A construction of the real number system based on almost homomorphisms of the integers 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).
"An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals." Notre Dame J. Formal Logic 53 (4) 557 - 570, 2012. https://doi.org/10.1215/00294527-1722755