Lectures on Non-Standard Analysis
- PDF / 5,221,567 Bytes
- 86 Pages / 576 x 785 pts Page_size
- 79 Downloads / 221 Views
94
Moshe Machover Chelsea College University of London, London
Joram Hirschfeld The Hebrew University, Jerusalem
Lectures on Non-Standard Analysis 1969
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer-Verlag Berlin' Heidelberg 1969 Library of Congress Catalog Card Number 75-94156 Printed in Germany. Title No. 3700
Foreword
These Lecture Notes are in two parts. Part I, written by M. Machover, is based on short courses of lectures delivered by him in Bristol, England (1966) and in Jerusalem, Israel (1966/61 and 1961/68). Part II, written by J. Hirschfeld, is based on his M.Sc. thesis, submitted to the Hebrew University of Jerusalem in May 1968, and on seminar lectures delivered by him in Jerusalem during 1961/68• Most of the discussion of filters is also based on Hirschfeld's M.Sc. thesis, but is included here in Part I for methodological reasons. The purpose of Part I is to make Abraham Robinson's theory and some of its methods more easily accessible to mathematicians, including those who have little or no knowledge of formal logic.
The
stress here is not on new results but on the simplified framework in which the theory is presented.
A
considerable number of theorems and methods in Part I are taken directly or adapted from Robinson's book Non-Standard Analysis (North-Holland, 1966). Robinson himself uses the language and "ontology" of type theory as a framework for nonstandard analysis.
This, we believe, has two disadvantages.
First, the vast majority of mathematicians think of the various branches of mathematics as imbedded in set theory;
actually, what they use is practically always a rather small and "non-
committed" portion of set theory which they do not explicitly specify.
Concepts in algebra, topology,
analysis, etc., are defined in terms of sets and the membership relation, or in terms of other concepts which, in turn, can be reduced - in a way familiar to every mathematician - to sets and membership. True, all this can also be done in the framework of type theory - but the point is that it is not done in this way, probably because most mathematicians are not familiar with type theory and because of some technical difficulties which that procedure involves.
Thus to many mathematicians Robinson's original
framework may seem unnatural and unnecessarily complicated. Second, the so-called compactness theorem - the main tool borrowed by non-standard analysis from mathematical logic - is somewhat complicated to state (let alone prove) for a theory formulated in terms of types, and its intuitive meaning is then a bit difficult to explain.
For a type-free language, on
the other hand, the theorem can easily be explained with adequate rigour even without going into a great
IV deal of formal and technical detail. The main aim of Part I is therefore to explain Robinson's theory in a way that departs as little as possible from the usuage and conventions accepted by most mathematicians. Part II carr
Data Loading...
