Euclidean integers, Euclidean ultrafilters, and Euclidean numerosities

TL;DR

This paper introduces the Euclidean integers ring, characterizes its algebraic and order properties, and applies it to define Euclidean size (numerosity) for ordinal sets, maintaining classical set operations.

Contribution

It axiomatizes the Euclidean integers ring, characterizes its order algebraically, and extends numerosity to ordinal sets while preserving classical set operations.

Findings

The Euclidean integers form an ultrapower of  with a specific ultrafilter.

Positivity is characterized by transfinite sums of natural numbers.

Sets of ordinals can be assigned Euclidean sizes consistent with classical operations.

Abstract

We introduce axiomatically the ring $\bf{Z}_\kappa$ of the Euclidean integers, that can be viewed as the ``integral part" of the field $\mathbb{E}$ of Euclidean numbers of [4], where the transfinite sum of ordinal indexed $\kappa$-sequences of integers is well defined. In particular any ordinal might be identified with the transfiite sum of its characteristic function, preserving the so called natural operations. The ordered ring $\bf{Z}_\kappa$ may be obtained as an ultrapower of $\mathbb{Z}$ modulo suitable ultrafilters, thus constituting a \it{ring of nonstandard integers.} Most relevant is the \it{algebraic} characterization of the ordering: a Euclidean integer is \it{positive} if and only if it is \it{the transfinite sum of natural numbers.} This property requires the use of special ultrafilters called Euclidean, here introduced to ths end. The ring $\bf{Z}_\kappa$ allows to assign a ``Euclidean" size (\it{numerosity}) to ``ordinal Punktmengen", i.e. sets of tuples of ordinals, as the transfinite sum of their characteristic functions: so every set becomes equinumerous to a set of ordinals, the Cantorian defiitions of \it{order, addition and multiplication} are maintained, while the Euclidean principle ``the whole is greater than the part" (\it{a set is (strictly) larger than its proper subsets}) is fulfilled.