Sizes of Countable Sets

Abstract: The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their subsets, unions, and Cartesian products are algorithmically enumerable up to one element as sequences of natural numbers. The method is similar to that of Theory of Numerosities of Benci and Di Nasso 2019) but in comparison, it is motivated by Bolzano's concept of infinite series from his Paradoxes of the Infinite, it is constructive because it does not use ultrafilters, and set sizes are uniquely determined. The results mostly agree with those of Theory of Numerosities, but some differ, such as the size of rational numbers. However, set sizes are just partially and not linearly ordered. \emph{Quid pro quo.}