Non-standard Analysis

310 Pages·1974·23.446 MB·English

by  Abraham Robinson

STUDIES IN LOGIC
FOUNDATIONS OF MATHEMATICS
L.E.J.BROUWER/A.HEYTING/_A.ROBINSONIP.SUPPES
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EDITORS
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Non-standard
Analysus
A. ROBINSON
.ORTH-HOLLAND/AMERICAN ELSEVIER
'AMSTERDAM.OXFORD o.NEW'VOR;K
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NON-STANDARD ANALYSIS
STUDIES IN LOGIC
AND
THE FOUNDATIONS OF MATHEMATICS
Editors
L. E. J. BROUWER, Laren (N.H.)
A. HEYTING, Amsterdam
A. ROBINSON, LosAngeles
P. SUPPES, Stanford
Advisory Editorial Board
Y. BAR-HILLEL,Jerusalem
K. L. DE BOUVERE, Amsterdam
H. HERMES, Miinster i/W.
J. HINTIKKA, Helsinki
A. MOSTOWSKI, Warszawa
J. C. SHEPHERDSON, Bristol
E. P. SPECKER, Ziirich
NORTH-HOLLAND PUBLISHING COMPANY —AMSTERDAM ∙ LONDON
AMERICAN ELSEVIER PUBLISHING COMPANY, INC.—NEW YORK
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NON-STANDARD
ANALYSIS
Revised edition
ABRAHAM ROBINSON
YaleUniversity,New Haven
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1974
NORTH-HOLLAND PUBLISHING COMPANY — AMSTERDAM �LONDON
AMERICAN ELSEVIER PUBLISHING COMPANY, INC. — NEW YORK
© North-Holland Publishing Company — 1966
AllRightsReserved.Nopart of thispublication maybereproduced,stored inaretrieval
systemortransmitted,inanyform orbyanymeans,electronic,mechanical,photocopying,
recordingorotherwise,without thepriorpermission of theCopyrightowner.
First edition1966
Secondprinting 1970
Revisededition1974
Library ofCongressCatalogCardNumber 73-93100
North-HollandISBNfor theseries0720422000
for thisvolume072042237x
AmericanElsevierISBN0444106103
Published by:
North-Holland Publishing Company-Amsterdam
North-Holland Publishing Company,Ltd.-London
Soledistributorsfor theU.S.A.andCanada:
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PREFACE
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juger deriensursagrandeur apparente.
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a‘dessubstancesquiparaissent si
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VOLTAIRE, Micromégas
In thefall of 1960it occurred to me that the concepts and methods of
contemporary Mathematical Logic are capable of providing a suitable
framework for thedevelopment of the Differential and Integral Calculus
by means of infinitely small and infinitely large numbers.I first reported
my ideasinaseminar talk atPrinceton University(November 1960)and,
later,inanaddressattheannual meeting of theAssociationfor Symbolic
Logic (January 1961)and inapaper published inthe Proceedings of the
Royal Academy of Sciences of Amsterdam (ROBINSON [1961]). The re-
sulting subject was called by me Non-standard Analysis since it involves
and was, in part, inspired by the so-called Non-standard models of
Arithmetic whose existence wasfirst pointed out by T. Skolem.
In the intervening years, Non-standard Analysis has developed con-
siderably in several directions. Since many of the results have been
reported sofar onlyincoursesorlectures,and inmimeographed reports,
it was thought that a book dedicated entirely to this subject would be
inorder.
Over the years, my thinking in this area has been enlivened by dis-
cussions withseveral colleagues,among whom I venture tomention here
R.Arens,C.C.Chang,A.Erdelyi, A.Horn, G. Kreisel, l. Lakatos,and
J.B.Rosser.Special thanksareduetoW.A.J.Luxemburg,whoselectures
and lecture notes on Non-standard Analysis have done much to make
the subject known among mathematicians.
VIII PREFACE
I acknowledge with thanks the support received from the National
Science Foundation (Grants GP-1812and GP-4038) in connectionwith
much of the research incorporated in the present book. [ also wish to
mention that Chapter VI is based on research sponsored inpart by the
Air Force Ofliceof Scientific Research andpublished previously inreport
form (ROBINSON [1962]).Finally,I wish toput onrecordmyindebtedness
to M. Machover who read the book at the proof stage and suggested
variousimprovements.
ABRAHAM ROBINSON
Universityof California,Los Angeles
April, 1965