Bulletin of Symbolic Logic 1 (3):317-326 (1995)

Abstract
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.
Keywords No keywords specified (fix it)
Categories (categorize this paper)
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

PhilArchive copy


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 58,398
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

No references found.

Add more references

Citations of this work BETA

Comparing Peano Arithmetic, Basic Law V, and Hume’s Principle.Sean Walsh - 2012 - Annals of Pure and Applied Logic 163 (11):1679-1709.
La herencia oscura del logicismo.José Ferreirós - 2020 - Metatheoria – Revista de Filosofía E Historia de la Ciencia 10 (2):19--30.

Add more citations

Similar books and articles

On Translating Frege's Die Grundlagen der Arithmetik.Matthias Schirn - 2010 - History and Philosophy of Logic 31 (1):47-72.
Frege's Theorem and His Logicism.Hirotoshi Tabata - 2000 - History and Philosophy of Logic 21 (4):265-295.

Analytics

Added to PP index
2014-01-21

Total views
17 ( #594,680 of 2,420,559 )

Recent downloads (6 months)
1 ( #542,979 of 2,420,559 )

How can I increase my downloads?

Downloads

My notes