{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,30]],"date-time":"2023-08-30T06:27:19Z","timestamp":1693376839793},"reference-count":8,"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>In [4] it is shown that if the structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052944_inline1\" \/> omits a type \u03a3, and \u03a3 is complete with respect to Th(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052944_inline2\" \/>), then there is a proper elementary extension of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052944_inline2\" \/> which omits \u03a3. This result is extended in the present paper. It is shown that Th(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052944_inline2\" \/>) has models omitting \u03a3 in all infinite powers.<\/jats:p><jats:p>A <jats:italic>type<\/jats:italic> is a countable set of formulas with just the variable <jats:italic>\u03bd<\/jats:italic> occurring free. A structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052944_inline3\" \/> is said to <jats:italic>omit<\/jats:italic> the type \u03a3 if no element of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052944_inline3\" \/> satisfies all of the formulas of \u03a3. A type \u03a3, in the same language as a theory <jats:italic>T<\/jats:italic>, is said to be <jats:italic>complete<\/jats:italic> with respect to <jats:italic>T<\/jats:italic> if (1) <jats:italic>T<\/jats:italic> \u222a \u03a3 is consistent, and (2) for every formula <jats:italic>\u03c6<\/jats:italic>(<jats:italic>\u03bd<\/jats:italic>) of the language of <jats:italic>T<\/jats:italic> (with just <jats:italic>\u03bd<\/jats:italic> free), either <jats:italic>\u03c6<\/jats:italic> or \u00ac<jats:italic>\u03c6<\/jats:italic> is in \u03a3.<\/jats:p><jats:p>The proof of the result of this paper resembles Morley's proof [5] that the Hanf number for omitting types is <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052944_inline4\" \/>. It is shown that there is a model of Th(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052944_inline2\" \/>) which omits \u03a3 and contains an infinite set of indiscernibles. Where Morley used the Erd\u00f6s-Rado generalization of Ramsey's theorem, a definable version of the ordinary Ramsey's theorem is used here.<\/jats:p><jats:p>The \u201comitting types\u201d version of the \u03c9-completeness theorem ([1], [3], [6]) is used, as it was in Morley's proof and in [4]. In [4], satisfaction of the hypotheses of the \u03c9-completeness theorem followed from the fact that, in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052944_inline2\" \/>, any infinite, definable set can be split into two infinite, definable sets.<\/jats:p>","DOI":"10.2307\/2272157","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:36:35Z","timestamp":1146936995000},"page":"317-320","source":"Crossref","is-referenced-by-count":3,"title":["Types omitted in uncountable models of arithmetic"],"prefix":"10.1017","volume":"40","author":[{"given":"Julia F.","family":"Knight","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200052944_ref008","first-page":"443","volume-title":"Proceedings of the 1969 Logic Colloquium","author":"Yates","year":"1971"},{"key":"S0022481200052944_ref005","first-page":"265","volume-title":"The theory of models, Proceedings of the 1963 Symposium at Berkeley","author":"Morley"},{"key":"S0022481200052944_ref006","first-page":"246","volume":"21","author":"Orey","year":"1956","journal-title":"On \u03c6-consistency and related properties"},{"key":"S0022481200052944_ref003","volume-title":"Model theory for infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200052944_ref001","first-page":"183","volume":"19","author":"Henkin","year":"1954","journal-title":"A generalization of the concept of \u03c9-consistency"},{"key":"S0022481200052944_ref007","first-page":"439","volume-title":"Proceedings of the 1969 Logic Colloquium","author":"Specker","year":"1971"},{"key":"S0022481200052944_ref004","first-page":"413","volume":"38","author":"Knight","year":"1973","journal-title":"Complete types and the natural numbers"},{"key":"S0022481200052944_ref002","first-page":"268","volume":"37","author":"Jockusch","year":"1972","journal-title":"Ramsey's theorem and recursion theory"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200052944","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T15:22:01Z","timestamp":1559143321000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200052944\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,9]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1975,9]]}},"alternative-id":["S0022481200052944"],"URL":"https:\/\/doi.org\/10.2307\/2272157","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,9]]}}}