Euclidean integers, Euclidean ultrafilters, and Euclidean numerosities

• Published in: arXiv.org
• Main Authors: Mauro Di Nasso, ti, Marco
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 05.12.2022
• Subjects: Integers; Multiplication; Number theory
• ISSN: 2331-8422

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.