{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T17:21:39Z","timestamp":1709832099036},"reference-count":4,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11334,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1983,3]]},"abstract":"<jats:p><jats:bold>Notations, conventions, and definitions.<\/jats:bold> {<jats:italic>\u03bc<jats:sub>i<\/jats:sub><\/jats:italic>\u2223<jats:italic>i<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic>} will be an effective enumeration of all partial recursive <jats:italic>\u03bc<jats:sub>i<\/jats:sub><\/jats:italic>{<jats:italic>\u03c9<\/jats:italic> \u2192 2. A type of a theory <jats:italic>T<\/jats:italic> will be a set of formulas in the language of <jats:italic>T<\/jats:italic>, in finitely many free variables, which is consistent with <jats:italic>T<\/jats:italic>. A complete type is a maximal type in some fixed number of free variables. A type is recursive if, relative to some effective enumeration of the formulas of the language, the characteristic function for the type is recursive. A set <jats:italic>\u03c8<\/jats:italic> of recursive types has property P if some set of indices of characteristic functions for all the types in <jats:italic>\u03c8<\/jats:italic> has property P. So, for example, we might say that a set of recursive types is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200043450_inline1\" \/>. If <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200043450_inline2\" \/> is an <jats:italic>L<\/jats:italic>-structure, then the type spectrum of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200043450_inline2\" \/>, denoted \u2018TySp(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200043450_inline2\" \/>)\u2019, is the set of complete types realized in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200043450_inline2\" \/> (we will assume that an <jats:italic>n<\/jats:italic>-type has formulas with free variables among {<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic>}). A type spectrum for a theory <jats:italic>T<\/jats:italic> is a type spectrum of some model of <jats:italic>T<\/jats:italic>. \u2018TySp<jats:sub>0<\/jats:sub>(<jats:italic>T<\/jats:italic>)\u2019 will denote the set of principal types of <jats:italic>T<\/jats:italic>.<\/jats:p><jats:p>We will assume that the reader is familiar with Henkin constructions of models, and of passing from a maximal consistent set of sentences, with \u201cHenkin constants\u201d, to a model. In particular, for a theory <jats:italic>T<\/jats:italic> in <jats:italic>L<\/jats:italic> we will let {<jats:italic>a<jats:sub>i<\/jats:sub><\/jats:italic>\u2223<jats:italic>i<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic>} be new distinct constant symbols, and {<jats:italic>\u03c6<jats:sub>i<\/jats:sub><\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic>} a list of all sentences in the expanded language. \u2018<jats:italic>\u0394<jats:sub>N<\/jats:sub><\/jats:italic>\u2019 will denote the elementary diagram constructed at stage <jats:italic>N<\/jats:italic>, and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200043450_inline3\" \/>.<\/jats:p>","DOI":"10.2307\/2273331","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:02:24Z","timestamp":1146952944000},"page":"171-181","source":"Crossref","is-referenced-by-count":14,"title":["Omitting types, type spectrums, and decidability"],"prefix":"10.1017","volume":"48","author":[{"given":"Terrence","family":"Millar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200043450_ref001","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200043450_ref002","unstructured":"Millar T. S. , Type structure complexity and decidability (to appear)."},{"key":"S0022481200043450_ref003","unstructured":"Millar T. S. , Counter-examples via model completions (to appear)."},{"key":"S0022481200043450_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90030-X"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200043450","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T19:36:10Z","timestamp":1558726570000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200043450\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,3]]},"references-count":4,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1983,3]]}},"alternative-id":["S0022481200043450"],"URL":"https:\/\/doi.org\/10.2307\/2273331","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,3]]}}}