Bonatti, Nicola (2022) A Reassessment of Cantorian Abstraction based on the εoperator. [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 Firstorder set theory extended with Hilbert's εoperator. Moreover, it is argued that the BK system and the εoperator 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 εoperator. 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 Zermelovon Neumann and FregeRussell. 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 εoperator in the BK definition of cardinal numbers.
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Item Type:  Preprint  

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Keywords:  Georg Cantor, Nicolas Bourbaki, εoperator, 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://philsciarchive.pitt.edu/id/eprint/20984 
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