{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T12:26:31Z","timestamp":1655123191009},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10328,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1985,12]]},"abstract":"In this paper we prove that if T<\/jats:italic> is the complete elementary diagram of a countable structure and is a theory as in the title, then Vaught's conjecture holds for T<\/jats:italic>. This result is Theorem 7, below. In the process of establishing this proposition, in Theorem 3 we give a sufficient condition for a superstable theory having only countably many types without parameters to be \u03c9<\/jats:italic>-stable. Familiarity with the rudiments of stability theory, as presented in [3] and [4], will be supposed throughout. The notation used is, by now, standard.<\/jats:p>We begin by giving a new proof of a lemma due to J. Saffe in [6]. For T<\/jats:italic> stable, recall that the multiplicity<\/jats:italic> of a type p<\/jats:italic> over a set A<\/jats:italic> \u2286 \u2133 \u22a8 T<\/jats:italic> is the cardinality of the collection of strong types over A<\/jats:italic> extending p<\/jats:italic>.<\/jats:p>Lemma 1 (Saffe). Let T be stable<\/jats:italic>, A<\/jats:italic> \u2286 \u2133 \u22a8 T<\/jats:italic>. If<\/jats:italic> t(b\u0304, A<\/jats:italic>) has infinite multiplicity and<\/jats:italic> t(c\u0304, A<\/jats:italic>) has finite multiplicity, then<\/jats:italic> t(b\u0304, A<\/jats:italic> \u222a {c\u0304}) has infinite multiplicity<\/jats:italic>.<\/jats:p>Proof. We suppose not and work for a contradiction. Let \u2039b\u0304\u03b3<\/jats:italic><\/jats:sub>:\u03b3<\/jats:italic> \u2264 \u03b1<\/jats:italic>\u203a, \u03b1<\/jats:italic> \u2265 \u03c9, be a list of elements so that t(b\u0304\u03b3<\/jats:italic><\/jats:sub>, A<\/jats:italic>) = t(b\u0304, A<\/jats:italic>) for all \u03b3<\/jats:italic> \u2264 \u03b1<\/jats:italic>, and st(b\u0304\u03b3<\/jats:italic><\/jats:sub>, A<\/jats:italic>) \u2260 st(b\u0304\u03b4<\/jats:italic><\/jats:sub>, A<\/jats:italic>) for \u03b3<\/jats:italic> \u2260 \u03b4<\/jats:italic>. Furthermore, let c\u0304\u03b3<\/jats:italic><\/jats:sub> satisfy t(b\u0304\u03b3<\/jats:italic><\/jats:sub>\u2227c\u0304\u03b3<\/jats:italic><\/jats:sub>, A<\/jats:italic>) = t(b\u0304 \u2227 c\u0304, A<\/jats:italic>) for each \u03b3<\/jats:italic> < \u03b1<\/jats:italic>.<\/jats:p>Since t(c\u0304, A<\/jats:italic>) has finite multiplicity, we may assume for all \u03b3<\/jats:italic>, \u03b4<\/jats:italic> < \u03b1<\/jats:italic>. that st(c\u0304\u03b3<\/jats:italic><\/jats:sub>, A<\/jats:italic>) = st(c\u0304\u03b4<\/jats:italic><\/jats:sub>, A<\/jats:italic>). For each \u03b3<\/jats:italic> < \u03b1<\/jats:italic> there is an automorphism f<\/jats:italic>\u03b3<\/jats:italic><\/jats:sub> of the so-called \u201cmonster model\u201d of T<\/jats:italic> (a sufficiently large, saturated model of T<\/jats:italic>) that preserves strong types over A<\/jats:italic> and is such that f<\/jats:italic>(c\u0304\u03b3<\/jats:italic><\/jats:sub>) = c\u03040<\/jats:sub>.<\/jats:p>","DOI":"10.2307\/2273987","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:15:32Z","timestamp":1146939332000},"page":"1020-1024","source":"Crossref","is-referenced-by-count":3,"title":["A note on nonmultidimensional superstable theories"],"prefix":"10.1017","volume":"50","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]},{"given":"Charles","family":"Steinhorn","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200031984_ref007","volume-title":"Classification theory and the number of non-isomorphic models","author":"Shelah","year":"1978"},{"key":"S0022481200031984_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760649"},{"key":"S0022481200031984_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760651"},{"key":"S0022481200031984_ref004","volume-title":"Introduction to stability theory","author":"Pillay","year":"1983"},{"key":"S0022481200031984_ref006","unstructured":"Saffe J. , On Vaught's conjecture for superstable theories (to appear)."},{"key":"S0022481200031984_ref001","unstructured":"Baldwin J. , Introduction to stability theory (book in preparation)."},{"key":"S0022481200031984_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757234"},{"key":"S0022481200031984_ref005","volume-title":"Groupe d'\u00e9tude: th\u00e9ories stables. III (1980\/82)","author":"Pillay","year":"1983"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200031984","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T16:55:48Z","timestamp":1558544148000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200031984\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,12]]},"references-count":8,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1985,12]]}},"alternative-id":["S0022481200031984"],"URL":"https:\/\/doi.org\/10.2307\/2273987","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,12]]}}}