# The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

17
The Finite and the Infinite in Frege's
Grundgesetze der Arithmetik

RICHARD G. JNR. HECK

1. OPENING

Some of the best-known work on the foundations of mathematics done in the late nineteenth century was concerned with the concepts of the finite and the infinite, rigorous characterizations of which were famously given by both Cantor and Dedekind. It is little known, however, that, in his Grundgesetze der Arithmetik,1 Frege also studied the notions of finitude and infinity. His work in Grundgesetze had little influence upon that of his contemporaries because, quite simply, almost no one appears to have read it. One obstacle was Frege's notation, which is notoriously difficult to learn to read (though easy enough to read, actually); following Russell's discovery that the formal system of Grundgesetze is inconsistent, mathematicians might justifiably have thought themselves to have had little reason to bother. Indeed, one might wonder why, given his system's inconsistency, the proofs Frege gives in it merit any serious study at all.

The answer, as I have argued elsewhere,2 is that Frege's proofs depend far less upon the axiom responsible for the inconsistency than one might have supposed. For present purposes, the relevant axiom, Axiom V, may be taken to be:
ἐ.Fε = ἐ.Gε iff ∀x(Fx ≡ Gx).

That is: the value-range (or extension) of the concept Fξ is the same as that of the concept Gξ just in case the Fs are exactly the Gs. Now, Frege uses value-ranges, and so Axiom V, throughout Grundgesetze, but he makes essential appeal to Axiom V only in the course of his proof of what may be

____________________
1
Gottlob Frege, Grundgesetze der Arithmetik, Hildesheim, Georg Olms Verlagsbuchhandlung, 1966. Further references are in the text, marked 'Gg' with a roman numeral, for the volume number, and a section number. The first volume was published in 1893; the second, in 1903.
2
See my "'The Development of Arithmetic in Frege's Grundgesetze der Arithmetik'", Journal of Symbolic Logic, 58 ( 1993), 579-601, esp. §1, repr. with a postscript in W. Demopoulos (ed.), Frege's Philosophy of Mathematics, Cambridge, Mass., Harvard University Press, 1995, 257-94.

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