{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T15:21:29Z","timestamp":1648999289530},"reference-count":1,"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":14072,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,9]]},"abstract":"<jats:p>Here we prove the following:<\/jats:p><jats:p>Theorem. <jats:italic>For every N<\/jats:italic> \u2264 \u03c9 <jats:italic>there is a complete theory T<jats:sub>n<\/jats:sub> having exactly n nonisomorphic rigid models and no uncountable rigid models. Moreover, each non-rigid model admits a nontrivial automorphism<\/jats:italic>.<\/jats:p><jats:p>The <jats:italic>T<jats:sub>n<\/jats:sub><\/jats:italic> are theories in the first-order predicate calculus and a rigid structure is a structure with no nontrivial endomorphisms, i.e., the only endomorphism of the structure into itself is the identity. The theorem answers a question of A. Ehrenfeucht.<\/jats:p><jats:p>For the most part we use standard model theoretic notation with Th <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052981_inline1\" \/> denoting the set of sentences true in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052981_inline1\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052981_inline2\" \/> meaning <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052981_inline2\" \/>. A complete set of sentences is one of the form Th <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052981_inline1\" \/> for some <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052981_inline1\" \/>. The universe of a structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052981_inline1\" \/> may be denoted by \u2223<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052981_inline1\" \/>\u2223. An <jats:italic>n<\/jats:italic>-ary relation on <jats:italic>X<\/jats:italic> is a set of <jats:italic>n<\/jats:italic> tuples (<jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>, \u2026, <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub>) with each <jats:italic>x<jats:sub>i<\/jats:sub><\/jats:italic> \u2208 <jats:italic>X<\/jats:italic>. All the structures encounted here will be relational. If <jats:italic>P<\/jats:italic> is an <jats:italic>n<\/jats:italic>-ary relation then <jats:italic>P<\/jats:italic> \u21be <jats:italic>Y<\/jats:italic>, the restriction of <jats:italic>P<\/jats:italic> to <jats:italic>Y<\/jats:italic>, is {(<jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>, \u2026, <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub>): (<jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>, \u2026, <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub>) \u2208 <jats:italic>P<\/jats:italic> and <jats:italic>x<\/jats:italic><jats:sub>0<\/jats:sub>, \u2026, <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub><jats:italic>Y<\/jats:italic>}.<\/jats:p>","DOI":"10.2307\/2272161","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:36:35Z","timestamp":1146936995000},"page":"389-392","source":"Crossref","is-referenced-by-count":1,"title":["Complete theories with countably many rigid nonisomorphic models"],"prefix":"10.1017","volume":"40","author":[{"given":"Jerome","family":"Malitz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200052981_ref001","first-page":"70","volume-title":"Theory of models","author":"Ehrenfeucht","year":"1965"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200052981","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T15:21:14Z","timestamp":1559143274000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200052981\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,9]]},"references-count":1,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1975,9]]}},"alternative-id":["S0022481200052981"],"URL":"https:\/\/doi.org\/10.2307\/2272161","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,9]]}}}