A Reassessment of Cantorian Abstraction based on the epsilon-operator

Bonatti, Nicola (2022): A Reassessment of Cantorian Abstraction based on the epsilon-operator. In: Synthese, Bd. 200, Nr. 5

Abstract

Cantor'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 epsilon-operator. Moreover, it is argued that the BK system and the epsilon-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 epsilon-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 epsilon-operator in the BK definition of cardinal numbers.

• Dokumententyp: Zeitschriftenartikel
• Fakultät: Philosophie, Wissenschaftstheorie und Religionswissenschaft > Munich Center for Mathematical Philosophy (MCMP)
• ISSN: 0039-7857
• Sprache: Englisch
• DFG: Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) - 491502892