Sizes of Countable Sets

The paper introduces the notion of size of countable sets, which preserves the Part-Whole Principle. The sizes of the natural and the rational numbers, their subsets, unions, and Cartesian products are algorithmically enumerable as sequences of natural numbers. The method is similar to that of Numerosity Theory, but in comparison it is motivated by Bolzano's concept of infinite series, it is constructive because it does not use ultrafilters, and set sizes are uniquely determined. The results mostly agree, but some differ, such as the size of rational numbers. However, set sizes are only partially, not linearly, ordered. Quid pro quo.

Automated summary

The paper introduces the concept of size of countable sets, which follows the Part-Whole Principle. The sizes of the natural and rational numbers, as well as their subsets, unions, and Cartesian products, can be algorithmically enumerated using sequences of natural numbers. This approach is constructive, motivated by Bolzano's concept of infinite series and does not rely on ultrafilters. Set sizes are uniquely determined. The results mostly agree, although there are some differences, such as the size of rational numbers. However, set sizes are only partially ordered, not linearly ordered. Quid pro quo.