Euclidean integers, Euclidean ultrafilters, and Euclidean numerosities

Mauro Di Nasso, Marco Forti

Published 2022-07-15Version 1

We introduce a "Euclidean" notion of size (numerosity) for "Punktmengen", i.e. sets of points of Euclidean (finitely dimensional) spaces over any "line" L, namely one that maintains the Cantorian defiitions of order, addition and multiplication, while preserving the ancient principle that "the whole is greater than the part" (a set is (strictly) larger than its proper subsets). These numerosities satisfy the five Euclid's common notions, thus enjoying a very good arithmetic, since they constitute the nonnegative part of the ordered ring of the Euclidean integers, here introduced by suitably assigning a transfinite sum to (ordinally indexed) kappa-sequences of integers (so generating a semiring of nonstandard natural numbers). Most relevant is the natural set theoretic definition of the set-preordering <: given any two sets X, Y of any cardinality, one has X<Y if and only if there exists a proper superset of X that is equinumerous to Y . Extending this "superset property" from countable to uncountable sets has been one of the main open question in this area from the beginning of the century.