{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T13:23:29Z","timestamp":1698499409715},"reference-count":4,"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":12337,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1980,6]]},"abstract":"<jats:p>We prove first that if <jats:italic>T<\/jats:italic> is a countable complete theory with <jats:italic>n(T)<\/jats:italic>, the number of countable models of <jats:italic>T<\/jats:italic>, equal to three, then <jats:italic>T<\/jats:italic> is similar to the Ehrenfeucht example of such a theory. Woodrow [4] showed that if <jats:italic>T<\/jats:italic> is in the same language as the Ehrenfeucht example, <jats:italic>T<\/jats:italic> has elimination of quantifiers, and <jats:italic>n(T)<\/jats:italic> = 3 then <jats:italic>T<\/jats:italic> is very much like this example. All known examples of theories <jats:italic>T<\/jats:italic> with <jats:italic>n(T)<\/jats:italic> finite and greater than one are based on the Ehrenfeucht example. We feel that such theories are a pathological case. Our second theorem strengthens the main result of [2]. The theorem in the present paper says that if <jats:italic>T<\/jats:italic> is a countable theory which has a model in which all the elements of some infinite definable set are algebraic of uniformly bounded degree, then <jats:italic>n(T)<\/jats:italic> \u2265 4. It is known [3] that if <jats:italic>n(T)<\/jats:italic> &gt; 1, then <jats:italic>n(T)<\/jats:italic> &gt; 3, so our result is the first nontrivial step towards proving that <jats:italic>n(T)<\/jats:italic> \u2265 \u2135<jats:sub>0<\/jats:sub>. We would also like, of course, to prove the result without the uniform bound on the finite degrees of the elements in the subset.<\/jats:p><jats:p>Theorem 2.1 is included in the author's Ph. D. thesis, as is a weaker version of Theorem 3.7. Thanks are due to Harry Simmons for his suggestions concerning the presentation of the material, and to Wilfrid Hodges for his advice while I was a Ph. D. student.<\/jats:p>","DOI":"10.2307\/2273190","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:52:48Z","timestamp":1146937968000},"page":"302-310","source":"Crossref","is-referenced-by-count":4,"title":["Theories with exactly three countable models and theories with algebraic prime models"],"prefix":"10.1017","volume":"45","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200046818_ref004","first-page":"672","volume":"41","author":"Woodrow","year":"1976","journal-title":"A note on complete theories having three isomorphism types of countable models"},{"key":"S0022481200046818_ref002","first-page":"492","volume":"43","author":"Pillay","year":"1978","journal-title":"Number of countable models"},{"key":"S0022481200046818_ref001","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200046818_ref003","volume-title":"Infinitistic methods","author":"Vaught","year":"1961"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200046818","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T15:12:40Z","timestamp":1558883560000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200046818\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,6]]},"references-count":4,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1980,6]]}},"alternative-id":["S0022481200046818"],"URL":"https:\/\/doi.org\/10.2307\/2273190","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,6]]}}}