Table Of ContentLecture Notes ni
Mathematics
detidE yb .A Dold dna .B nnamkcE
454
maroJ Hirschfeld
William .H Wheeler
Forcing, Arithmetic,
noisiviD sgniR
Springer-Verlag
Berlin-Heidelberg • New York 1975
Authors
.rD Joram Hirschfeld
Department of Mathematics
leT Aviv University
Aviv Ramat
leT Aviv
learsI
.rD William .H Wheeler
Bedford College
University of London
London
England
Permanent address:
Department of Mathematics
Indiana University
Bloomington, Indiana 47401
USA
Library of Congress Cataloging in Publication Data
Hirschfeld, Joram.
Forcing, arithmetic, and division rings.
(Lecture notes in mathematics ; 454)
Bibliography: p.
Includes index.
1. Forcing (Model theory) 2. Model theory.
3. Division rings. .I Wheeler, William H., 1946-
joint author. II. Title. III. Series: Lecture notes
in mathematics (Berlin) ; 454.
QA3.L28 no. 454 [QAg.7] 510'.8 [511'.8] 75-12981
AMS Subject Classifications (1970): 02 H 05, 02 H ,31 02 H ,51 02 H 20,
08A20, 10N10, 10N15, 16A40
ISBN 3-540-07157-1 Springer-Verlag Berlin- Heidelberg" New York
ISBN 0-387-07157-1 Springer-Verlag New York • Heidelberg • 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,
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© by Springer-Verlag Berlin • Heidelberg 1975. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
In memory of
Abraham Robinson
ACKNOWLEDGEMENTS
We gratefully acknowledge the contributions of our colleagues
and coworkers to this volume. Deserving of special mention are
Mrs. S. Mandel for typing Part Two; L. Manevitz, D. Saracino,
.P van Praag, and P. M. Cohn for their valuable comments and criticism
of the manuscript; the members of the 1973-1974 Bedford College
(University of London) logic seminar, to whom much of the material
herein was presented, for their questions and comments; and the members
of the Department of Mathematics of Yale University, 1969-1972,
especially N. Jacobson, J. Barwise, G. Sacks, .E Fisher, M. Lerman,
J. Schmerl, and S. Simpson, for their instruction and suggestions.
We are indebted especially to Abraham Robinson, our adviser,
for his guidance and encouragement and for the proposal of this volume.
Finally, we thank our wives for their patience and moral support
during the writing and preparation of this volume; the second author
acknowledges in particular his gratitude to his wife for her labor
of love in typing Parts One and Three.
Joram Hirschfeld
William H. Wheeler
CONTENTS
INTRODUCTION
PART ONE ii
FORCING
CHAPTER 1 EXISTENTIALLY COMPLETE STRUCTURES AND 15
EXISTENTIALLY UNIVERSAL STRUCTURES
§ i Existentially Complete Structures 15
§ 2 Existential Types and Existentially 28
Universal Structures
CHAPTER 2 MODEL-COMPLETIONS AND MODEL-COMPANIONS 44
CHAPTER 3 INFINITE FORCING IN MODEL THEORY 55
§ i Infinite Forcing 55
§ 2 Model-companions and Infinitely Generic 69
Structures
§ 3 Subclasses of ~ 72
CHAPTER 4 APPROXIMATING CHAINS FOR ~ 76
CHAPTER 5 FINITE FORCING IN MODEL THEORY 86
§ 1 Finite Forcing 86
§ 2 Finite Forcing by Structures 93
§ 3 Finitely Generic Structures and 98
the Finite Forcing Companion
§ 4 Model-companions and Finite Forcing iOO
Companions
§ 5 Approximating Theories for 104
the Finite Forcing Companion
CHAPTER 6 AXIOMATIZATIONS iii
CHAPTER 7 FORCING AND RECURSION THEORY 117
§ 1 Degree of Unsolvability of F T 119
§ 2 Degree of Unsolvability of f T 124
§ 3 Consequences ef the Joint Embedding Property 12S
§ 4 Non-elementarily Equivalent Existentially 129
Complete Structures
S UMMARY 132
IV
PART TWO 137
ARITHMETIC
CHAPTER EXISTENTIALLY COMPLETE MODELS 141
§ 1 Models of T 141
w2
§ 2 Existentially Complete Models for Arithmetic 146
§ 3 The Definition of N in Existentially 150
Complete Models
CHAPTER 9 SIMPLE MODELS AND R. E. ULTRAPOWERS 155
CHAPTER iO REGULAR MODELS 160
Biregular Models 166
CHAPTER ii REGULAR MODELS AND SECOND ORDER MODELS 168
FOR ARITHMETIC
Biregular Models and Models 172
of Arithmetical Comprehension
CHAPTER 12 GENERIC MODELS AND THE ANALYTIC HIERARCHY 179
§ 1 Generic and Existentially Universal Models 179
§ 2 The Approximating Chains for ~ 182
§ 3 The Analytic Hierarchy 184
CHAPTER 13 APPLICATIONS TO COMPLETE EXTENSIONS 187
OF PEANO'S ARITHMETIC
PART THREE 191
DIVISION RINGS
CHAPTER 14 EXISTENTIALLY COMPLETE DIVISION RINGS 198
§ 1 Inner Automorphisms 199
§ 2 Transcendental Elements and Subfields 202
§ 3 Centralizers and Maximal Subfields 207
§ 4 Embeddings and Extensions 215
§ 5 The Number of Finitely Generated 219
Division Algebras
CHAPTER 15 NULLSTELLENSATZ 223
CHAPTER 16 CLASSES OF EXISTENTIALLY COMPLETE 232
DIVISION ALGEBRAS
§ 1 Structures for Second Order Arithmetic 233
§ 2 An Interpretation of the Language 236
of Second Order Arithmetic
§ 3 Second Order Arithmetic and Classes 241
of Existentially Complete Division Algebras
IIV
BIBLIOGRAPHY 253
SUBJECT INDEX 261
INTRODUCTION
Forcing in model theory is a recent development in the
metamathematics of algebra. The context of this development has three
principal features: the importance of algebraically closed fields in
commutative algebra and the existence of analogues of algebraically closed
fields for other algebraic systems, earlier work on model-completeness
and model-completions by Abraham Robinson and others, and Paul Cohen's
forcing techniques in set theory.
Algebraically closed fields serve a useful function in commutative
algebra, algebraic number theory, and algebraic geometry. Certain
arithmetical questions can be settled conclusively in an algebraically
closed field. Examples are well-known. For instance, a system of
polynomials has a common zero in some extension of their coefficient
field if and only if they have a common zero in the algebraic closure
of their coefficient field. In algebraic number theory, the study of
the prolongations of a valuation from its base field to a finite dimen-
sional extension field reduces to the consideration of the embeddings
of the extension field into the algebraic closure of the completion of
the base field. A third example is the use of universal domains in
algebraic geometry as the proper setting for the study of algebraic
varieties over fields. In these and other instances, the existence and
use of algebraically closed fields simplify the treatment of many
mathematical problems.
The usefulness of algebraically closed fields has motivated the
development of analogues for other algebraic systems. The best known
analogue is the class of real closed ordered fields, introduced by Artin
and Schreier for the solution of Hilbert's seventeenth problem on
ordered fields. Another important analogue is the Henselization of a
discrete, nonarchimedean valued field (see Ax & Kochen (4)). Analogues
have been introduced also for other algebraic systems, including groups
.W( R. Scott (iO1)), abelian groups .r( Szele (106)), modules .P( Eklof &
.G Sabbagh (34)), commutative rings .G( Cherlin (17)), commutative rings
without nilpotent elements .D( Saracino & .L Lipshitz (98), A. Carson
(12)), and on a more abstract level, universal algebra .P( M. Cohn (20),
B. Jonsson (49)).
While some analogues have been quite productive, others have
presented almost insuperable difficulties. Algebraically closed groups
are an example of the latter. In 1951 W. R. Scott defined a group G
to be algebraically closed if each finite system of equations and
inequalities of the form
I = w 2 e, = w e, ..., n = w I e, ~ v e, 2 ~ v e, ..., m ~ v e
(where the i w and vj are words in indeterminates and elements of G)
with a solution in some extension of G has a solution in G itself.
During the next year, .B H. Neumann (70) showed that only equations
need be considered in the above definition and that every algebraically
closed group is simple. No more work was done on these groups until
1969, when Neumann (71) proved that each finitely generated, recursively
absolutely presented group is a subgroup of every algebraically closed
group. Since algebraically closed groups are simple, they have not
been amenable to the usual methods of group theory. However, they can
be investigated through forcing techniques.
Algebraically closed fields have a special significance for
mathematical logic also. The theory of algebraically closed fields is
the canonical example of each of the following concepts: a complete
theory, a model-complete theory, a totally transcendental theory, and
an ~l-categorical but not ~O-categorical theory. Moreover,
algebraically closed fields were the starting point for the
metamathematics of algebra. Relationships between the theory of fields
and the theory of algebraically closed fields led to the concepts of
model-completeness and model-completion .A( Robinson).
From the point of view of algebra, the concept of a model-completion has
been the most useful metamathematical concept introduced by logicians.
For example, algebraically closed fields are the model-completion of
commutative fields; real closed ordered fields are the model-completion
of ordered fields; Hensel fields are the model-completion of discrete,
nonarchimedean valued fields; and Szele's algebraically closed abelian
groups are the model-completion of abelian groups. Both the Nullstellen-
satz and Hilbert's seventeenth problem are consequences of the model-
completeness of algebraically closed fields and of real closed ordered
fields, respectively. Furthermore, the notion of a model-completion
led A. Robinson to the definition and proof of the existence of
differentially closed fields.
The notion of a model-completion was weakened in 1969 by Eli Bers
to that of a model-companion. A theory T* is called the
model-companion of a theory T if )i( T and T* are mutually
model-consistent, i.e., each model of T is contained in a model of T*
and vice versa, and (ii) T* is model-complete, i.e., whenever M and
M' are models of T* and M' contains M, then any sentence defined
in M is true in M if and only if it is true in M'. Any model-
completion is also a model-companion. However, some theories, for
example, formally real fields, have a model-companion but not a
model-completion. This phenomenon occurs when the original theory does
not have the amalgamation property. The concept of a model-companion
encompasses all of the common, useful analogues in algebra of
algebraically closed fields.
The second requirement in the definition of a model-companion,
that T* must be model-complete, is the essence of "being algebraically
closed" and is also the more difficult of the two requirements to
satisfy. Robinson's model-completeness test demonstrates the first
assertion. A formula ~(Vo,... , Vn) in a first order logic is called
primitive if it consists of a string of existential quantifiers
