{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T04:13:43Z","timestamp":1649132023278},"reference-count":29,"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>The study of countable theories categorical in some uncountable power was initiated by \u0141o\u015b and Vaught and developed in two stages. First, Morley proved (1962) that a countable theory categorical in some uncountable power is categorical in every uncountable power, a conjecture of \u0141o\u015b. Second, Baldwin and Lachlan confirmed (1969) Vaught's conjecture that a countable theory categorical in some uncountable power has either one or countably many isomorphism types of countable models. That result was obtained by pursuing a line of research developed by Marsh (1966). For certain well-behaved theories, which he called <jats:italic>strongly minimal<\/jats:italic>, Marsh's method yielded a simple proof of \u0141o\u015b's conjecture and settled Vaught's conjecture.<\/jats:p><jats:p>In recent years efforts have been made to extend these results to uncountable theories. The generalized \u0141o\u015b conjecture states that a theory <jats:italic>T<\/jats:italic> categorical in some power greater than \u2223<jats:italic>T<\/jats:italic>\u2223 is categorical in every such power. It was settled by Shelah (1970). Shelah then raised the question of the models in power \u2223<jats:italic>T<\/jats:italic>\u2223 = \u2135<jats:sub>\u03b1<\/jats:sub> of a theory <jats:italic>T<\/jats:italic> categorical in \u2223<jats:italic>T<\/jats:italic>\u2223<jats:sup>+<\/jats:sup>, conjecturing in [S3] that there are exactly \u2223\u03b1\u2223 + \u2135<jats:sub>0<\/jats:sub> such models, up to isomorphism. This conjecture provided the initial motivation for the present work. We define and study <jats:italic>semi-minimal<\/jats:italic> theories analogous in some ways to Marsh's strongly minimal (countable) theories. We describe the models of a semi-minimal theory <jats:italic>T<\/jats:italic> which contain an infinite indiscernible set. Besides throwing some light on Shelah's conjecture, our method gives simple proofs of the \u0141o\u015b conjecture and of the Morley conjecture on categoricity in \u2223<jats:italic>T<\/jats:italic>\u2223, in the case of a semi-minimal theory <jats:italic>T<\/jats:italic>. Other results as well as some examples are provided.<\/jats:p>","DOI":"10.2307\/2272166","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:36:35Z","timestamp":1146936995000},"page":"419-438","source":"Crossref","is-referenced-by-count":2,"title":["Semi-minimal theories and categoricity"],"prefix":"10.1017","volume":"40","author":[{"given":"Daniel","family":"Andler","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200053032_ref006","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200053032_ref013","doi-asserted-by":"publisher","DOI":"10.4064\/cm-3-1-58-62"},{"key":"S0022481200053032_ref024","first-page":"272","article-title":"Stability, the f.c.p. and superstability","volume":"3","author":"Shelah","year":"1971","journal-title":"Annals of Mathematical Logic"},{"key":"S0022481200053032_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771568"},{"key":"S0022481200053032_ref022","first-page":"73","volume":"35","author":"Shelah","year":"1970","journal-title":"On theories T categorical in \u2223T\u2223"},{"key":"S0022481200053032_ref003","unstructured":"Baldwin J. T. , Countable theories categorical in power, Doctoral Dissertation, Simon Fraser University, 1970."},{"key":"S0022481200053032_ref011","first-page":"240","volume":"36","author":"Keisler","year":"1971","journal-title":"On theories categorical in their own power"},{"key":"S0022481200053032_ref018","doi-asserted-by":"publisher","DOI":"10.1007\/BF02771623"},{"key":"S0022481200053032_ref004","first-page":"487","volume":"37","author":"Baldwin","year":"1972","journal-title":"Almost strongly minimal theories"},{"key":"S0022481200053032_ref021","volume-title":"Saturated model theory","author":"Sacks","year":"1972"},{"key":"S0022481200053032_ref001","volume":"19","author":"Andler","year":"1972","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200053032_ref002","unstructured":"Andler D. M. , Models of uncountable theories categorical in power, Doctoral Dissertation, University of California, Berkeley, 1973."},{"key":"S0022481200053032_ref028","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(54)50058-2"},{"key":"S0022481200053032_ref007","doi-asserted-by":"publisher","DOI":"10.4064\/fm-43-1-50-68"},{"key":"S0022481200053032_ref016","unstructured":"Marsh W. E. , On \u03c91-categorical but not \u03c9-categorical theories, Doctoral Dissertation, Dartmouth College, 1966."},{"key":"S0022481200053032_ref008","unstructured":"Goldhaber and Ehrlich , Algebra , 1970."},{"key":"S0022481200053032_ref009","unstructured":"Harnik V. , Stability and related concepts, Doctoral Dissertation, Hebrew University, 1971."},{"key":"S0022481200053032_ref015","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19690150705"},{"key":"S0022481200053032_ref014","first-page":"257","volume-title":"Infinitistic Methods","author":"MacDowell","year":"1961"},{"key":"S0022481200053032_ref019","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10648"},{"key":"S0022481200053032_ref020","unstructured":"Park D. M. R. , Set-theoretic construction in model theory, Doctoral Dissertation, M.I.T., 1964."},{"key":"S0022481200053032_ref023","volume-title":"Proof of the Los conjecture for uncountable theories","author":"Shelah","year":"1970"},{"key":"S0022481200053032_ref012","volume-title":"Proceedings of the IVth International Congress on Logic, Methodology and the Philosophy of Science","author":"Lachlan","year":"1971"},{"key":"S0022481200053032_ref025","volume-title":"Proceedings of Symposia in Pure Mathematics","author":"Shelah","year":"1974"},{"key":"S0022481200053032_ref026","volume-title":"Mathematical logic","author":"Shoenfield","year":"1967"},{"key":"S0022481200053032_ref027","first-page":"81","article-title":"Arithmetical extensions of relational systems","volume":"13","author":"Tarski","year":"1957","journal-title":"Composite Mathematica"},{"key":"S0022481200053032_ref029","first-page":"303","volume-title":"Infinitistic methods","author":"Vaught","year":"1961"},{"key":"S0022481200053032_ref017","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1965-0175782-0"},{"key":"S0022481200053032_ref005","first-page":"79","volume":"36","author":"Baldwin","year":"1971","journal-title":"On strongly minimal sets"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200053032","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T15:21:37Z","timestamp":1559143297000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200053032\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,9]]},"references-count":29,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1975,9]]}},"alternative-id":["S0022481200053032"],"URL":"https:\/\/doi.org\/10.2307\/2272166","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,9]]}}}