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# Non-standard analysis

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 Title: Non-standard analysis Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Non-standard analysis

Gottfried Wilhelm Leibniz argued that idealized numbers containing infinitesimals be introduced.

The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis[1][2][3] instead reformulates the calculus using a logically rigorous notion of infinitesimal number.

Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson.[4][5][6] He wrote:

[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latter

Robinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:

However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.[7]

Robinson continues:

It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.

In 1973, intuitionist Arend Heyting praised non-standard analysis as "a standard model of important mathematical research".[8]

## Contents

• Introduction 1
• Basic definitions 2
• Motivation 3
• Historical 3.1
• Pedagogical 3.2
• Technical 3.3
• Approaches to non-standard analysis 4
• Robinson's book 5
• Invariant subspace problem 6
• Other applications 7
• Applications to calculus 7.1
• Critique 8
• Logical framework 9
• Internal sets 10
• First consequences 11
• κ-saturation 12
• References 15
• Bibliography 16

## Introduction

A non-zero element of an ordered field \mathbb F is infinitesimal if and only if its absolute value is smaller than any element of \mathbb F of the form \frac{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.[9] 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.

## Basic definitions

In this section we outline one of the simplest approaches to defining a hyperreal field ^*\mathbb{R}. Let \mathbb{R} be the field of real numbers, and let \mathbb{N} be the semiring of natural numbers. Denote by \mathbb{R}^{\mathbb{N}} the space of sequences of real numbers. A field ^*\mathbb{R} is defined as a suitable quotient of \mathbb{R}^\mathbb{N}, as follows. Take a nonprincipal ultrafilter F \subset P(\mathbb{N}). In particular, F contains the Fréchet filter. Consider a pair of sequences

u = (u_n), v = (v_n) \in \mathbb{R}^\mathbb{N}

We say that u and v are equivalent if they coincide on a set of indices which is a member of the ultrafilter, or in formulas:

\{n \in \mathbb{N} : u_n = v_n\} \in F

The quotient of \mathbb{R}^\mathbb{N} by the resulting equivalence relation is a hyperreal field ^*\mathbb{R}, a situation summarized by the formula ^*\mathbb{R} = \frac{\mathbb{R}^\mathbb{N}}{F}.

## Motivation

There are at least three reasons to consider non-standard analysis: historical, pedagogical, and technical.

### Historical

Much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity. As noted in the article on

• The Ghosts of Departed Quantities by Lindsay Keegan.