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Frege's Theorem and the Peano Postulates

Published online by Cambridge University Press:  15 January 2014

George Boolos
George Boolos*
Affiliation:
Department of Linguistics and Philosophy, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. E-mail: boolos@mit.edu
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Extract

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.

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 Grundgesetze der Arithmetik and his earlier exoteric book Die Grundlagen der Arithmetik. We may describe the two central results of the recent re-evaluation of his work in the following way: Let Frege arithmetic be the result of adjoining to full axiomatic second-order logic a suitable formalization of the statement that the Fs and the Gs have the same number if and only if the F sand the Gs are equinumerous.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 1 , Issue 3 , June 1995 , pp. 317 - 326
Copyright
Copyright © Association for Symbolic Logic 1995

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References

The author is grateful to the Alexander von Humboldt Foundation for its support while work on this paper was begun.

1 Demopoulos, William, ed., Frege's Philosophy of Mathematics, Harvard University Press, Cambridge, Massachusetts, 1995 Google Scholar.

2 We use ‘precedes’ to mean ‘immediately precedes’ (rather than ‘is less than’).

3 Henkin, Leon, “On mathematical induction”, American Mathematical Monthly 67, 04 1960, 323–338 CrossRefGoogle Scholar.

4 One of Frege's examples is “Jupiter has four moons”.

5 FE is further discussed in my “On the proof of Frege's theorem”, to appear in a Festschrift for Paul Benacerraf edited by Adam Morton and Stephen Stich.

6 Frege realized that he needed this condition on d only in Grundsetze der Arithmetik and not in Die Grundlagen der Arithmetik. For more details on his error, see “Die Grundlagen der Arithmetik, §§82–3, by George Boolos and Richard G. Heck, Jr., to appear in Philosophy of Mathematics Today, edited by Matthias Schirn.