Sizes of countable sets

Kateřina Trlifajová, 21 Nov 2022

According to Cantor, two sets have the same size, i.e. cardinality, if there is a one-to-one correspondence between their elements. From this point of view, all countable sets have the same size, namely \(\aleph_0\). In contrast, Bolzano insisted on the Part-Whole Principle stating that the whole is greater than its part.

We introduce a theory of sizes of countable sets motivated by Bolzano’s concept such that the cardinality of finite sets is preserved and the Part-Whole Principle holds. We determine the sizes of natural numbers, integers, and rational numbers, their subsets, unions, and Cartesian products. Our method is similar to Benci’s and Di Nasso’s Theory of Numerosities (TN), but unlike it is constructive and unambiguous and it does not need ultrafilters. Although the set sizes are only partially and not linearly ordered, our results mostly agree with TN, in some cases, they are more subtle.