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2012 An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals
Alexandre Borovik, Renling Jin, Mikhail G. Katz
Notre Dame J. Formal Logic 53(4): 557-570 (2012). DOI: 10.1215/00294527-1722755


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).


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Alexandre Borovik. Renling Jin. Mikhail G. Katz. "An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals." Notre Dame J. Formal Logic 53 (4) 557 - 570, 2012.


Published: 2012
First available in Project Euclid: 8 November 2012

zbMATH: 1266.03074
MathSciNet: MR2995420
Digital Object Identifier: 10.1215/00294527-1722755

Primary: 26E35
Secondary: 03C20

Rights: Copyright © 2012 University of Notre Dame


Vol.53 • No. 4 • 2012
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