Real sets

Abstract

After reviewing a universal characterization of the extended positive real numbers published by Denis Higgs in 1978, we define a category which provides an answer to the questions:

• what is a set with half an element?

• what is a set with $\pi$ elements?

The category of these extended positive real sets is equipped with a countable tensor product. We develop somewhat the theory of categories with countable tensors; we call the commutative such categories series monoidal and conclude by only briefly mentioning the non-commutative possibility called $\omega$-monoidal. We include some remarks on sets having cardinalities in $[−\infty, \infty]$.