{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T20:16:29Z","timestamp":1775506589783,"version":"3.50.1"},"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":11424,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,12]]},"abstract":"<jats:p>A set <jats:italic>J<\/jats:italic> of Turing degrees is called an <jats:italic>ideal<\/jats:italic> if (1) <jats:italic>J<\/jats:italic> \u2260 \u2205, (2) for any pair of degrees \u00e3, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline01\"\/>, if \u00e3, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline01\"\/> \u03f5 <jats:italic>J<\/jats:italic>, then \u00e3 \u22c3 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline01\"\/> \u03f5<jats:italic>J<\/jats:italic>, and (3) for any \u22c3 \u03f5 <jats:italic>J<\/jats:italic> and any <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline01\"\/>, if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline01\"\/> &lt; \u22c3, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline01\"\/> \u03f5 <jats:italic>J<\/jats:italic>. A set <jats:italic>J<\/jats:italic> of degrees is said to be <jats:italic>closed<\/jats:italic> if for any theory <jats:italic>T<\/jats:italic> with a set of axioms of degree in <jats:italic>J, T<\/jats:italic> has a completion of degree in <jats:italic>J<\/jats:italic>.<\/jats:p><jats:p>Closed ideals of degrees arise naturally in the following way. If <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/> is a recursively saturated structure, let <jats:italic>I<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/>) = {<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline02\"\/> for some \u0101 \u03f5 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/>}. Let <jats:italic>D<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/>) = {<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline03\"\/>: <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/> is recursive in <jats:italic>d<\/jats:italic>-saturated}. (Recursive in <jats:italic>d<\/jats:italic>-saturation is defined like recursive saturation except that the sets of formulas considered are recursive in <jats:italic>d<\/jats:italic>.) These two sets of degrees were investigated in [2]. It was shown that if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/> is a recursively saturated model of <jats:italic>P<\/jats:italic>, Pr = Th(\u03c9, +), or Pr\u2032 = Th(<jats:italic>Z<\/jats:italic>, +, 1), then <jats:italic>I<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/>) = <jats:italic>D<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/>), and this set is a closed ideal. Any closed ideal <jats:italic>J<\/jats:italic> can be represented as <jats:italic>I<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/>) = <jats:italic>D<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/>) for some recursively saturated model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/> of Pr\u2032. For sets <jats:italic>J<\/jats:italic> of power at most \u2135<jats:sub>1<\/jats:sub>, Pr\u2032 can be replaced by <jats:italic>P<\/jats:italic>.<\/jats:p><jats:p>Assuming CH, all closed ideals have power at most \u2135<jats:sub>1<\/jats:sub>, but if CH fails, there are closed ideals of power greater than \u2135<jats:sub>1<\/jats:sub>, and it is not known whether these can be represented as <jats:italic>I<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/>) = <jats:italic>D<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/>) for a recursively saturated model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004370X_inline04\"\/> of <jats:italic>P<\/jats:italic>.<\/jats:p><jats:p>In the present paper, it will first be shown that information about representation of closed ideals provides new information about an old problem of MacDowell and Specker [6] and extends an old result of Scott [8] in a natural way. It will also be shown that the representation results from [2] answer a problem of Friedman [1]. This part of the paper is aimed at convincing the reader that representation problems are worth investigating.<\/jats:p>","DOI":"10.2307\/2273102","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:01:43Z","timestamp":1146938503000},"page":"833-840","source":"Crossref","is-referenced-by-count":8,"title":["Models of arithmetic and closed ideals"],"prefix":"10.1017","volume":"47","author":[{"given":"Julia","family":"Knight","sequence":"first","affiliation":[]},{"given":"Mark","family":"Nadel","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120004370X_ref007","first-page":"612","volume":"45","author":"Nadel","year":"1980","journal-title":"On a problem of MacDowell and Specker"},{"key":"S002248120004370X_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1978-0491158-5"},{"key":"S002248120004370X_ref004","unstructured":"Knight J. , Additive structure in uncountable models for a fixed completion of P (preprint)."},{"key":"S002248120004370X_ref006","first-page":"257","volume-title":"Infinitistic methods","author":"MacDowell","year":"1961"},{"key":"S002248120004370X_ref008","first-page":"117","volume-title":"Proceedings of Symposia in Pure Mathematics","volume":"5","author":"Scott","year":"1962"},{"key":"S002248120004370X_ref003","unstructured":"Knight J. , Amalgamation of recursively saturated structures (preprint)."},{"key":"S002248120004370X_ref001","first-page":"539","volume-title":"Lecture Notes in Mathematics","volume":"337","author":"Friedman","year":"1973"},{"key":"S002248120004370X_ref002","first-page":"587","volume":"47","author":"Knight","year":"1982","journal-title":"Expansions of models and Turing degrees"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120004370X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T16:18:14Z","timestamp":1558714694000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120004370X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,12]]},"references-count":8,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1982,12]]}},"alternative-id":["S002248120004370X"],"URL":"https:\/\/doi.org\/10.2307\/2273102","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,12]]}}}