{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T13:49:46Z","timestamp":1778593786726,"version":"3.51.4"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7589,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1993,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Frege's development of the theory of arithmetic in his<jats:italic><jats:bold>Grundgesetze der Arithmetik<\/jats:bold><\/jats:italic>has long been ignored, since the formal theory of the<jats:italic>Grundgesetze<\/jats:italic>is inconsistent. His derivations of the axioms of arithmetic from what is known as<jats:italic>Hume's Principle<\/jats:italic>do not, however, depend upon that axiom of the system\u2014Axiom V\u2014which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be \u201c<jats:italic>The<\/jats:italic>Basic Laws of Cardinal Number\u201d, as Frege understood them.<\/jats:p><jats:p>Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever.<jats:italic>Grundgesetze<\/jats:italic>is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establish may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, \u201cWhy did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?\u201d<\/jats:p>","DOI":"10.2307\/2275220","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:48:26Z","timestamp":1146955706000},"page":"579-601","source":"Crossref","is-referenced-by-count":41,"title":["The development of arithmetic in Frege's<i><b>Grundgesetze der arithmetik<\/b><\/i>"],"prefix":"10.1017","volume":"58","author":[{"suffix":"Jr.","given":"Richard G.","family":"Heck","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200021332_ref010","first-page":"180","volume-title":"Philosophy in America","author":"Parsons","year":"1965"},{"key":"S0022481200021332_ref011","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093636853"},{"key":"S0022481200021332_ref008","volume-title":"Philosophical and mathematical correspondence","author":"Frege","year":"1980"},{"key":"S0022481200021332_ref004","volume-title":"The Foundations of Arithmetic","author":"Frege","year":"1980"},{"key":"S0022481200021332_ref009","volume-title":"Foundational Problems in Frege and Modern Logic","author":"Heck"},{"key":"S0022481200021332_ref001","first-page":"3","volume-title":"On being and saying: essays in honor of Richard Cartwright","author":"Boolos","year":"1987"},{"key":"S0022481200021332_ref005","volume-title":"Grundgesetze der Arithmetik","author":"Frege","year":"1966"},{"key":"S0022481200021332_ref002","first-page":"261","volume-title":"Meaning and method","author":"Boolos","year":"1990"},{"key":"S0022481200021332_ref006","doi-asserted-by":"crossref","DOI":"10.1525\/9780520312364","volume-title":"The basic laws of arithmetic: exposition of the system","author":"Frege","year":"1964"},{"key":"S0022481200021332_ref003","volume-title":"Frege: philosophy of mathematics","author":"Dummett","year":"1992"},{"key":"S0022481200021332_ref007","volume-title":"Translations from the philosophical writings of Gottlob Frege","author":"Frege","year":"1970"},{"key":"S0022481200021332_ref012","volume-title":"Frege's conception of numbers as objects","author":"Wright","year":"1983"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200021332","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T20:28:40Z","timestamp":1627331320000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200021332\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,6]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1993,6]]}},"alternative-id":["S0022481200021332"],"URL":"https:\/\/doi.org\/10.2307\/2275220","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,6]]}}}