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

Notre Dame Journal of Formal Logic 53 (4):557-570 (2012)

Authors
Alexandre Borovik
University of Manchester
Abstract
A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ 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)
Keywords Eudoxus   hyperreals   infinitesimals   limit ultrapower   universal hyperreal field
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DOI 10.1215/00294527-1722755
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Cauchy's Continuum.Karin U. Katz & Mikhail G. Katz - 2011 - Perspectives on Science 19 (4):426-452.

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