An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals

Abstract

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).

17 pages, 1 figure

References in corpus (6)

• Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond
• Who Gave you the Cauchy-Weierstrass Tale? The Dual History of Rigorous Calculus
• Ten Misconceptions from the History of Analysis and Their Debunking
• Leibniz's Laws of Continuity and Homogeneity
• Cauchy's continuum
• Commuting and noncommuting infinitesimals

Cited by in corpus (8)

• Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond
• Ten Misconceptions from the History of Analysis and Their Debunking
• Almost equal: Method of adequality from Diophantus to Fermat and beyond
• Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics
• Leibniz's Laws of Continuity and Homogeneity
• A Cauchy-Dirac delta function
• Commuting and noncommuting infinitesimals
• A footnote to The crisis in contemporary mathematics