Single Idea 13527

[catalogued under 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle]

Full Idea

Frege defined a cardinal as an equivalence class of one-one correspondences. The cardinal 3 is the class of all sets with three members. This definition is intuitively appealing, but it is not permissible in ZFC.

Gist of Idea

Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC

Source

report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2

Book Reference

Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.64


A Reaction

This is why Frege's well known definition of cardinals no longer figures in standard discussions of the subject. His definition is acceptable in Von Neumann-Bernays-Gödel set theory (Wolf p.73).