Frege's Cardinals Do Not Always Obey Hume's Principle. Issue 2 (3rd April 2017)

• Record Type: Journal Article
• Authors: Landini, Gregory
• Is Part Of: History and philosophy of logic. Volume 38:Issue 2(2017)
• Page Start: 127
• Page End: 153
• Publication Date: 2017-04-03
• Subjects: Logic -- Periodicals; 160.905
• DOI: 10.1080/01445340.2016.1270635
• Languages: English
• ISSNs: 0144-5340

Abstract

Abstract : Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations (where non-homogeneous relations are allowed) that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's 'no-classes' theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze . Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle.