Sizes of Countable Sets.

This article develops a theory of sizes for certain countable sets that preserves the Part-Whole Principle (PW), which states that a whole is greater than its part, contrasting with the classical Hume’s Principle (HP) based on one-to-one correspondence. Inspired by Bernard Bolzano’s concept of infinite series and “determining ground,” the theory defines sizes of natural numbers, integers, rational numbers, their subsets, unions, and Cartesian products as algorithmically enumerable sequences of natural numbers, called size sequences. These size sequences form a partially ordered, non-Archimedean semiring under equality and ordering defined via the Fréchet filter, avoiding non-constructive ultrafilters used in Numerosity Theory (NT). The paper shows that this approach uniquely determines set sizes, preserves PW, and ensures homogeneity of rational intervals, while acknowledging that the partial order means some infinite set sizes remain incomparable. It also contrasts this constructive framework with NT, which relies on ultrafilters and yields linear but non-unique orderings, and responds to critiques claiming no “good” PW-preserving theory exists by demonstrating a consistent, algorithmically definable model grounded in Bolzano’s principles.