Non-standard Analysis - Introduction

Introduction

A non-zero element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a standard natural number. Ordered fields that have infinitesimal elements are also called non-Archimedean. More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the transfer principle for real numbers is a hyperreal field, and non-standard real analysis uses these fields as non-standard models of the real numbers.

Robinson's original approach was based on these non-standard models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print. On page 88, Robinson writes:

The existence of non-standard models of arithmetic was discovered by Thoralf Skolem (1934). Skolem's method foreshadows the ultrapower construction

Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it is not enough to construct an ordered field with infinitesimals. See the article on hyperreal numbers for a discussion of some of the relevant ideas.