# Real sets

## TL;DR

The paper introduces a novel category of extended positive real sets, exploring their properties, tensor products, and potential for defining sets with unconventional 'cardinalities', expanding categorical set theory.

## Contribution

It defines a new category of extended positive real sets with a countable tensor product, and develops the theory of series monoidal categories, including non-commutative variants.

## Key findings

- Introduction of the category of extended positive real sets
- Development of the theory of series monoidal categories
- Brief discussion on sets with extended cardinalities

## 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: \begin{itemize} \item what is a set with half an element? \item what is a set with $\pi$ elements? \end{itemize} 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 {\em series monoidal} and conclude by only briefly mentioning the non-commutative possibility called {\em $\omega$-monoidal}. We include some remarks on sets having cardinalities in $[-\infty,\infty]$.