{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,11]],"date-time":"2025-07-11T10:41:01Z","timestamp":1752230461815},"reference-count":10,"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":6951,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1995,3]]},"abstract":"<jats:p>A key idea in both Frege's development of arithmetic in the<jats:bold><jats:italic>Grundlagen<\/jats:italic><\/jats:bold>[7] and Zermelo's 1904 proof [10] of the well-ordering theorem is that of a \u201ctype reducing\u201d correspondence between second-level and first-level entities. In Frege's construction, the correspondence obtains between<jats:italic>concept<\/jats:italic>and<jats:italic>number<\/jats:italic>, in Zermelo's (through the axiom of choice), between<jats:italic>set<\/jats:italic>and<jats:italic>member<\/jats:italic>. In this paper, a formulation is given and a detailed investigation undertaken of a system \u2131 of many-sorted first-order logic (first outlined in the Appendix to [6]) in which this notion of type reducing correspondence is accorded a central role and which enables Frege's and Zermelo's constructions to be presented in such a way as to reveal their essential similarity. By adapting Bourbaki's version of Zermelo's proof of the well-ordering theorem, we show that, within \u2131, any correspondence<jats:italic>c<\/jats:italic>between second-level entities (here called<jats:italic>concepts<\/jats:italic>) and first-level ones (here called<jats:italic>objects<\/jats:italic>) induces a well-ordering relation<jats:italic>W<\/jats:italic>(<jats:italic>c<\/jats:italic>) in a canonical manner. We shall see that, when<jats:italic>c<\/jats:italic>is the \u201cFregean\u201d correspondence between concepts and cardinal numbers,<jats:italic>W<\/jats:italic>(<jats:italic>c<\/jats:italic>) is (the well-ordering of) the ordinal<jats:italic>\u03c9<\/jats:italic>+ 1, and when<jats:italic>c<\/jats:italic>is a \u201cZermelian\u201d choice function on concepts,<jats:italic>W<\/jats:italic>(<jats:italic>c<\/jats:italic>) is a well-ordering of the universal concept embracing all objects.<\/jats:p><jats:p>In \u2131 an important role is played by the notion of<jats:italic>extension<\/jats:italic>of a concept. To each concept<jats:italic>X<\/jats:italic>we assume there is assigned an object<jats:italic>e<\/jats:italic>(<jats:italic>X<\/jats:italic>) in such a way that, for any concepts<jats:italic>X, Y<\/jats:italic>satisfying a certain predicate<jats:italic>E<\/jats:italic>, we have<jats:italic>e<\/jats:italic>(<jats:italic>X<\/jats:italic>) =<jats:italic>e<\/jats:italic>(<jats:italic>Y<\/jats:italic>) iff the same objects fall under<jats:italic>X<\/jats:italic>and<jats:italic>Y<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2275518","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:54:10Z","timestamp":1146956050000},"page":"209-221","source":"Crossref","is-referenced-by-count":7,"title":["Type reducing correspondences and well-orderings: Frege's and Zermelo's constructions re-examined"],"prefix":"10.1017","volume":"60","author":[{"given":"J. L.","family":"Bell","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200018983_ref009","volume-title":"Sets: an introduction","author":"Potter","year":"1990"},{"key":"S0022481200018983_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF01445300"},{"key":"S0022481200018983_ref006","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/1.2.139"},{"key":"S0022481200018983_ref004","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1093\/oso\/9780195079296.003.0003","volume-title":"Mathematics and mind","author":"Boolos","year":"1994"},{"key":"S0022481200018983_ref005","volume-title":"Th\u00e9orie des ensembles","author":"Bourbaki","year":"1963"},{"key":"S0022481200018983_ref008","volume-title":"Cantorian set theory and limitation of size","author":"Hallett","year":"1984"},{"key":"S0022481200018983_ref002","doi-asserted-by":"publisher","DOI":"10.1093\/aristotelian\/87.1.137"},{"key":"S0022481200018983_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01049178"},{"key":"S0022481200018983_ref003","first-page":"261","volume-title":"Meaning and method: essays in honor of Hilary Putnam","author":"Boolos","year":"1990"},{"key":"S0022481200018983_ref007","volume-title":"Die Grundlagen der Arithmetik","author":"Frege","year":"1884"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200018983","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T07:28:12Z","timestamp":1707031692000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200018983\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1995,3]]}},"alternative-id":["S0022481200018983"],"URL":"https:\/\/doi.org\/10.2307\/2275518","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,3]]}}}