Real sets.G. Janelidze, R. StreetAfter 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 π 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 ω-monoidal. We include some remarks on sets having cardinalities in [-∞, ∞]. Tbilisi Mathematical Journal, Vol. 10(3) (2017), pp. 23-49 |