{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,19]],"date-time":"2024-03-19T10:23:28Z","timestamp":1710843808624},"reference-count":2,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":6711,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1995,9]]},"abstract":"<jats:p>Two thoughts about the concept of number are incompatible: that any zero or more things have a (cardinal) number, and that any zero or more things have a number (if and) only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any (zero or more) things have a number is Frege's; the thought that things have a number only if they are the members of a set may be Cantor's and is in any case a commonplace of the usual contemporary presentations of the set theory that originated with Cantor and has become ZFC.<\/jats:p><jats:p>In recent years a number of authors have examined Frege's accounts of arithmetic with a view to extracting an interesting subtheory from Frege's formal system, whose inconsistency, as is well known, was demonstrated by Russell. These accounts are contained in Frege's formal treatise <jats:italic>Grundgesetze der Arithmetik<\/jats:italic> and his earlier exoteric book <jats:italic>Die Grundlagen der Arithmetik<\/jats:italic>. We may describe the two central results of the recent re-evaluation of his work in the following way: Let <jats:italic>Frege arithmetic<\/jats:italic> be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the <jats:italic>Fs<\/jats:italic> and the <jats:italic>Gs<\/jats:italic> have the same number if and only if the <jats:italic>F<\/jats:italic> sand the <jats:italic>Gs<\/jats:italic> are equinumerous.<\/jats:p>","DOI":"10.2307\/421158","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:08:06Z","timestamp":1146985686000},"page":"317-326","source":"Crossref","is-referenced-by-count":10,"title":["Frege's Theorem and the Peano Postulates"],"prefix":"10.1017","volume":"1","author":[{"given":"George","family":"Boolos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600008118_ref001","volume-title":"Frege's Philosophy of Mathematics","author":"Demopoulos","year":"1995"},{"key":"S1079898600008118_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/2308975"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600008118","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T21:47:25Z","timestamp":1557784045000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600008118\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,9]]},"references-count":2,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1995,6]]}},"alternative-id":["S1079898600008118"],"URL":"https:\/\/doi.org\/10.2307\/421158","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,9]]}}}