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

@article{Borovik2012AnIC, title={An Integer Construction of Infinitesimals: Toward a Theory of Eudoxus Hyperreals}, author={Alexandre V. Borovik and Renling Jin and Mikhail G. Katz}, journal={Notre Dame J. Formal Log.}, year={2012}, volume={53}, pages={557-570} }

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…

## 19 Citations

### Commuting and Noncommuting Infinitesimals

- MathematicsAm. Math. Mon.
- 2013

Some of the hyperreal concepts are reviewed, and they are compared with some of the concepts underlying noncommutative geometry.

### Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics

- Philosophy
- 2012

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model $${\mathcal{S}}$$ as ammunition against non-standard analysis, but the…

### Contemporary Infinitesimalist Theories of Continua and their late 19th- and early 20th-century forerunners

- Philosophy
- 2018

The purpose of this paper is to provide a historical overview of some of the contemporary infinitesimalist alternatives to the Cantor-Dedekind theory of continua. Among the theories we will consider…

### Proofs and Retributions, Or: Why Sarah Can’t Take Limits

- Mathematics
- 2015

The small, the tiny, and the infinitesimal (to quote Paramedic) have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that…

### Proofs and Retributions, Or: Why Sarah Can’t Take Limits

- MathematicsFoundations of Science
- 2014

The small, the tiny, and the infinitesimal (to quote Paramedic) have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that…

### Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus

- Mathematics
- 2012

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the…

### Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond

- MathematicsPerspectives on Science
- 2013

We analyze some of the main approaches in the literature to the method of ‘adequality’ with which Fermat approached the problems of the calculus, as well as its source in the παρισότης of Diophantus,…

### Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow

- Philosophy
- 2014

Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential…

### Leibnizian and Nonstandard Analysis: Philosophical Problematization of an Alleged Continuity Ivano Zanzarella

- Philosophy
- 2020

In the present paper the philosophical and mathematical continuity alleged by A. Robinson in Nonstandard Analysis (1966) between his theory and Leibniz’s calculus is investigated. In Section 1, after…

## 58 References

### The Eudoxus Real Numbers

- Mathematics
- 2004

This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real…

### Two ways of obtaining infinitesimals by refining Cantor's completion of the reals

- Mathematics
- 2011

Cantor's famous construction of the real continuum in terms of Cauchy sequences of rationals proceeds by imposing a suitable equivalence relation. More generally, the completion of a metric space…

### An Irrational Construction of R from Z

- MathematicsTPHOLs
- 2001

It turns out that the Dedekind cuts provide many routes one can travel to get from the ring of integers to the field of real numbers, R, and going via certain rings of algebraic numbers can provide a pleasant alternative to the more well-trodden track.

### The Absolute Arithmetic Continuum and the Unification Of all Numbers Great and Small

- MathematicsThe Bulletin of Symbolic Logic
- 2012

The simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum.

### Nonstandard Analysis

- MathematicsAm. Math. Mon.
- 2005

The construction of nonstandard extensions could provide a rigorous foundation for the use of infinitesimals in basic analysis, and NSA has become an active branch of research in its own right.

### Commuting and Noncommuting Infinitesimals

- MathematicsAm. Math. Mon.
- 2013

Some of the hyperreal concepts are reviewed, and they are compared with some of the concepts underlying noncommutative geometry.

### Lectures on the hyperreals : an introduction to nonstandard analysis

- Mathematics
- 1998

I Foundations.- 1 What Are the Hyperreals?.- 1.1 Infinitely Small and Large.- 1.2 Historical Background.- 1.3 What Is a Real Number?.- 1.4 Historical References.- 2 Large Sets.- 2.1 Infinitesimals as…

### Zooming in on infinitesimal 1–.9.. in a post-triumvirate era

- Philosophy
- 2010

The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity, is discussed via a proceptual analysis of the meaning of the ellipsis “...” in the real formula.

### Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics

- Philosophy
- 2012

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model $${\mathcal{S}}$$ as ammunition against non-standard analysis, but the…

### Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus

- Mathematics
- 2012

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the…