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The Euclidean numbers

We introduce axiomatically a Nonarchimedean field E, called the field of the Euclidean numbers, where a transfinite sum is defined that is indicized by ordinal numbers less than the first inaccessible Ω. Thanks to this sum, E becomes a saturated hyperreal field isomorphic to the so called Kiesler field of cardinality Ω, and suitable topologies can be put on E and on Ω \cup {Ω} so as to obtain the transfinite sums as limits of a suitable class of their finite subsums. Moreover there is a natural isomorphic embedding into E of the semiring Ω equipped by the natural sum and product. Finally a notion of numerosity satisfying all Euclidean common notions is given, whose values are nonnegative nonstandard integers of E. Then E can be charachterized as the hyperreal field generated by the real numbers and together with the semiring of numerosities (and this explains the name Euclidean numbers).