On the limits of comparing subset sizes within $\mathbb{N}$

TL;DR

This paper compares six formal methods for assigning sizes to subsets of natural numbers, revealing fundamental limitations and differences in their properties, especially regarding constructiveness and uniqueness.

Contribution

It introduces and analyzes a new formalism called c-numerosity, providing insights into the intrinsic limitations of size comparison methods within alculus.

Findings

Generalised densities and -numerosities lack uniqueness.

C-numerosity is fully constructive but only partially ordered.

All formalisms relate through limit operations, highlighting inherent limitations.

Abstract

We review and compare five ways of assigning totally ordered sizes to subsets of the natural numbers: cardinality, infinite lottery logic with mirror cardinalities, natural density, generalised density, and $\alpha$-numerosity. Generalised densities and $\alpha$-numerosities lack uniqueness, which can be traced to intangibles: objects that can be proven to exist in ZFC while no explicit example of them can be given. As a sixth and final formalism, we consider a recent proposal by \citet{Trlifajova:2024}, which we call c-numerosity. It is fully constructive and uniquely determined, but assigns merely partially ordered numerosity values. By relating all six formalisms to each other in terms of the underlying limit operations, we get a better sense of the intrinsic limitations in determining the sizes of subsets of $\mathbb{N}$.