Real sets

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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 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 [-∞;∞].

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