Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 53, Number 4 (2012), 557-570.
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 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).
Notre Dame J. Formal Logic, Volume 53, Number 4 (2012), 557-570.
First available in Project Euclid: 8 November 2012
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 26E35: Nonstandard analysis [See also 03H05, 28E05, 54J05]
Secondary: 03C20: Ultraproducts and related constructions
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. https://projecteuclid.org/euclid.ndjfl/1352383232