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Cantor on Frege's Foundations of Arithmetic : Cantor's 1885 Review of Frege's Die Grundlagen der Arithmetik

2009; Taylor & Francis; Volume: 30; Issue: 4 Linguagem: Inglês

10.1080/01445340903102813

1464-5149

Philip A. Ebert, Marcus Rossberg,

Computability, Logic, AI Algorithms

Abstract In 1885, Georg Cantor published his review of Gottlob Frege's Grundlagen der Arithmetik. In this essay, we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantor's review, and a new translation of Frege's brief response to Cantor. In recent years, it has become philosophical folklore that Cantor's 1885 review of Frege's Grundlagen already contained a warning to Frege. This warning is said to concern the defectiveness of Frege's notion of extension. The exact scope of such speculations varies and sometimes extends as far as crediting Cantor with an early hunch of the paradoxical nature of Frege's notion of extension. William Tait goes even further and deems Frege ‘reckless’ for having missed Cantor's explicit warning regarding the notion of extension. As such, Cantor's purported inkling would have predated the discovery of the Russell–Zermelo paradox by almost two decades. In our introductory essay, we discuss this alleged implicit (or even explicit) warning, separating two issues: first, whether the most natural reading of Cantor's criticism provides an indication that the notion of extension is defective; second, whether there are other ways of understanding Cantor that support such an interpretation and can serve as a precisification of Cantor's presumed warning. Acknowledgements We thank Mike Beaney, Peter Milne, Stephen Read, Crispin Wright and an anonymous referee for their helpful comments on earlier drafts of the translations and the introductory essay. Notes 1 See Cantor 1885 and Frege 1884. 2 See Frege 1885; an earlier translation of Frege's response by Hans Kaal is published in McGuinness 1984, 122. 3 Compare Thiel 1986, LII. 4 Compare Frege 1884, 80 fn, and Frege's remark in the last sentence of his reply to Cantor. 5 See Rang and Thomas 1981. 6 Russell's characterisation of the paradox in his 1902 letter is, in fact, not well formed in Frege's system. However, Frege himself, in his 1902 reply to Russell, 213, provides the proper reformulation of the paradox. 7 In the translations we use ‘number’ as a translation of both ‘Zahl’ and ‘Anzahl’, providing the German term parenthetically in the text. ‘Anzahl’ should be translated as ‘cardinal number’ in Frege; Cantor, however, understands ‘Anzahl’ as ordinal number. Later parts of his criticisms are based on just this confusion. In order not to render this confusion even more perplexing, we opted for the neutral term ‘number’ in the translation. In our discussion here, we use ‘cardinal number’ instead for the sake of precision. 8 This is in sharp contrast to Tait's claim that Cantor explicitly warns against Frege's assumption that every concept has an extension. There is simply no explicit warning of this kind. Later we discuss a reading of Cantor that could be regarded as an implicit warning to this effect. 9 See Boolos 1987, 1989 and also Cook 2003. 10 Published in German in Deutsche Litteraturzeitung, VI, no. 20, 16 May 1885, columns 728–729. Reprinted in Cantor 1932, 440–441, and in Thiel 1986, 117–119. Zermelo's edition, Cantor 1932, adds various emphases. We here follow the original publication. 11 See Ueberweg 1857/51882. [Translators' note.] 12 See footnote 7. 13 See Cantor 1883. [Translators' note.] 14 In his edition of Cantor's Gesammelte Abhandlungen, 1932, 141–142. Reprinted in Frege 1986, 119. 15 Gottlob Frege: ‘Erwiderung auf Georg Cantors Rezension der Grundlagen’, in: Deutsche Litteraturzeitung, VI, no. 28, 11 July 1885, column 1030. Reprinted in Frege 1986, 120.