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 { \Omega}. Thanks to this sum, E becomes a saturated hyperreal field isomorphic to the so called Kiesler field of cardinality { \Omega}, and suitable topologies can be put on E and on { \Omega} \cup { { \Omega}} 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 { \Omega} 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).
Categories: math.LO math.RA