Non-standard Analysis

A course to be given, summer 2010, at the Nesin Matematik Köyü.

Abstract

A distinction between calculus as a computational tool, and analysis as a rigorous mathematical discipline, was recognized by Archimedes. (We know this from the Archimedes Palimpsest, discovered in Istanbul by Heiberg in 1906.) Calculus as we know it today was developed in the 17th century by means of infinitesimals, but the notion of an infinitesimal was not then made rigorous. In the 19th century, calculus was made rigorous by means of limits: thus began analysis. In 1960, Abraham Robinson used the tools of mathematical logic to make the notion of infinitesimals rigorous: this begins non-standard analysis. In his book on the subject, Robinson quotes Kurt Gödel:

non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results…there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future.

This course will review some of the work of Archimedes, along with Dedekind's construction of the real numbers, before constructing the “non-standard real numbers”—among which are the infinitesimals—and proving theorems with them.

Prerequisites

Some knowledge of

standard analysis (how to prove theorems using the definition of a limit), and
algebra (for example, that the quotient of a commutative ring by a maximal ideal is a field).

Notes

I have thoroughly revised the notes from last year's course for use this year; copies will be distributed at the Village.