Table Of ContentCOMPLETE THEORIES
ABRAHAM ROBINSON
Associate Prof..cor of Mathematics
University of Toronto
1 9 5 6
NORTH-HOLLAND PUBLISHING CONIPAN~
AMSTERDAM
Cozzyright 1956
by
-
N. V. N 0 0 R D H 0 L LAN D S C H E UIT QE VER S Bl A ATS C H AP P IJ
AhtSTERDAM
Printed in the Nclltnlandr
PREFACE
The present treatise belongs to the metamathematics of Algebra,
The fact that this subject has in recent years attracted the active
attention of various eminent logicians has encouraged the present
author to contribute to it yet another monograph. A general
survey of the contents will be found in the first section of the first
chapter. Except where references to earlier papers are given, the
methods and results described here were obtained within the last
twelve months. Some of the work was reported upon in lectures
in Zurich (June, 1954) and Jerusalem (July, 1954) and at the
Symposium on the mathematical interpretation of formal systems,
which was held in conjunction with the International Congress of
Mathematicians (Amsterdam, September, 1954).
The author is indebted to B. Eckmann, L. Henkin, A. H. Light-
stone, and P. C. Gilmore for stimulating conversations on some
of the topics considered here. He also wishes to express his gratitude
to L. E. J. Brouwer, E. W. Beth, and A. Heyting, and to the
North-Holland Publishing Company for agreeing to publish the
present book in the series “Studies in Logic and the Foundations
of Mathematics”.
University of Toronto
February, 1955
ABRAHAM ROBINSON
I
GENERAL INTRODUCTION
1.1. Preliminary Remarks
It is known that a close integration of the methods of Abstract
Algebra and of Mathematical Logic can lead to results which are
of considerable interest to both subjects. The present work, like
another monograph which preceded it in the same series (ref. 16)
is based on this fundamental idea.
The book is chiefly concerned with completeness problems for
algebraic structures. The two most interesting algebraic notions
for which completeness has been established in the past are, the
concept of an algebraically closed field of given characteristic,
and the concept of a real-closed field. By introducing a modified
version of completeness, we are able to establish the property for
further important types of structures, in particular for certain
classes of algebraically closed fields with ib non-archimedean
valuation in an ordered abelian group. At the same time, we obtain
new proofs of the completeness of the concepts mentioned earlier.
The modified version of completeness alluded to above is called
“model-completeness)’. It owes its central position in our analysis
to the fact that there exists a relatively simple necessary and
sufficient condition for a set of axioms in the lower predicate
calculus to be model-complete, and another simple condition which
ensures in various interesting cases that a model-complete set is
also complete in the ordinary sense. However, the study of model-
complete systems also possesses a considerable interest of its own.
For example, it can be shown that any predicate which is defined
in the lower predicate calculus for a model-complete system can be
reduced to a particularly simple form. Another problem which is
tackled in the present book for certain particular cases is, how
and in how many ways an incomplete set of axioms can be made
complete by the adjunction of new axioms.
2 GENERAL INTRODUCTION
The last chapter of the monograph is concerned with a class of
syntactical transforms. In order to introduce this notion, we
observe that certain statements on normed algebras (groups, rings,
fields) can be obtained from statements on the corresponding
purely algebraic structures by a prescribed formal (syntactical)
transformation. For example, the statement that the sum x of
two elements x and y exists (in a ring, say) is transformed in this
way into the statement that the sums of any two elements x and y
exist and are bounded uniformly, for given bounds on x and y.
The operation which produces the latter statement from the
former is called a bounding transformation. It can be shown that,
for certain classes of statements, the provability of a statement
entails the deducibility of its bounding transform. The theory of
syntactical transforms is essentially independent of the com-
pleteness considerations contained in the earlier chapters of this
book but the two subjects possess common fields of application,
and there exist interesting arguments which make use of a com-
bination of the two theories.
It is known that metamathematical results of the type considered
here can be used to prove purely mathematical theorems, and a
restricted number of such applications is included. However,
it appears impracticable to try and exhaust all interesting examples
of this kind, and at any rate, this cannot be the task of a meta-
mathematical treatise. Rather, it is to be expected that in the
course of time, those active in certain branches of Mathematics
will train themselves to give due consideration to the application
of metamathematical arguments.
We shall suppose that the reader is familiar with the principal
methods and results of the lower predicate calculus. Individual
versions of this calculus differ as to the propositional connectives
which are adopted as primary, or as to the choice of the axioms,
or as to the inclusion of functors, or as to whether an axiom
involves particular formulae or all formulae of a certain type (so
that the axiom really is an axiom scheme), etc. However, all these
differences are inessential, in view of Godel’s completeness theorem
for the lower predicate calculus. Thus, in order to follow all but
LOWER PREDICATE CALCULUS 3
the last chapter of the present book, familiarity with a standard
text, e.g. ref. 11, is sufficient, In the last chapter, we make use of
Herbrand’s method in which functors play an essential part.
1.2. The Lower Predicate Calculus (LPC)
The particular version of the lower, or restricted, predicate
-
calculus which we shall use, here includes the propositional con-
nectives (negation), A (conjunction), A (disjunction), 3 (im-
plication), and = (equivalence). Relations, including relations of
order zero, will be denoted by Roman capitals, constants by small
Roman letters near the beginning of the alphabet, a, b, c, ...
(except in the case of standard mathematical entities such as
0, 1, 2, ...) and variables by small Roman letters near the end of
the alphabet, x,y , 2, . . .. Quantification takes place with respect to
variables only and is denoted by (x),( y) ... - for all x, y, ... - and
by (Rx)(,R y) ... - there exists an x, y, .... The domain of well-
formed formulae is determined by inductive definitions in the
usual way (compare ref. 16, section 2.3). Square brackets will be
used to indicate the order of concatenation in a well-formed
formula, and these will be omitted in accordance with accepted
conventions, e.g. between successive quantifiers, and between
successive disjunctions or conjunctions, in view of the associative
laws which hold for these operations. Brackets will also be omitted
in cases where a negation is followed by any other connective, or
where a disjunction or conjunction is followed by an implication
or an equivalence, on the understanding that the former connectives
tie the constituent formulae more closely than the latter. Well-
formed formulae without free variables are called statements, all
other well-formed formulae are called predicates. Functors (oper-
ators, or mathematical functions) are not included in our formal
language at this stage. For all practical purposes, an operator
of n variables can be replaced by a relation of order nf 1, provided
a relation of equality (i.e. a substitutive relation of equivalence)
is defined in the given system. For example, the operation of
addition can be expressed by means of the relation of order 3,
X(x, y, z) - read “z is the sum of x and y”. Thus, the concept of
4 GENERAL INTF&ODUCTION
Of these, the first three axioms state that E(z,y ) -read “x equals
y”-is a relation of equivalence while the fourth ensures sub-
stitutivity of equal elements. The fifth and sixth axioms state
the existence and uniqueness of the sum of any two elements and
these are followed by the associated and commutative laws of
addition. Finally, the ninth axiom asserts that the equation
x+z=y
always possesses a solution z, for given x, y.
For many of the subsequent arguments, it is only necessary to
know that a particular mathematical concept can be expressed by
a finite or infinite system of axioms in the LPC while the particular
form of the axioms is irrelevant. A concept which can be so for-
mulated is said to be elementary (ref. 24), or elementarily dehable,
in terms of certain relations and constants. Thus, the concept of
an abelian group is elementarily definable in terms of the relations
of addition, S(x,y , z), and of equality, E(x, y).
Among other concepts which are elementary we mention:
- The concept of a commutative field, definable in terms of the
relations E(x,y ) (“x equals y”), S(x,y , z) (“z is the sum of x and y”)
and P(x,y , z) (“2 is the product of x and y”), and without con-
STRUCTURES, MODELS, DIAGRAMS 6
stants. It will be taken for granted that a field contains at least
two elements.
- The concept of an algebraically closed field, definable in
terms of the same relations. The number of axioms required to
formulate this concept is infinite.
- The concept of an algebraically closed field of given oharac-
teristic p. For p>O, this requires only the adjunction of tb single
axiom Xvw hich states, translated into ordinary language, that the
successive addition of any element to itself, n times, yields the
neutral element with respect to addition. For characteristic zero, we
-
include instead the sequence of- s tatements
{- x,,- x3, x,,.. . x,,. . ]
,
where Xv is defined as above.
Similarly, the concept of a commutative ordered field is elemen-
tarily definable in terms of the relations E, X, P, and of the
additional relation &(x, y) (read “x is smaller than OP equal to y”),
and without constants.
Finally, a real-closed ordered field can be cha.racterised as a
commutative ordered field in which every positive element possesses
a square root, and every polynomial of one variable of odd degree
possesses at least one root. These two conditions can be formulated
without diEiculty in the LPC, in terms of the relations E, 8, P,Q ,
although an infinite sequence of axioms is required to express the
second condition. Thus, the notion of a real-closed ordered field
also is elementarily definable.
Other elementary concepts will be introduced in due course.
1.3. Structures, Models, Diagrams
A mathematical structure M of the type to be considered here
is based on a non-empty set of relations P and a (non-empty) set
of constants C such that for every relation of order n which is
contained in P, R(xl, ..., xn) say, and for every ordered n-uple of
elements of C, a,, ..., a,, say, R(%, ..., a,) either holds or does not
hold in Af. We may, if we wish, assume that P includes relations
of order zero, and in that case C may be empty. However, no such
cases are considered in standard algebra.
6 GENERAL INTRODUCTION
Given a structure M, one assumes implicitly that the formal
language of the LPC is sufficiently comprehensive, so that the
relations and constants of M can be correlated in one-one cor-
respondence with some of the relations and constants of the
language. To simplify matters, one may even assume that the
relations and constants of the structure actually belong to the
language and denote themselves (compare ref. 17) and this course
will be adopted here. Then the expressions R(a,, ..., a,,), R E P,
a, E C, i = 1, .. . , n, become statements of the LPC and starting
from these one determines in a natural and well-defined way
whether a given statement X which is defined in M, does or does
not hold in M. For example, a statement X v Y holds iri M, by
-
definition, if and only if at least one of the statements X or Y holds
in M, and a statement X holds in M if X does not hold in M
although it is defined in that structure. In this connection we say
that a statement X is defined in a structure Jl, if all the relations
or constants which occur in X belong to the sets P and C on which
M is based, respectively. A statement (3x)Y(x)h olds in M if
there exists a constant a EC such that Y(a) holds in M, etc.
A structure M is said to be a model of a set of statements K
if all the statements of K are defined in M and hold in that
structure.
Let M, M be two structures, and let P and C, and P’ and C’ be
the sets of relations and constants on which these two structures
are based, respectively. Then M’ is said to be an extension of M if
P C P’, C C C’, and if for all R(x,, ..., x,) E P, a,, ... a, E C, the
relation R(a,, .. . a,) holds in M‘ if and only if it holds in M. Also,
under these conditions M is called a partial structure of M‘.
For a given structure M,l et N be the set o-f s tatements R(a,, ... a,)
whenever the statement holds in M, and of R(a,, . . . a,) whenever
R(a,, ... a,) does not hold in M although it is defined in that
structure, i.e. although R and a,, ... a, belong to P and C respec-
tively. N is called the diagram of M (“complete diagram” in
ref. 16). If we are willing to accept the idea that SL mathematical
structure consists of a set of statements, we may even identify a
structure with its diagram.
UODEL’S THEOREM AND ITS EXTENSIONS 7
1.4. Godel’s Theorem and its Extensions
A statement in the LPC is said to be “valid” or “provable” if
it is a consequence of the axioms and rules of deduction of the
calculus. Godel’s theorem (ref. 4) states that this calculus is com-
plete in the sense that for every statement X which is not provable
there exists a structure M such that N X holds in M.
A set of statements K is said to be contradictory if it includes
statements XI,.. ., X,, such that for any statement Y,
1.4.1. X1A ...
is provable. It follows from the rules of the propositional calculus
that in order to prove a set K contradictory it is sufficient to
establish 1.4.1. for a single statement Y which is of the form
Y =Z A N 2. The extended completeness theorem states that a
set of statements K is either contradictory or else it possesses a
model M (compare refs. 1, 7, 14, 16). Moreover, if K is finite then
we may assume that the cardinal number of M (i.e. the cardinal
number of the constants of M) does not exceed &,, while if R is
infinite then K possesses a model whose cardinal number does not
exceed the cardinal number of all statements in K.
A set of statements K is said to be consistent if it is not contra-
dictory. The extended completeness theorem then amounts to
saying that a set of statements is consistent if and only if it
possesses a model. It follows from the finitary character of the
rules of the LPC that a set of statements is consistent if and only
if all its finite subsets are consistent. Hence, a necessary and
sufficient condition for a set of statements R to possess a model is
that every finite subset of K possesses a model.
A statement Y is said to be deducible from a set of statements
K if K contains statements X,, ..., X,, such that 1.4.1 is provable.
We include the possibility that X,, .. . , X,, is empty, in which case
1.4.1 is supposed to reduce to Y. It follows from the extended
completeness theorem that if a statement Y holds in all the models
of a set of statements R, then Y is deducible from K.
The term “completeness”, which was used above in connection
