A Reassessment of Cantorian Abstraction based on the $$\varepsilon $$-operator

Abstract

AbstractCantor’s abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor’s proposal based upon the set theoretic framework of Bourbaki—called BK—which is a First-order set theory extended with Hilbert’s $$\varepsilon $$ ε -operator. Moreover, it is argued that the BK system and the $$\varepsilon $$ ε -operator provide a faithful reconstruction of Cantor’s insights on cardinal numbers. I will introduce first the axiomatic setting of BK and the definition of cardinal numbers by means of the $$\varepsilon $$ ε -operator. Then, after presenting Cantor’s abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo–von Neumann and Frege–Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege’s objections to Cantor’s proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the $$\varepsilon $$ ε -operator in the BK definition of cardinal numbers.