An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals

TL;DR

This paper presents a novel one-step construction of hyperreal number systems from integers using ultrapower methods, unifying various existing hyperreal models within a set-theoretic framework.

Contribution

It introduces a unified construction method for hyperreal fields directly from integers, including maximal hyperreals, using ultrapower techniques.

Findings

Any hyperreal field with a set universe can be constructed from integers.

Maximal hyperreal systems can be obtained via a definable ultrapower in NBG.

The construction generalizes previous approaches to hyperreal number systems.

Abstract

A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter).