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

Abstract

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).

A footnote to The crisis in contemporary mathematics

“…3 [Dieudonné 1975] A small group of followers has successfully pursued Bishop's program of what he defined as constructive mathematics. The most notable of his disciples are Douglas Bridges and Fred Richman who have published widely in constructive mathematics; see e.g., [Bridges-Richman 1987]. There have also been attempts to bridge a perceived gap between constructive mathematics and Robinson's framework for infinitesimals; see e.g., [Schuster et al 2001].…”

“…It seems very difficult to construct e.g., infinitesimals in terms of ordinary integers (but see [Borovik et al 2012]), i.e., it seems the former cannot be reduced to the latter in any way acceptable to Bishop. Hence, the very core of Robinson's framework for infinitesimal analysis deals with objects (seemingly) unacceptable 6 to Bishop, which is what presumably led him to the 'debasement' comment.…”

Leibniz’s Laws of Continuity and Homogeneity

“…the finite realm, to the "realm" of the infinite is precisely the content of Leibniz's law of continuity. 8 We therefore apply Leibniz's law of continuity to equation (4) for an infinite H. The resulting entity is still an ellipse of sorts, to the extent that it satisfies all of the equations (1) through (4). However, this entity is no longer finite.…”

“…We then have the standard part function (10) st : IIR <∞ → R, illustrated in Figure 1. Note that the hyperreals can be constructed out of integers (see [4]). The traditional quotient construction using Cauchy sequences, usually attributed to Cantor (and actually due to Méray [43], who published three years earlier than E. Heine), can be factored through the hyperreals (see [17]).…”

A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

“…23 The LPO is still unacceptable to a constructivist, but could have served as a basis for a meaningful dialog between Brouwer and Hilbert (see [19]), that could allegedly have changed the course of 20th century mathematics.…”