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Corollaries of the theorems answer some open questions of Friedman and Stanley in an L<jats:sub><jats:italic>\u03c9<\/jats:italic>1<jats:italic>\u03c9<\/jats:italic><\/jats:sub>-interpretability theory. We also give a survey of this theory.<\/jats:p>","DOI":"10.2307\/2695052","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:05:52Z","timestamp":1146938752000},"page":"902-922","source":"Crossref","is-referenced-by-count":6,"title":["Some dichotomy theorems for isomorphism relations of countable models"],"prefix":"10.1017","volume":"66","author":[{"given":"Su","family":"Gao","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200011087_ref016","first-page":"A390","article-title":"Vaught's conjecture for linear orderings","volume":"24","author":"Rubin","year":"1977","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200011087_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(74)90017-5"},{"key":"S0022481200011087_ref013","first-page":"301\u2013318","volume":"46","author":"Makkai","year":"1981","journal-title":"An example concerning Scott heights"},{"key":"S0022481200011087_ref012","volume-title":"Studies in logic and foundations of mathematics","volume":"62","author":"Keisler","year":"1971"},{"key":"S0022481200011087_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"S0022481200011087_ref010","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S0022481200011087_ref008","first-page":"1273\u20131300","volume":"60","author":"Hjorth","year":"1995","journal-title":"Analytic equivalence relations and Ulm-type classifications"},{"key":"S0022481200011087_ref004","first-page":"894\u2013914","volume":"54","author":"Friedman","year":"1989","journal-title":"A Borel reducibility theory for classes of countable structures"},{"key":"S0022481200011087_ref003","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511735264"},{"key":"S0022481200011087_ref002","unstructured":"Becker H. , Vaught's conjecture for complete left-invariant Polish groups, 1996, handwritten notes."},{"key":"S0022481200011087_ref001","volume-title":"Perspectives in mathematical logic","author":"Barwise","year":"1975"},{"key":"S0022481200011087_ref018","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0069301"},{"key":"S0022481200011087_ref005","unstructured":"Gao S. , The isomorphism relation between countable models and definable equivalence relations, Ph.D. thesis , UCLA, 1998."},{"key":"S0022481200011087_ref014","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S0022481200011087_ref017","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1994-1022169-2"},{"key":"S0022481200011087_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1990-1057041-5"},{"key":"S0022481200011087_ref006","doi-asserted-by":"crossref","first-page":"25\u201337","DOI":"10.4064\/fm-163-1-25-37","article-title":"A dichotomy theorem for mono-unary algebras","volume":"163","author":"Gao","year":"2000","journal-title":"Fundamenta Mathematicae"},{"key":"S0022481200011087_ref009","unstructured":"Hjorth G. and Solecki S. , Vaught's Conjecture and the Glimm-Effros property for Polish transformation groups, preprint, 1995."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200011087","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T20:12:28Z","timestamp":1627330348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200011087\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,6]]},"references-count":18,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2001,6]]}},"alternative-id":["S0022481200011087"],"URL":"https:\/\/doi.org\/10.2307\/2695052","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,6]]}}}