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

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