Bonatti, Nicola (2022) A Reassessment of Cantorian Abstraction based on the ε-operator. [Preprint]

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

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Item Type:	Preprint
Creators:	Creators Email ORCID Bonatti, Nicola nicola.bonatti@campus.lmu.de 0000-0002-1442-4336
Keywords:	Georg Cantor, Nicolas Bourbaki, ε-operator, abstraction, arbitrary reference.
Subjects:	Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > History of Philosophy Specific Sciences > Mathematics > History Specific Sciences > Mathematics > Logic Specific Sciences > Mathematics
Depositing User:	Mr. Nicola Bonatti
Date Deposited:	01 Aug 2022 18:56
Last Modified:	01 Aug 2022 18:56
Item ID:	20984
Subjects:	Specific Sciences > Mathematics > Epistemology Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > History of Philosophy Specific Sciences > Mathematics > History Specific Sciences > Mathematics > Logic Specific Sciences > Mathematics
Date:	2022
URI:	https://philsci-archive.pitt.edu/id/eprint/20984

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