{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T02:50:36Z","timestamp":1648867836932},"reference-count":7,"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":10054,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,9]]},"abstract":"<jats:p>By a \u201cpartly numerical structure\u201d (p.n.s.) we shall here mean a quadruple <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline1\" \/>, where <jats:italic>M<\/jats:italic> is a set, <jats:italic>\u03c9<\/jats:italic> = the natural numbers, <jats:italic>\u03c9<\/jats:italic> \u2286 <jats:italic>M<\/jats:italic>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline2\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline3\" \/> are disjoint sets, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline2\" \/> is a set of <jats:italic>relations<\/jats:italic> (of various positive integral arities) <jats:italic>on M<\/jats:italic>, and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline3\" \/> is a set of <jats:italic>functions<\/jats:italic> (of various positive integral arities) <jats:italic>with arguments and values in M<\/jats:italic>. Thus, in calculated disharmony with common practice, we do not (except as noted below, in connection with naming the elements of <jats:italic>\u03c9<\/jats:italic>) fix a similarity type as part of our notion of a \u201cstructure\u201d. Suppose a finitary first-order language <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline4\" \/> (with identity) has been specified, with constant symbols <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline5\" \/>, <jats:italic>n<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic>, and with exactly enough relation and function symbols of each arity to enable us to interpret <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline4\" \/> in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline6\" \/>. We wish to consider the variation in the <jats:italic>degree<\/jats:italic> (relative to a fixed G\u00f6del-numbering of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline4\" \/>) of the complete <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline4\" \/>-theory of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline6\" \/> as we vary the way in which elements of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline2\" \/> \u222a <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline3\" \/> are assigned as interpretations to the relation and function symbols of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline4\" \/>. We shall in fact, therefore, be concerned exclusively with p.n.s.'s <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline6\" \/> for which <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline2\" \/> \u222a <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline3\" \/> is countable. More: we assume <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline6\" \/> to be such that we can effectively tell, uniformly in <jats:italic>n<\/jats:italic> &gt; 0, exactly how many <jats:italic>n<\/jats:italic>-ary relations <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline2\" \/> has and exactly how many <jats:italic>n<\/jats:italic>-ary functions <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030905_inline3\" \/> has.<\/jats:p>","DOI":"10.2307\/2274027","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:18:32Z","timestamp":1146939512000},"page":"732-747","source":"Crossref","is-referenced-by-count":0,"title":["Some elementary degree-theoretic reasons why structures need similarity types"],"prefix":"10.1017","volume":"51","author":[{"given":"T. G.","family":"McLaughlin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200030905_ref007","volume-title":"Degrees of unsolvability","author":"Shoenfield","year":"1971"},{"key":"S0022481200030905_ref006","volume-title":"Degrees of unsolvability","author":"Sacks","year":"1963"},{"key":"S0022481200030905_ref005","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200030905_ref004","first-page":"723","volume":"46","author":"Richter","year":"1981","journal-title":"Degrees of structures"},{"key":"S0022481200030905_ref001","volume-title":"Models and ultraproducts: an introduction","author":"Bell","year":"1971"},{"key":"S0022481200030905_ref003","volume-title":"A mathematical introduction to logic","author":"Enderton","year":"1972"},{"key":"S0022481200030905_ref002","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093890996"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200030905","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T00:41:06Z","timestamp":1558485666000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200030905\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1986,9]]}},"alternative-id":["S0022481200030905"],"URL":"https:\/\/doi.org\/10.2307\/2274027","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,9]]}}}