Download ON THE APPLICATION OF SYMBOLIC LOGIC TO ALGEBRA1 1

ON THE APPLICATION OF SYMBOLIC LOGIC TO ALGEBRA1
ABRAHAM ROBINSON
1. Symbolic logic has been used to clarify a variety of subjects in philosophy
and epistemology; it has permeated directly or indirectly some of the new "operational" sciences; the calculi of classes and of relations which, at least historically,
were evolved as branches of logic, have been applied to various combinatorial
problems and to biology. However, in the present paper we shall be concerned
with the effective application of symbolic logic to mathematics proper, more
particularly, to abstract algebra. Thus, we may hope to find the answer to a
genuine mathematical problem by applying a decision procedure to a certain
formalised statement. While the practical possibilities in this direction, though
limited, may be quite real, the present paper will be concerned with applications
which are rather less intimately connected with any particular deductive procedure, and rather more with the general relations between a system of formal
statements and the mathematical structures which it describes. The argument
will be developed from the point of view of a fairly robust philosophical realism
in mathematics, and it is left to those to whom this point of view is unacceptable
to interpret our undoubtedly positive results according to their individual outlook. Thus we shall attribute full "reality" to any given mathematical structure,
and we shall use our formal language merely to describe the structure, but not to
justify its "reality" or "existence" whichfis taken for granted.
There is no room here for a detailed historical survey, but perhaps we may
mention the names of K. Godei, L. Henkin, and A. Tarski as representative of
those who either directly or indirectly contributed towards the establishment of
symbolic logic as an effective tool in mathematical research. In particular, it is
understood that the theorem on algebraically closed fields which is proved below
and which was stated by the present author in 1948 (see J. Symbolic Logic
vol. 14 (1949) p. 74) has also been found independently by Professor Tarski.
2. The particular mathematical structures which we shall consider are exemplified by the ordered field of all real numbers, any specific group, or any specific
ring, and are, more generally, sets of objects a,b,c, • • • and of relations R(, • • • ),
S(, • • • ), T(, • • • ), and (possibly) of functors $(, • • • ), \p(, • • • ) such that
any particular R(a, b, • • • ), etc., either holds or does not hold, and such that
there is just one object a which is the functional value of any given <ß(b, • • • ).
Our formal language, on the other hand, will coincide roughly with the lower
predicate calculus, that is to say, in more detail, it will contain the following
atomic symbols:
(i) Object symbols a, b, c, • • • and dummy symbols u, v, w, • • • where the
former are used as "free variables" and the latter as "bound variables." The
rigid distinction between the two classes is important in the present context.
1
This address was listed in the printed program under the title Applied symbolic logic.
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APPLICATION OF SYMBOLIC LOGIC TO ALGEBRA
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(ii) Relative symbols R(, • • • ), $ ( , • • • ) , T(, • » • ), and functor symbols
•*(, • • • ) , * ( , • • • ) •
We use the word "symbol" in (i) and (ii) in order to differentiate between the
entities in question which belong to our formal language and the corresponding
entities of specific mathematical structures. Though not unavoidable, this distinction is very convenient.
(iii) Propositional copulae ^ , A. V, Z>, ==. and quantifiers (x), (Sx).
(iv) Brackets.
Well formed formulae are produced in the usual way, and statements are defined as well formed formulae which do not include unbound dummy symbols.
All other well formed formulae are called predicates. In order to have in mind
a concrete example we may write down the commutative law of addition for an
abelian group. Its form still depends on the choice of our relative symbols and,
possibly, functor symbols. For instance, if we introduce the relative symbol
JS(x, y, z) ("z is th® sum of x and y") as well as a relative symbol for equality,
E(x, y), the commutative law may be expressed as
(x)(y)(z)(t)[S(x, y, z) A 8(y, x} t) 3 E(z, t)].
On the other hand, if we replace S by the functor o~(x, y) (i.e., "sum of x and
2/"), then the law in question may be written as
(x)(y)[E(a(x, y), a(y, x))].
There is no fundamental distinction in this formulation between the relative
symbol of equality and any other relative symbol.
Having established the domain of well formed formulae, we may then develop
the deductive calculus of "valid" or "analytic" statements, basing it on a set of
axioms and rules of inference in the usual way, except that we have to take
the inclusion of functor symbols into account. On the other hand, we still have
to interpret our statements, which so far are merely sequences of symbols, i.e.,
we have to define recursively under what conditions a statement will be said to
hold in a given structure. It is clear that the question whether a particular statement holds in a specified structure may still depend on the correspondence between the relative symbols of the language and the relations of the structure;
and some reflection shows that it may also depend on the correspondence between
the object symbols and objects of language and structure respectively.
Finally, we have to establish the correlation between deductive and semantic
concepts as given above. Thus, it must and can be shown that a set of statements
K which is deductively contradictory cannot possess a model, i.e., a structure
in which all its statements hold. But the converse also is true, i.e., any set K that
is deductively consistent (not contradictory) possesses a model. This is the extended "completeness theorem" which holds whatever the cardinal number of
the set K and of the object, relative, and functor symbols contained in it. It
follows that if a statement X holds in all structures in which a set of statements
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K holds, then X can be deduced from K; and by the rules of the deductive calculus this implies that X can even be deduced from a finite subset of K.
3. A convenient starting point for the application of the logical framework
sketched above is obtained when we reflect that in "orthodox" mathematics it is
our normal business to prove a particular statement or theorem either for a
specific structure, e.g., for the ring of rational integers, or for all structures which
obey a specific set of axioms, e.g., for all groups. It is now natural to try and
establish principles which do not refer to any specific theorem, but apply to all
theorems or statements which belong to a certain class. For instance, we may
try to prove that any statement of a certain class which is true for one particular
type of mathematical structure is also true for another type. Such a metamathematical theorem may be called a "transfer principle." It is exemplified by the
classical principle of duality in projective geometry which is logically so simple
that no formal apparatus was required to establish it. However, in other cases,
the use of symbolic logic becomes indispensible, if only in order to delimit the
class of theorems to which the principle applies. We shall now state three transfer
principles which belong to different types, sketch their proofs, and apply one of
them to the demonstration of an actual mathematical theorem whose proof by
more conventional methods is not apparent.
"Any statement X, formulated in the lower predicate calculus in terms of the
relation of equality and the operations of addition and multiplication, which is
true for all commutative fields of characteristic 0, is true for all commutative
fields of characteristic p è Po where po is a constant depending on the statement
X."
To prove the theorem, let if be a (finite) set of axioms for the concept of a
commutative field, formulated after the manner indicated above, in terms of
E(x> y) ("x equals y"), S(x, y, z) ("z is the sum of x and y"), and P(x, y, z)
("z is the product of x and y"). It is easy to construct such a set K, and we may
or may not include object symbols for 0 and 1, and functor symbols for sum
and product, having already introduced S and P. Furthermore, it is not difficult
to formulate a sequence of statements Yn such that any Yn is interpreted semantically as:
"There exists an element x such that x + • * • + x (pn times) is different from
0, where pn is the nth. prime number."
A set of axioms for the concept of a commutative field of characteristic 0 is
now given by H = Kl) {Yi, Y2, • • • }, so that X holds in all models of H, by
assumption. It then follows from the extended completeness theorem that X
can be deduced from H, and hence that it can be deduced from some finite subset of H. Thus, X can be deduced from some set H' = Kl) [ Yx, Y2, • • • , Yn,},
for some nf. But Hf is satisfied by all commutative fields of characteristic
greater than pn* and so X holds in all such fields.
In spite of the simplicity of its proof the above principle can be used to es-
APPLICATION OF SYMBOLIC LOGIC TO ALGEBRA
689
tablish mathematical theorems whose demonstration by other methods is not
apparent. For example,
"Let Qi(xi, • • • , xn) = 0, i — 1, 2, • • • , k, be a system of polynomials with
integral coefficients which have no more than m roots (£i, • • • , £„) .in common
in any extension in the field of rational numbers. Then if we take the coefficients
of the qi modulo p, the resulting polynomials cannot have more than m roots
in common in any field of characteristic p *z p0, where pQ depends on the given
system."
Taldng into account the above transfer principle, we have to show only that
the statement, "There cannot be more than m different solutions to the system
of equations qi(x\, • • • , xn) = 0" can be formalised within our language, where
the integral cogfficients are taken as operators indicating continued addition.
A transfer principle of a different type is the following:
"Any statement formulated in terms of the relations of equality, addition,
and multiplication, as above, that holds in the field of all complex numbers, holds
in any other algebraically closed commutative field of characteristic 0."
This is equivalent to the statement that an axiomatic system corresponding
to the concept of an algebraically closed field of characteristic 0 (or indeed of any
other characteristic) is complete in Tarski's sense, as defined in his calculus of
systems. However, the statement of this fact as a transfer principle is more suggestive from the point of view of a working mathematician. In the proof, considerable use will be made of the results of Steinitz' field theory and this may be
regarded as an interesting example of what can be done by the close integration
of logic and algebra, or as a flaw in purity, according to taste.
Let X be any statement formulated in terms of equality, addition, and multiplication, as above, and let H be a set of axioms for the concept of a commutative
field of characteristic 0, as before. It is not difficult to construct statements Zn
which assert that "every equation of nih degree whose highest coefficient does
not vanish possesses a root," n — 2, 3, • • • . Then J = H\i{Z2, Z%, • • • } is a
set of axioms for the concept of an algebraically closed field of characteristic 0,
where we may assume for simplicity that J does not contain any object symbols.
Let a1, a2, • • • , an , • • • be a countable sequence of object symbols, and let
F be a set of statements affirming that no polynomial relation with integral
coefficients, p(a\, • • • , an) = 0, exists between these an , or rather, between the
objects corresponding to them. Then F is countable and so therefore is the set
G = FÖJ. G expresses the concept of an algebraically closed field whose degree
of transcendence is è No, or more precisely, every model of G must be such a
field while every such field becomes a model of G if we match (let correspond)
the ai, 0.2 , • • • with algebraically independent numbers of the field.
If the set Gf = 6rU {X} is consistent, then it follows from the theorem of Skolem
and Löwenheim (or from the construction required for the proof of the completeness theorem), that Gf is satisfied in a countable field M', whose degree of transcendence therefore is exactly N 0 . Similarly, if the set G" = GU[~X] is con-
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sistent, it possesses a model Mn whose degree of transcendence is exactly No.
But by a theorem due to Steinitz any two algebraically closed fields of equal
characteristicm and of equal degree of transcendence are isomorphic, e.g., Mf
and M" are isomorphic, and this would imply that both G' and G" hold in both
Mf and M". This is impossible and so either G' or G,f is contradictory, i.e., either
~X or X is deducible from G, more precisely it is deducible from a finite subset
of G. Assuming for the sake of argument that the second alternative applies,
this signifies that X holds in all algebraically closed fields of characteristic 0 which
contain elements satisfying a finite number of inequalities Pi(a\, • • • , ak) 5^ 0.
It is easy to find elements in any algebraically closed field of characteristic 0
which satisfy this condition. Thus either X holds in all algebraically closed fields
of characteristic 0, or the same applies to ^X, and this }is equivalent to the
theorem stated above.
Next we come to a transfer principle of yet another type, which requires a
more difficult method for its proof.
"Let R be the field of rational numbers, and let S = R[x±, x2, • • • ] be the
field obtained by adjoining to R a sequence of indeterminate elements (S is the
field of all rational functions of Xi, x2, • • • with rational coefficients). Let K
be the set of all statements formulated in terms of equality, addition, and multiplication, as above, which hold in R. Then there exists a field S' =! S such that
the elements of S' — £ (if any) are transcendental with respect to S, and such
that all the statements of K hold in £'."
I t may be conjectured that we might take S' = S, but the available proof
appears to be inadequate for establishing this. 2
We observe that we are now referring to a specific field (for which an explicit
set of axioms is unknown) and not, as formerly, to all the models of a given set
of axioms. It is for this reason that the theorem cannot be obtained by the direct
application of a completeness theorem. The proof depends on the introduction
of additional functor symbols by means of which the statements of K are replaced by statements containing universal quantifiers only. Use is made of a purely
algebraic theorem, due to Skolem, which states that if a polynomial p(x, y,z, • • • ;
t) with integral coefficients has a rational integral root t for integral values of
x, y, z, " • which are "dense" in a certain sense, then p possesses a linear factor
t ~ fffo y, • • • )Let p(xi, x2, • • • ; 2/1, 2/2, • • • ) be a polynomial with integral coefficients,
where the 2/1, 2/2, • • • are regarded as variables and the Xi, x2, • • • as parameters. The property of any such problem to be (relatively) irreducible in R, S,
or S' may be regarded as a predicate of the Xi, x2, • • • . Using this fact, we
may show by means of a simple application of the above transfer principle that
if p—taken as a polynomial of the variables y\, y2, • • • —is irreducible for indeterminate Xi, x2, • • • , then it must be irreducible for an infinite number of
rational values of x±, x2, • • • . This result is slightly weaker than Hubert's
well-known irreducibility theorem, according to which integral values may be
2
Added in proof. More precisely it can be shown that S' must be different from S.
APPLICATION OF SYMBOLIC LOGIC TO ALGEBRA
691
taken for Xi, x2, • • • , but it is all that is needed for the algebraic applications.
However, the elegance of our proof is reduced somewhat by the fact that we
have to show first that if p is irreducible in S, then it must be irreducible in Sf.
Also, it should be mentioned that the above mentioned lemma was formulated
by Skolem precisely in order to prove Hubert's theorem, or rather to improve it.
Let öL be any irrational algebraic number. Then it is not difficult to show
that all the statements formulated as above that hold in R(a) also hold in S'(a).
From this fad we may again deduce a modified version of the counterpart of
Hilbert's theorem for algebraic extensions of the field of rational numbers.
4. Another train of thoughts leading to the application of symbolic logic to
mathematics is prompted by the idea that instead of discussing the properties
of algebraic structures defined by specific sets of axioms, we may consider
structures given by different sets of axioms simultaneously. We define an "algebra of axioms" as any set of axioms formulated in the lower predicate calculus
which includes a relative symbol of equality, i.e., a binary relative symbol which
satisfies axioms of equivalence as well as substitutivity. A model of an algebra
of axioms will be called an "algebraic structure". We may compare these definitions with the concept of a general algebra as given by G. Birkhoff. Birkhoff's
definition includes only axioms of an equational type, such as the associative
and distributive laws. We may extend the domain of admissible axioms slightly
by including "relations" of constants, e.g., a¥ = e, although it is customary, in
algebra, to differentiate between these "relations" and the "axioms". Even so,
it can be shown that no system of axioms for a Birkhoff algebra as defined can
realize the concept of an algebraic field. The proof of this assertion depends on
the fact that one of the ordinary field axioms, that which stipulates the existence
of an inverse for multiplication, includes a disjunction, and for that reason
cannot be deduced from any set of axioms for a Birkhoff algebra as defined
above. Being more restricted, the concept of a Birkhoff algebra is correspondingly
more definite, but the fact that it does not include the concept of a field shows
that a more general theory is justified. Since an algebra of axioms contains a
relative symbol for equality, it is clear that all functor symbols can be replaced
by relative symbols and the former may therefore be omitted for the sake of
simplicity.
The question now arises whether the concept of an algebra of axioms as defined above is sufficiently definite to permit the development of a general theory.
More specifically, we may take a standard concept of algebra and may ask
whether it can be defined jointly for all algebras of axioms (and algebraic structures) in such a way that it is not merely analogous to the particular concept
from which it is taken, but that it actually reduces to its prototype for the
particular case in question. For example, let us consider the concept of a polynomial ring of n variables adjoined to a given ring. A close investigation of the
concept of a polynomial in algebra shows—in contradistinction to the homonymous concept in the theory of functions—that it cannot be regarded simply as a
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certain t y p e of function in a specific structure. I n fact, t w o polynomials w i t h
coefficients in a field of characteristic p m a y well take t h e same values in t h a t
field without being identical. Accordingly, we prefer t o define polynomials as
constructs of a formal language. T h u s , given a n y algebra of axioms K (e.g.,
formulated in terms of a relative symbol for equality EQ a n d of two t e r n a r y
relative symbols S(„) a n d P ( „ ) , for t h e case of a commutative ring), we m a y
consider all predicates Q(x±, • • • , xn, y) formulated in terms of the constants
of K, such t h a t y is "uniquely" determined b y Xi, • • • , xn , i.e., such t h a t
(xi) • • • (xn)(3y)(z)[Q(x1,
-•- ,Xn,y)
A [Qfa , • • • , xn , z) => E(y, z)]]
can be deduced from K. W e now define a n algebraic structure MK whose objects
are t h e predicates just selected, a n d whose relations are determined in t h e following way. If R(„) is a n y relative symbol in K (ternary, say), then a relation
R*(Qi j Q2, Q3) shall hold between predicates Q i , Q2, Q3 as defined above whenever t h e statement
(xi) - < • (xn)(yi)(y2)(yz)[Qi(xi,
• • • , xn , 2/1) A Q2(XX , • • • , xn , y2)
A Qs(si, • • • , xn , 2/3) => R(yi,
2/2, yùì
is deducible from K. T h e structure MK t h u s defined does not always satisfy all
t h e statements of K<(ìn a n a t u r a l correspondence), b u t it does so for wide classes
of algebras. I n particular, if K is a set of axioms for t h e concept of a commutative ring, MK also is a commutative ring. MK is n o t itself isomorphic t o t h e
ring of polynomials of n variables with integral coefficients b u t it contains a
substructure MK which possesses this property a n d which can again be characterised in a perfectly general formal w a y .
Another important concept which can be generalised in t h e sense detailed
above is t h a t of a n ideal. I t would appear natural t o take t h e homomorphisms
of a n y algebraic structure as t h e counterparts of t h e ideals of a ring, b u t it is
difficult t o formulate some of t h e familiar notions of ideal theory in t h a t framework, e.g., t h e notion of a basis. Here again it appears t o be more convenient t o
formulate t h e generalised concept b y means of deductive considerations within
a formal language, with respect t o a given algebra of axioms, K. For instance,
we m a y define an ideal as t h e set of statements J which can be deduced from K
together with an additional set of statements which "identify" (i.e., make equal)
certain individual constants, or objects, which were previously supposed different, E(a, a'), E(b, &'), etc. This yields a general concept which reduces t o t h e
notion of an ideal, or of a normal subgroup, for rings a n d groups respectively.
Other examples of concepts which can b e so generalised are: t h e concept of a n
algebraic number and, more generally, of a number which is algebraic with
respect t o a given commutative field; t h e distinction between a separable a n d
a n inseparable extension of a given field; t h e concept of a power series ring
associated with a given ring. T h u s , t h e generalisation of t h e last concept is
based on t h e following definitions.
L e t JQ 2 «/"i 3 • • • Z> J « 2 • • • be a fixed descending chain of sets of s t a t e -
APPLICATION OF SYMBOLIC LOGIC TO ALGEBRA
693
ments formulated in the lower predicate calculus. A sequence of statements
{Xn} = {X 0 , X i , • • • } will be said to be deducible from {/„}, if for every n,
Xp is deducible from Jn for sufficiently high p. The sequence will be called convergent with respect to {Jn) if for every n, Xp ss Xq is deducible from Jn for
sufficiently high p and q. There are similar definitions for sets and sequences of
predicates, and from one of those the concept of a power series ring adjoined to
a given ring can be obtained by specialization (and not merely by analogy).
As in the algebraic case, the general theory indicated here can be expounded
conveniently by the introduction of a valuation.
5. Since all the above concepts were formulated in terms of statements, or
axioms, in a formal language, the question may be asked how we can relate
them to particular structures. Let K be a set of axioms as above, e.g., for the
notion of a commutative ring, and let M be one of its models. We now add to
the symbols of K, object symbols corresponding to all the objects of M which
are not already designated by object symbols in K, and we add to the statements
of K, all statements of the form R(ai, a^ , • • • ) or ~R(a±, a2, • • • ) according
as the corresponding relation does or does not hold in M. The set of such statements may be called a "diagram" D of M. Every model of M' of K' = K U D
is an extension of M, or more precisely it contains a substructure isomorphic
to M, and if we construct the structures MK> and M'K> as above, then MK> turns
out to be the polynomial ring of n variables adjoined to M.
The concept of a diagram, in conjunction with the completeness theorem also
permits us to establish algebraic theorems whose proof by conventional methods
is not apparent. A simple example is as follows.
"Let Po, P\, P2, • • • be an infinite sequence of finite sets of polynomials of n
variables in a skew field F (where the variables do not necessarily commute
with the elements of F or with each other). Then if in every skew extension
F' of F, in which all the polynomials of P 0 have a joint zero, all the polynomials
of at least one Pi, i ^ 1, have a joint zero, it follows that there exists a finite
set P i , • • • , Pfc , such that in every Fr __2 F in which all the polynomials of P 0
have a joint zero, all the polynomials of at least one P . , 1 ^ i ^ /c, have a
joint zero also."
There is a similar theorem for nonassociative fields. It was stated at the
beginning that we attribute an equal degree of reality to a mathematical structure and to the language within which it is described. Accordingly we may introduce notions which are defined partly with reference to a given algebra of
axioms, and partly with reference to its models. For example, an algebra of
axioms will be called convex if, roughly speaking, the intersection of any number
of its models is again a model. A number of interesting theorems can be proved
concerning such algebras; thus it can be shown that the union of an increasing
chain of models M0 £__. Mi £__. • • • ÇL Mn £1 • • • of a convex algebra of axioms
K is again a model of K.
A considerable number of the results of standard algebra can be extended to
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ABRAHAM • ROBINSON
the more general concepts mentioned above, as shown in a 1949 London thesis
which will be published shortly. However, the concrete examples produced in
the present paper will have shown that contemporary symbolic logic can produce useful tools—though by no means omnipotent ones—for the development
of actual mathematics, more particularly for the development of algebra and,
it would appear, of algebraic geometry. This is the realisation of an ambition
which was expressed by Leibnitz in a letter to Huyghens as long ago as 1679.
COLLEGE OF AERONAUTICS,
CRANFIELD, ENGLAND.