italic>-categorical stable pseudoplanes of Hrushovski do not have the finite submodel property.<\/jats:p>Let's call a consistent first-order sentence without finite models an axiom of infinity<\/jats:italic>. Can we somehow describe the axioms of infinity? Two standard examples are:<\/jats:p>\u03d5<\/jats:italic>1<\/jats:sub>: A first-order sentence which expresses that a binary relation < on a nonempty universe is transitive and irreflexive and that for every x<\/jats:italic> there is y<\/jats:italic> such that x<\/jats:italic> < y<\/jats:italic>.<\/jats:p>\u03d5<\/jats:italic>2<\/jats:sub>: A first-order sentence which expresses that there is a unique x<\/jats:italic> such that, (0) for every y, s(y)<\/jats:italic> \u2260 x<\/jats:italic> (where s<\/jats:italic> is a unary function symbol),<\/jats:p>and, for every x<\/jats:italic>, if x<\/jats:italic> does not satisfy (0) then there is a unique y<\/jats:italic> such that s(y)<\/jats:italic> = x<\/jats:italic>.<\/jats:p>Every complete theory T<\/jats:italic> such that \u03d5<\/jats:italic>1<\/jats:sub> \u03f5 T<\/jats:italic> has the strict order property (as defined in [10]), since the formula x<\/jats:italic> < y<\/jats:italic> will have the strict order property for T<\/jats:italic>. Let's say that if \u03a8 is an axiom of infinity and every complete theory T<\/jats:italic> with \u03a8 \u03f5 T<\/jats:italic> has the strict order property, then \u03a8 has the strict order property<\/jats:italic>.<\/jats:p>Every complete theory T<\/jats:italic> such that \u03d5<\/jats:italic>2<\/jats:sub> \u03f5 T<\/jats:italic> is not \u03c9<\/jats:italic>-categorical. This is the case because a complete theory T<\/jats:italic> without finite models is \u03c9<\/jats:italic>-categorical if and only if, for every 0 < n<\/jats:italic> < \u03c9<\/jats:italic>, there are only finitely many formulas in the variables x<\/jats:italic>1<\/jats:sub>,\u2026,x<\/jats:italic>n<\/jats:italic><\/jats:sub>, up to equivalence, in any model of T<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1082418529","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T16:33:18Z","timestamp":1109781198000},"page":"329-339","source":"Crossref","is-referenced-by-count":3,"title":["On first-order sentences without finite models"],"prefix":"10.1017","volume":"69","author":[{"given":"Marko","family":"Djordjevi\u0107","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200007763_ref005","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S0022481200007763_ref011","volume-title":"Automorphisms of First-Order Structures","author":"Wagner","year":"1994"},{"key":"S0022481200007763_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(01)00109-9"},{"key":"S0022481200007763_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-97-01869-2"},{"key":"S0022481200007763_ref004","first-page":"353","article-title":"Weight \u03c9 in stable theories with few types","volume":"60","author":"Herwig","year":"1995","journal-title":"this Journal"},{"key":"S0022481200007763_ref006","volume-title":"Logic Colloquium '92 (Vezprem, 1992)","author":"Ivanov","year":"1995"},{"key":"S0022481200007763_ref007","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-59.3.439"},{"key":"S0022481200007763_ref008","unstructured":"Lippel D. , Finitely axiomatizable \u03c9-categorical theories and the Mazoyer hypothesis, preprint."},{"key":"S0022481200007763_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(85)90023-5"},{"key":"S0022481200007763_ref010","volume-title":"Classification theory and the number of non-isomorphic models","author":"Shelah","year":"1978"},{"key":"S0022481200007763_ref009","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093635744"},{"key":"S0022481200007763_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-3002-0"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200007763","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T17:02:39Z","timestamp":1557162159000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200007763\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,6]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2004,6]]}},"alternative-id":["S0022481200007763"],"URL":"http:\/\/dx.doi.org\/10.2178\/jsl\/1082418529","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,6]]}}}