On the Limits of Comparing Subset Sizes within N

Abstract

Wereviewandcomparefivewaysofassigningtotallyorderedsizestosubsetsofthenatural numbers: cardinality, infinite lottery logic with mirror cardinalities, natural density, generalised density, and α-numerosity. Generalised densities and α-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 Trlifajová (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 N.