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Abstraction and Infinity$
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Paolo Mancosu

Print publication date: 2016

Print ISBN-13: 9780198746829

Published to Oxford Scholarship Online: January 2017

DOI: 10.1093/acprof:oso/9780198746829.001.0001

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Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable?

Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable?

Chapter:
(p.116) 3 Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable?
Source:
Abstraction and Infinity
Author(s):

Paolo Mancosu

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198746829.003.0004

The standard Cantorian assignment of cardinal numbers to sets uses the criterion of one–one correspondence. The fruitfulness of this definition in mathematics and the lack of mathematically worked-out alternatives seemed to make any other approach impossible. Indeed, Gödel went as far as to offer an argument for why a generalization of arithmetic from the finite to the infinite would inevitably lead to Cantor’s theory of cardinals. Chapter 3 presents a theory of counting with infinite sets, the theory of numerosities, which assigns sizes to sets according to a different criterion from Cantor’s. The numerosities in this theory satisfy the part-whole axiom: if a set A is strictly included in a set B, then the numerosity of A is strictly smaller than the numerosity of B. The existence of a mathematical alternative to Cantor’s theory for capturing the notion of “size” for sets leads to deep mathematical, historical, and philosophical problems. In particular, Mancosu discusses in this chapter various philosophical claims that have been made in connection to Cantor’s theory of cardinal numbers (e.g. Gödel’s inevitability claim and Kitcher’s discussion of rational transitions between mathematical practices) showing that the new theory of numerosities leads to a much deeper appreciation of the issues involved.

Keywords:   Cantor, Gödel, theory of numerosities, part–whole principle

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