Bolzano and the Part-Whole Principle

TL;DR

This paper revisits Bolzano's early ideas to develop a constructive method for comparing infinite set sizes that respects the Part-Whole Principle, challenging standard set theory's reliance on the Cantor Principle.

Contribution

It extends Bolzano's theory to create a simpler, constructive approach for determining countable set sizes that aligns with the Part-Whole Principle, differing from traditional cardinality.

Findings

Sizes of countable sets are uniquely determined.

The set sizes are not linearly ordered.

The approach is closer to numerosity theory but simpler.

Abstract

The embracing of actual infinity in mathematics leads naturally to the question of comparing the sizes of infinite collections. The basic dilemma is that the Cantor Principle (CP), according to which two sets have the same size if there is a one-to-one correspondence between their elements, and the Part-Whole Principle (PW), according to which the whole is greater than its part, are inconsistent for infinite collections. Contemporary axiomatic set-theoretic systems, for instance, ZFC, are based on CP. PW is not valid for infinite sets. In the last two decades, the topic of sizes of infinite sets has resurfaced again in a number of papers. A question of whether it is possible to compare the sizes to comply with PW has been risen and researched. Bernard Bolzano in his 1848 Paradoxes of the Infinite dealt with principles of introducing infinity into mathematics. He created a theory of infinite quantities that respects PW and which is based on sums of infinite series. We extend Bolzano's theory and create a constructive method for determining the set size of countable sets so that the cardinality of finite sets is preserved and PW is valid. Our concept is close to the numerosity theory from the beginning of this century but it is simpler and more intuitive. In the results, we partly agree with the theory of numerosities. The sizes of countable sets are uniquely determined, but they are not linearly ordered.