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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. 686 APPLICATION OF SYMBOLIC LOGIC TO ALGEBRA 687 (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 688 ABRAHAM ROBINSON 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- 690 ABRAHAM ROBINSON 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 692 ABRAHAM ROBINSON 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 694. 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.