{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,21]],"date-time":"2024-07-21T02:48:56Z","timestamp":1721530136599},"reference-count":34,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":5799,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1998,3]]},"abstract":"<jats:p><jats:bold>\u00a71. Introduction<\/jats:bold>. In this report we wish to describe recent work on a class of first order theories first introduced by Shelah in [32], the <jats:italic>simple<\/jats:italic> theories. Major progress was made in the first author's doctoral thesis [17]. We will give a survey of this, as well as further works by the authors and others.<\/jats:p><jats:p>The class of simple theories includes stable theories, but also many more, such as the theory of the random graph. Moreover, many of the theories of particular algebraic structures which have been studied recently (pseudofinite fields, algebraically closed fields with a generic automorphism, smoothly approximable structures) turn out to be simple. The interest is basically that a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. Stable theories will be defined formally in the next section. An exhaustive study of them is carried out in [33]. Without trying to read Shelah's mind, we feel comfortable in saying that the importance of stability for Shelah lay partly in the fact that an unstable theory <jats:italic>T<\/jats:italic> has 2<jats:sup>\u03bb<\/jats:sup> many models in any cardinal \u03bb \u2265 \u03c9<jats:sub>1<\/jats:sub> + |<jats:italic>T<\/jats:italic>| (proved by Shelah). (Note that for \u03bb \u2265 |<jats:italic>T<\/jats:italic>| 2<jats:sup>\u03bb<\/jats:sup> is the maximum possible number of models of cardinality \u03bb.)<\/jats:p>","DOI":"10.2307\/421004","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T03:11:08Z","timestamp":1146971468000},"page":"17-36","source":"Crossref","is-referenced-by-count":15,"title":["From Stability to Simplicity"],"prefix":"10.1017","volume":"4","author":[{"given":"Byunghan","family":"Kim","sequence":"first","affiliation":[]},{"given":"Anand","family":"Pillay","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600007368_ref031","volume-title":"Groupes stables","author":"Poizat","year":"1987"},{"key":"S1079898600007368_ref028","article-title":"Definability and definable groups in simple theories","author":"Pillay","journal-title":"The Journal of Symbolic Logic"},{"key":"S1079898600007368_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF02758643"},{"key":"S1079898600007368_ref011","unstructured":"Hrushovski E. , Pseudofinite fields and related structures, preprint, 1992."},{"key":"S1079898600007368_ref025","unstructured":"Lascar D. and Pillay A. , Hyperimaginaries and automorphism groups, in preparation."},{"key":"S1079898600007368_ref032","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(80)90009-1"},{"key":"S1079898600007368_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(85)90023-5"},{"key":"S1079898600007368_ref029","doi-asserted-by":"publisher","DOI":"10.2307\/2274390"},{"key":"S1079898600007368_ref013","unstructured":"Hrushovski E. , The Manin-Mumford conjecture and difference fields, preprint, 1996."},{"key":"S1079898600007368_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090164"},{"key":"S1079898600007368_ref023","doi-asserted-by":"publisher","DOI":"10.2307\/2273140"},{"key":"S1079898600007368_ref021","unstructured":"Kim B. and Pillay A. , Simple theories, Annals of Pure and Applied Logic (to appear)."},{"key":"S1079898600007368_ref022","first-page":"45","volume-title":"Logic, methodology, philosophy of science","volume":"IV","author":"Lachlan","year":"1973"},{"key":"S1079898600007368_ref016","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-59.3.439"},{"key":"S1079898600007368_ref034","unstructured":"Wagner F. O. , Groups in simple theories, preprint, 1997."},{"key":"S1079898600007368_ref003","unstructured":"Casanovas E. , The number of types in simple theories, preprint, 1997."},{"key":"S1079898600007368_ref033","volume-title":"Classification theory and the number of non-isomorphic models","author":"Shelah","year":"1990"},{"key":"S1079898600007368_ref020","article-title":"On the number of countable models of a countable supersimple theory","author":"Kim","journal-title":"Journal of the London Mathematical Society"},{"key":"S1079898600007368_ref005","volume-title":"Annals of Pure and Applied Logic","author":"Chatzidakis"},{"key":"S1079898600007368_ref002","unstructured":"BuechLer S. , Lascar strong types in simple theories, preprint, 1997."},{"key":"S1079898600007368_ref006","first-page":"107","article-title":"Definable sets over finite fields","volume":"427","author":"Chatzidakis","year":"1992","journal-title":"Journal f\u00fcr die Reine und Angewandte Mathematik"},{"key":"S1079898600007368_ref007","volume-title":"Proceedings of 1996 NATO ASI meeting at Fields Institute","author":"Cherlin"},{"key":"S1079898600007368_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-2080-7_12"},{"key":"S1079898600007368_ref030","doi-asserted-by":"publisher","DOI":"10.2307\/2275848"},{"key":"S1079898600007368_ref017","unstructured":"Kim B. , Simple first order theories, Ph.D. thesis , University of Notre Dame, 1996."},{"key":"S1079898600007368_ref027","volume-title":"An introduction to stability theory","author":"Pillay","year":"1983"},{"key":"S1079898600007368_ref018","article-title":"Forking in simple unstable theories","author":"Kim","journal-title":"Journal of the London Mathematical Society"},{"key":"S1079898600007368_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/1970573"},{"key":"S1079898600007368_ref010","unstructured":"Hart B. , Kim B. , and Pillay A. , Coordinatization and canonical bases in simple theories, in preparation."},{"key":"S1079898600007368_ref015","first-page":"69","article-title":"Definable subgroups of algebraic groups over finite fields","volume":"462","author":"Hrushovski","year":"1995","journal-title":"Journal f\u00fcr die Reine und Angewandte Mathematik"},{"key":"S1079898600007368_ref004","unstructured":"Chatzidakis Z. and Hrushovski E. , Model theory of difference fields, preprint, 1996."},{"key":"S1079898600007368_ref019","author":"Kim","journal-title":"The Journal of Symbolic Logic"},{"key":"S1079898600007368_ref026","doi-asserted-by":"publisher","DOI":"10.2307\/2273127"},{"key":"S1079898600007368_ref024","first-page":"313","volume-title":"Logic: from foundations to applications","author":"Lascar","year":"1993"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600007368","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,10]],"date-time":"2019-05-10T17:12:14Z","timestamp":1557508334000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600007368\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,3]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1998,3]]}},"alternative-id":["S1079898600007368"],"URL":"https:\/\/doi.org\/10.2307\/421004","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,3]]}}}