  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
$\mathbb{N}$.
