An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals

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).