Notre Dame Journal of Formal Logic

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

Alexandre Borovik, Renling Jin, and Mikhail G. Katz


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).

Article information

Notre Dame J. Formal Logic, Volume 53, Number 4 (2012), 557-570.

First available in Project Euclid: 8 November 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26E35: Nonstandard analysis [See also 03H05, 28E05, 54J05]
Secondary: 03C20: Ultraproducts and related constructions

Eudoxus hyperreals infinitesimals limit ultrapower universal hyperreal field


Borovik, Alexandre; Jin, Renling; Katz, Mikhail G. An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals. Notre Dame J. Formal Logic 53 (2012), no. 4, 557--570. doi:10.1215/00294527-1722755.

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