Model Theory and Algebra: A Memorial Tribute to Abraham Robinson

469 Pages·1975·5.6 MB·English

by  Jon Barwise

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
498
Model Theory dna Algebra
A Memorial Tribute
to Abraham Robinson
Edited yb ,D .H Saracino dna .V .B Weispfenning
SCIHTE BIB-HTE
5937230000010O
galreV-regnirpS
Berlin. .grebledieH weN kroY 5791
Editors
.rD Daniel H. Saracino
Department of Mathematics
Colgate U ytisrevin
Hamilton, New York 13346
USA
.rD Volker .B Weispfenning
Mathematisches Institut
der Universit~t Heidelberg
mI Neuenheimer Feld 288
69 Heidelberg/BRD
Library of Congress Cataloging in Publication Data
Main e~try under title:
Model theory and algebra.
(Lecture notes in mathematics ; 498)
"Bibliography of Robinson's works": p.
Includes index.
CONTENTS: Biograpb7 of Abraham Robinson.--Robinson~
A. Algorithms in algebra.--Barwise, J. and Schlipf, J.
On recursively saturated models of arithmetic. etc.
1. Model theory--Addresses, essays, lectures.
2. Algebra--Addresses, essays, lectures. 3. Robinson,
Abraham, 1918-1974. .I R~binson, Abraham~ 1918-1974.
.II Weispfenning, V., 1944- III. Saracino ~ D.,
1947- .VI Series: Lecture notes in mathematics
(Berlin) ; 498.
8_21.3AQ no. 498 QA9.7 510'.8s 511'.8 75-40483
AMS Subject Classifications (1970): 01A70, 02B25, 02E10, 02F50,
02H05, 02H13, 02H15, 02H20, 02H25, 10N15, 12A20, 12D15, 12E05,
21 E05, 12J15, 21 L10,12 L15,13A15,13 B 20,13 B25,13 L05,14 H99,16A40,
18A25, 20A10, 20E05, 20K10
ISBN 3-540-0?538-0 Springer-V.erlag Berlin (cid:12)9 Heidelberg (cid:12)9 New York
ISBN 0-387-07538-0 Springer-Verlag New York (cid:12)9 Heidelberg (cid:12)9 Berlin
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation,
reprinting, re-use of illustrations, broadcasting, reproduction by photo-
copying machine or similar means, dna storage in data banks.
Under w 54 of the German Copyright Law where copies are made for other
than private use, a fee is payable to the publisher, the amount of the fee to
be determined by agreement with the publisher.
(cid:14)9 by Springer-Verlag Berlin - Heidelberg 1975
Printed ni Germany
Offsetdruck: Julius Bettz, Hemsbach/Bergstr.
Abraham Robinson was "the one mathematical logician who
accomplished incomparably more than anybody else in making
this science fruitful for mathematics. I ma sure his name
will be remembered by mathematicians for centuries."
-- Kurt GSdel
MAHARBA NOSNIBOR
October 6, 1918 - April 11, 1974
Foreword
The sudden fatal illness of Abraham Robinson came as a great shock to many
people around the world. For Robinson was more than an excellent mathematician. eH
was also a person whom one came very quickly to like very much.
It was a wonderful thing to find in one person the combination of Abraham
Robinson - cofounder of model theory and inventor of nonstandard analysis - on the
one hand, and "Abby" - warm and humane human being - on the other. What a pleasure
it was to have him stop by one's office in the morning and ask if one could spare
the time for a walk to Naples Pizza for a cup of coffee. And on the way back one
would be almost hesitantly asked if one could spare the time for a detour to the
newsstand so he could pick up his weN York Times.
Those swift sad months of November 1973 - April 1974 were for those at Yale
tinged with a sense of unreality. eH was gone before anyone could come to grips
with what was happening. e41 sought a way of expressing our respect and our sense
of personal loss. This volume was the best way ew knew.
Perhaps a word is in order about the deliberately limited scope of the book.
Surely many more people than those represented here would want to contribute to a
collection in Robinson's honor. oT keep a volume of reasonable size we restricted
the contents to papers in "model theory and algebra", a subject with which he was
deeply involved for most of his career. Furthermore, in attempting to create a
personal tribute we sought papers primarily from young people who had worked with
him in this area. Particularly noticeable is the omission of papers in nonstandard
analysis. This omission has already been partly compensated for by some of the
papers presented at the Robinson memorial conference held at Yale in May, 1975. The
proceedings of this conference will appear separately as a special issue of the
Israel Journal of Mathematics.
eW would like to express our gratitude to Mrs. Ren#e Robinson for providing
us with the photograph at the beginning of the book and for giving us permission to
publish a version of Robinson's last paper. eW also wish to thank Professor Kurt
G~del for allowing us to include the quotation on page v.
While we were in the very early stages of planning this volume, Professor G.H.
MUller of Heidelberg suggested that the Springer Lecture Notes series might provide
an appropriate format. eW wish to thank him for arranging the publication of the
book with Springer-Verlag, and to thank Springer for providing su with secretarial
assistance. eW are also grateful for the characteristic swiftness with which the
manuscript was published.
Heidelberg, August 1975 D.S. and V.W.
TABLE FO STNETNOC
Biography of Abraham Robinson ............................................ I
Bibliography of Robinson's works ......................................... 4
Algorithms in Algebra , by A. Robinson .................................. 14
Contributed papers:
J. Barwise and J. Schlipf, nO recursively saturated models of arithmetic 42
.M Boffa, A note on existentially complete division rings .............. 56
.G Cherlin, Ideals of integers in nonstandard number fields ............. 60
P. Eklof, Categories of local functors .................................. 19
S. Feferman, Impredicativity of the existence of the largest divisible
subgroup of an abelian p-group ............................ 711
.E Fisher, .H Simmons, & .W Wheeler, Elementary equivalence classes of
generic structures and existentially complete structures .. 131
J. Schmerl, The number of equivalence classes of existentially complete
structures ................................................ 071
J. Hirschfeld, Finite forcing and generic filters in arithmetic ........ 271
A. Macintyre, Dense embeddings I : A theorem of Robinson in a general
setting ................................................... 002
K. McKenna, weN facts about Hilbert's seventeenth problem .............. 022
.P Roquette, Nonstandard aspects of Hilbert's irreducibility theorem ... 132
.G Sacerdote, Projective model theory and coforcing .................... 672
D. Saracino & V. Weispfenning, nO algebraic curves over commutative
regular rings ............................. 703
.S Shelah, Existence of rigid-like families of abelian p-groups ........ 483
.H Simmons, ehT complexity of T f and omitting types in F T .............. 304
.P Winkler, Model-completeness and Skolem expansions ................... 804
BIOGRAPHY
Abraham Robinson was born on October 6, 1918, in Waldenburg,
Germany. He spent his boyhood in Germany and Palestine (now Israel)
and graduated from the Jerusalem Gymnasium (Grammar School) in 1936.
He was a student at the Hebrew University, Jerusalem from 1936 to
1943, including a term at the Sorbonne, Paris. He obtained the degree
of M.Sc. ~rom the Hebrew University in 1946 and the degrees of Ph.D.
and D.Sc. from the University of London in 1949 and 1957 respectively.
During the second world war he served in the Free French Air Force
and later became a Scientific Officer at the Royal Aircraft Establish-
ment, Farmborough, England. From 1946 to 1951 he was Senior Lec~urer
in Mathematics, later Deputy Head of the Department of Aerodynamics
at the College of Aeronautics, Cranfield, England. Subsequently he was
Associate Professor, then Professor, of Applied Mathematics at the
University of Toronto, Canada (1951-1957), Professor of Mathematics
and sometime chairman of the department of Mathematics, at the Hebrew
University, Jerusalem, Israel (1957-1962), Professor of Mathematics
and Philosophy at the University of California, Los Angeles (1962-1967)
and Professor of Mathematics at Yale University 1967 - 1974 (Sterling
Professor of Mathematics since 1971). He was at various times a Vis(cid:127)
ting Professor at the Universities of Princeton, Paris, Rome, THbingen,
Heidelberg, at the California Institute of Technology, and at the
Weizmann Institute, Rehovoth, and a Visiting Fellow at St. Catherine's
College, Oxford. His activities also included membership of the Fluid
Notion Committee of the Aeronautical Research Council of Great Britain.
In 1972 he was elected a Fellow of the American Academy of Arts and
Sciences and in 1973 he received the Brouwer Medal from the Dutch
Mathematical Society. He published nine books and over one hundred
papers in Pure and Applied Mathematics. In 1944 Robinson married Ren~e
Kopel of Vienna, Austria.
Robinson worked in widely separated areas of science. However,
the common denominator to much of his research was his interest in
apFlications. He was always fascinated by the problem of fashioning or
refashioning a formal framework in order to fit a given problem, whether
in Physics or in Pure Mathematics. Within classical Applied Mathematics,
he was concerned chiefly with Fluid Mechanics, more particularly with
the determination of the pressures and forces that act on a body in
flight, under steady or unsteady conditions, from subsonic to super-
sonic speeds (ref. 3)
Some of his better known contributions in this area were concerned
with delta wings and related shapes, while other papers dealt with
the motion of small bodies in a viscous fluid and with the propagation
of disturbances in fluids and solids. One of these led to an early
example of a precise theory for a mixed boundary value problem for
hyperbolic differential equations.
However, Robinson's major effort went into the study of the re-
lations between Logic and Mathematics proper. In his Ph.D. dissertation
"On the Metamathematics of Algebraic Systems" 1949 (published in 1951,
ref. )I he helped to lay the foundations of the branch of Logic now
known as Model Theory. He discussed, generally and in special cases,
the mutual relationship between sets of axioms and the classes of
structures (models) which satisfy them. The dissertation also contains
a number of effective applications to Algebra. Among them is the theo-
rem that an assertion X of the first order predicate calculus which
is true in all commutative fields of characteristic zero is true also
in all fields of characteristic P > Po where the natural number Po
depends on X . Among the basic tools which were introduced in the
same work is the "method of diagrams".
In 1954 Robinson produced a widely applicable test (the model
competeness test) for proving the completeness of various algebraic
theories ( a theory is complete if for any sentence in its vocabulary
it contains either that sentence or its negation). An outgrowth of the
line of thought which led to this test was the introduction of concepts
which provide far-reaching generalizations of the notion of an algebrai-
cally closed field relative to the class of commutative fields and which
embrace both previously known concepts, such as real closed fields and
new concepts such as differentially closed fields. Beginning in 1969,
Robinson introduced further generalizations of these notions by using
the forcing methods introduced originally by Paul Cohen in Set Theory.
By these means, Robinson was able to establish that even in Arithmetic
one can introduce structures which are analogous to algebraically clo-
sed fields, and that the theory of these structures is complete (or,
which ic the same, such that any two of these structures are elemen-
tarily equivalent). In another direction, Robinson showed that the
compactness principle of the lower predicate calculus implies the exis-
tence of certain numerical bounds, in particular for the representa-
bility of positive definite functions as sums of squares (1955) where
it had not been known previously, as well as in other cases, some
known, some unknown.
Perhaps Robinson's best known contribution is Nonstandard Analysis
(ref. 7). This area, which was introduced by him from 196o on, makes
use of model theoretic notions and contributions in order to provide
for the first time a satisfactory solution to the ancient problem of
developing the Differential and Integral Calculus by means of infini-
tesimals. It turned out that the ideas which led to Nonstandard Ana-
lysis can be generalized so as to apply also to topological spaces and
many other areas of Mathematics. The method has been used successfully
e.g. for the solution of problems in Functional Analysis and in Complex
Variable Theory and, more recently, in Mathematical Economics. It is,
in many cases, an alternative to familiar classical methods, but it is
too early to say how many mathematicians will choose to use it in their
field.
As a logician, Robinson was also keenly interested in the Philosophy
of Mathematics, although he published only a few papers in this area.
He was opposed to the so-called "Platonic realism" which holds that
mathematical objects and structures, even infinite ones, lead an inde-
pendent existence which defines their properties uniquely in all cases.
-- Adapted from the official biography
published by the Yale News Bureau.