An Euclidean measure of size for mathematical universes

Vieri Benci, Mauro Di Nasso, Marco Forti


We show that a measure of size satisfying the five common notions of Euclid’s Elements can be consistently assumed for all sets in the universe of “classical" mathematics. In particular, such a universal Euclidean measure maintains the ancient principle that “the whole is greater than the part". Values are taken in the positive part of a discretely ordered ring (actually, into a set of hypernatural numbers of nonstandard analysis) in such a way that measures of disjoint sums and Cartesian products correspond to sums and products, respectively. Moreover, universal Euclidean measures can be taken in such a way that they satisfy a natural continuity property for suitable (normal) approximations.

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