Metamathematical Problems

Metamathematical ProblemsAuthor(s): Abraham RobinsonSource: The Journal of Symbolic Logic, Vol. 38, No. 3 (Dec., 1973), pp. 500-516Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2273049 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 38, Number 3, Sept. 1973

METAMATHEMATICAL PROBLEMS

ABRAHAM ROBINSON

Retiring presidential address presented at the annual meeting of the Association for Symbolic Logic in Dallas, January 1973.

When a logician approaches the world of mathematics, he may have in mind one or more of several purposes. He may try to find in mathematics a framework for formalizing commonly accepted laws of thought or perhaps laws of thought that are not commonly accepted. He may want to assist the mathematician by providing him with firm foundations for his theories. But it may also be the case that the logician wishes to use his own characteristic tools-formalized languages, explicit relations between symbols and objects, rigidly expressed and controlled rules of deduction-in order to gain a better understanding of the various and variegated kinds of structures, methods, theories and theorems that are to be found in mathematics. We may then expect him to adopt the attitude of the physicist or psycholo- gist who (whatever his professed philosophy) feels that he deals with phenomena of the external world, whose rules cannot be imposed by him arbitrarily. He, or those that come after him, may indeed use the understanding thus gained in order to modify these phenomena, but as a scientist he would not regard this possibility as his only justification.

For many years now, I have concentrated on the third of the lines of approach sketched above, and it seemed natural that I should discuss it again on the present occasion. However, today I do not wish to emphasize past developments but, using some of them as a background, I propose to enumerate a number of open problems. These problems seemed to me of some interest not only for their own sake but also because their solution might well require weapons whose introduction would close definite gaps in our armory. A few of my questions are classical and are well known to many workers in our field while others are put on the map here for the first time. Again, some of the problems require only a clear "yes " or " no " as answers while others call for the development, or expansion, of a whole topic, and with reference to them it would be idle to speak of the answer.

I have selected my problems from twelve different, though sometimes interrelated areas of mathematics. Without further ado, here they are.

?1. Decision problems in field theory. We shall be concerned with decision problems for individual fields, more particularly fields of characteristic zero. The

Received March 19, 1973. Research supported in part by the National Science Foundation, grant no. GP-29218. The

author is indebted to K. Godel and S. Kochen for valuable conversations on some of the topics discussed in this paper.

500 ? 1973, Association for Symbolic Logic



METAMATHEMATICAL PROBLEMS 501

question is either to produce a program, or systematic procedure, by which it can be decided whether any given sentence formulated in the lower predicate calculus in terms of addition, multiplication, and equality is true or false for the field in question-or else, to prove that no such procedure can possibly exist. Conspicuous among the decidable fields are the field of complex numbers, the field of real numbers, and the field of p-adic numbers, for any prime p. Among the undecidable fields are the field of rational numbers Q. and all finite algebraic extensions of Q. A number of years ago, Tarski raised the question whether the field of constructible numbers is'decidable. In the sense in which the term is used here, this is the field whose elements measure the length of segments obtained by ruler and compass from an initial segment, of length 1, by convention. In algebraic terms, the same field is obtained by the indefinitely repeated adjunction of square roots of positive numbers to the field of rationals Q. where Q and all the intermediate fields are now regarded as ordered. There is a variant of this problem, in which square roots of all numbers are adjoined successively. If the resulting field is undecidable, so is the field of constructible numbers. However both of these problems are still unanswered. In fact, I do not know of any proper subfield of the field of algebraic numbers, other than the fields of algebraic real or p-adic numbers, that has been shown to be decidable. A particular class of fields that merits investigation in this connection is the maximal abelian extensions of finite algebraic extensions of the rationals. More specifically-Is the maximal abelian extension of the field of rational numbers (i.e., the compositum of all cyclotomic fields) decidable? A more general question which belongs to this area is the following: Is a finitely generated extension of an undecidable field always undecidable?

The standard tool available in order to prove the undecidability of a given field is to show that the natural numbers N are definable in it (and it would be sufficient to show that N is decidable in terms of it). In fact, no field has been proved undecidable in any other way. On the other hand, a decision method may be worked out directly by means of an effective procedure for the elimination of quantifiers, or its existence may be proved by means of model theoretic results related to notions of completion or model completion. When a structure is proved decidable because it can be shown that it satisfies a recursively enumerable and complete set of sentences this provides a (general) recursive decision procedure. By contrast, all the decision procedures for fields that have been worked out directly by elimination of quantifiers are primitive recursive. It would be interesting to have a satisfactory explanation of this phenomenon.

?2. Generic arithmetic. Among all commutative fields, the algebraically closed ones can be singled out, intuitively speaking, by a special kind of completeness. It was first realized by Artin and Schreier that this phenomenon has an analogue else- where and it was thus that they introduced the notion of a real closed field. It is a familiar fact that when one tries to make an intuitive notion precise, one may be faced with several reasonable "explications." Such is the case here also. Thus, let. Z be the class of models given by a first order theory supposed inductive (i.e., such that I is closed under union of chains). A structure M E Z is called existentially complete (in A) if for any existential sentence X in the vocabulary of M (i.e.,



502 ABRAHAM ROBINSON

including constants for the individuals of M) such that X is satisfied in some M' - M, M' E X, the sentence X is satisfied already in M. The class of existentially complete structures E is a subclass of E. If E is the class of commutative fields then E is the class of algebraically closed fields. Returning to the general case, let G be a subclass of E which satisfies the following conditions: (i) Every M E E can be embedded in an M' E G; (ii) for any M, M' E G, M c M' entailsM -< M', Mis an elementary substructure of M'; (iii) if M E E, M' E G, and M -< M' then M E G. It turns out that there is one and only one subclass of E which satisfies these conditions. If E is the class of commutative fields, then G is again the class of algebraically closed fields. However, generally speaking, G may differ from E although it is always a subclass of E. If either one of E and G is axiomatizable (given by a set of first order axioms) then E and G coincide. Even if they are not axiomatizable in the first order sense they can both be axiomatized by sentences in some infinitary language L,,,,. G has the property that all the structures which belong to it are elementarily equivalent if and only if I has the joint embedding property. This is the case in particular if E is the class of all subrings of nonstandard models of the full first order theory of rational integers T. The theory TG of G (for reasons that we need not go into here) is then called generic arithmetic. It follows from what has been said that TG is complete. However TG #0 T. In fact, up to a certain point TG coin- cides with ordinary rational arithmetic; in particular, it satisfies a limited form of the axiom of induction. In other respects, TG-as might be expected from its gene- sis-has affinities with the theory of algebraically closed fields, and includes ana- logues of the theory of polynomial ideals and their varieties, and of the theory of resultants. These two sides of generic arithmetic are quite well developed but, as of now, not closely related to each other. The problem is to develop the theory of generic arithmetic by combining its arithmetical and algebraic aspects.

?3. Effective procedures in differential algebra. Differential algebra was introduced and developed by J. F. Ritt and his associates. It is a beautiful example of the formulation of a theory by analogy with another branch of mathematics, at a place where this is both needed and natural. A differential field is a commutative field with a binary operation D which is additive and satisfies the multiplicative rule of differentiation D(xy) = (Dx)y + xDy. One can define differential polynomials over such a field and one then obtains a purely algebraic theory of algebraic differential equations which, more particularly for characteristic zero, is largely analogous to the familiar theory of polynomial ideals and algebraic varieties for commutative fields. (For fields of characteristic p > 0 the analogy is less perfect.) The theory fits well into the framework described in the previous section. Thus, if E is the class of differential fields of characteristic 0, the corresponding classes E and G coincide and are axiomatizable by a complete theory. However, there is one important class of problems that can be largely ignored if one is concerned only with the structural theory of the subject. These are the questions which are concerned with the effective construction of particular objects of the theory, or with the decision, effectively, "in a finite number of steps," whether an object possesses a given property. For orientation, let us discuss some of these problems first for the case of an ordinary commutative field. Consider the following: (i) To




decide whether a given polynomial belongs to a given ideal; (ii) to find the absolutely irreducible components of a given variety; (iii) to decide whether a given polynomial vanishes on a given variety; (iv) to decide whether a given polynomial ideal is prime. In order to deal with these questions we have, as a preliminary step, to explicate what exactly we mean by saying that an object is given. For a polynomial the obvious interpretation is that we are given bounds on the number of variables and degree of polynomials and that we know the coefficients of the several monomials within these bounds. For an ideal we mean any set of polynomials that generates the ideal; and for a variety we mean any ideal that determines the variety. We then assume that we are able to carry out the elementary arithmetical operations on given field elements (and perhaps, in certain cases, to solve algebraic equations involving them). We do not want to assume that the fields with which we are concerned are themselves computable, hence countable, since our problems make perfectly good sense even for uncountable structures. They may be considered within the framework of a generalized recursion theory which for the present algebraic context was developed in the UCLA dissertation of W. Lambert. A closely related problem is whether the property under consideration can be expressed by a single formula in the lower predicate calculus. For example, with reference to the above problem (i), suppose that we are given g(x1, -., xn),

f1(x1, *, XJ), . *fk(xl,* , * Xn), n > 1, k ? 1, as general polynomials with indeterminate coefficients, of degree d ? 1 say, the coefficients being ranged in an arbitrary but definite manner as Yi, , ym for g(x1, *, xn) and as z7), ., z.. , for fi(x1, , xn), I = 1, , k. In order to answer problem (i) we have to decide whether there exist polynomials hj(x1, -, xn) such that g = h1f1 +.-- +hkfk.

This is an existential assertion for the yi, z"i, but as long as we do not have a bound on the degrees of the hi in terms of n, k and d the assertion cannot be expressed within the lower predicate calculus. The existence of such a bound was proved in effective form by G. Herrmann, and this reduces the question to the solv- ability of a system of linear operations and disposes of it. For problem (iii) a combination of Hilbert's Nullstellensatz with a model theoretic argument, which is by now classical, proves the existence of the required effective bounds, but for (i), which seems simpler on the face of it, only a beginning has been made in the effort to replace G. Herrmann's method by model theoretic arguments. As for (ii), a procedure for computing the irreducible varieties in question, for an algebraically closed field of coefficients, say, was given many years ago by Kronecker, and from it one can extract the existence of effective bounds for the number and degrees of the intervening polynomials. Using this fact one can show that the property of a variety being irreducible is definable (in the lower predicate calculus) and decidable. Finally, in (iv) the problem is to decide whether there exist polynomials p and q such that pq belongs to the given ideal, but neither p nor q belongs to it. In this case, the existence of the required bounds results from the work of Hentzelt and Noether but again does not follow from any known model theoretic arguments.

Similar problems arise in differential algebra. Here we are concerned with differential polynomial ideals, i.e., ideals which are closed under differentiation. In this case, problem (iii)-to decide whether a given differential polynomial vanishes on a given differential variety-involves the so-called Ritt Nullstellensatz and it




can be solved with the aid of model theoretic arguments, as in ordinary field theory. By contrast, the counterparts of problem (i)-to decide whether a differential polynomial belongs to a differential ideal given by a set of generators-and of (iv)- to decide whether a given differential ideal is prime-are still open. Here again one would expect a positive solution to come about either by algebraic or by model theoretic methods or by a combination of both. However, if one or the other of these problems is unsolvable then even an exact statement of this fact would by necessity involve recursion theory or one of its equivalents. To appreciate the rela- tive difficulty of these problems compared with the algebraic case, observe that we now have to control in addition to the number of degrees of the intervening polynomials, also the number of differentiations which may be involved. The problem of differential algebra which corresponds to (iii) was reduced by Ritt to the question of finding a bound to the number of differentiations required for the decomposition. Related to it is the following problem which was first tackled by Laplace in 1772. Etant donnee une equation diffirentielle d'un ordre quelconque .. . determiner si une equation d'un ordre inflrieur qui y satisfait est comprise on non dans son integrate generate. Having come to the conclusion that Laplace's proposed solution does not meet contemporary standards, Ritt reformulated the problem as follows: Let F and A be two differential polynomials which are algebraically irreducible and such that F vanishes at all points of the general solution of A. Decide whether the general solution of A is contained in the general solution of F.

This problem is still open even for the case of one variable. It might turn out to be one of the oldest algorithmically undecidable problems on record.

?4. Topological model theory. At present this subject is in statu nascendi, so I shall indicate what I mean by the term. Topological methods have been used in model theory for many years, e.g., in the case of Stone spaces, in the theory of n- types, in intuitionism. In the opposite direction, various topological structures have been considered from a model theoretic point of view. In particular, they are all subject to the methods of nonstandard analysis. In this approach, the higher order notions of a topological structure (e.g., open sets, sets of open sets) become amenable to treatment by model theoretic methods by being regarded, effectively, as first order notions, either within the framework of axiomatic set theory, or in type theory, by means of a Henkin interpretation. This is not what I have in mind here. What I have in mind is a theory which is related to algebraic-topological structures, such as topological groups and fields as ordinary model theory is related to algebraic structures (e.g., groups and fields). There are signs in several places (e.g., Illinois, Wisconsin, and Yale) that such a theory is beginning to emerge. Instead of trying to give a precise delimitation of the area, I shall produce a simple example whose proof is immediate.

A topological structure is a set of individuals M which is endowed (i) with an ordinary model theoretic structure and (ii) with a topology. The connection between (i) and (ii) is given by the condition that every basic relation R(x1, .., xn) of the structure is either open or closed in Mn, and that every basic function f(x1,... , Xn): Mn -? M is continuous. In this connection, equality is counted among the basic relations. Then a topological group is a topological structure since




its basic operations-multiplication and reciprocation-are continuous, by definition, and its only relation, equality, is closed. An ordered field with the internal topology is a topological structure, if we take as its basic relations equality (which is closed) and order (which is open) and as its basic functions addition and multiplication. The field of complex numbers with the usual topology and the same basic functions is a topological structure.

Let M be a topological structure, and let Q(x1, , xn) be a predicate (well- formed formula) which is meaningful for M. Then a point P = (1, . , en) e Mn is stable for Q if it has an open neighborhood Up such that M satisfies Q(a1,... , an) for all points A E Up, A = (a,, .., an). If P is stable for all predicates Q(X, * x , Xn) in a specified language L such that M k Q(61, * n) then we say that P is stable for L. The the following theorem holds.

Let M be a topological structure in a complete metric space whose theory K, in a countable language L, permits elimination of quantifiers. Then the set of points p e Mn which are stable for L is dense in Mn, n = 1, 2, 3, *

For illustration, let M be the field of complex numbers, and let the vocabulary of L consist of symbols for the relation of equality for the operations of addition and multiplication and for 0 and 1 (or, perhaps, for all rational numbers). Then the points P E Mn which are of stable L are just the points whose coordinates are algebraically independent of one another over Q.

The task I wish to specify here is of a general nature. It is to develop topological model theory in the direction exemplified above or in any other direction.

?5. Fields of power series. For any commutative field F, let F((t)) be the field of power series (Laurent series) of one variable t adjoined to F. In 1957, I posed the question whether for any two commutative fields, F1 and F2, the elementary equivalence of F, and F2 entails that F1((t)) and F2((t)) also are elementarily equivalent as fields. For fields of characteristic 0, this question was answered in the affirmative by Ax and Kochen as one of many results in their classical series of papers and also by Ershov. The solution employed relatively technical results from the theory of valuations. For fields of characteristic p > 0 the problem is still open. Perhaps its investigation would not only answer this particular question but also throw some light on the theory of structures which are defined by inductive or recursive rules on given first order structures.

?6. Solution of equations by functions with few -arguments. As No. 13 in his famous collection, Hilbert discusses the following problem: Consider a quadratic equation ax2 + bx + c = 0. Its solution can be obtained by the composition of functions of one or two variables (since x = (-b + (b2 - 4ac)112)/2a although we still have to consider the case a = 0 separately). The same applies to cubic and biquadratic equations since they are solvable by the adjunction of nth roots. But although the equation of degree 5 and 6 can no longer be solved in this manner, they can still be reduced, by the extraction of suitable roots to equations with two parameters, so that even there it is true that they can be obtained by the composition of functions of at most two variables, with the coefficients of the equation as the initial arguments. Hilbert conjectured that this kind of thing is no longer possible




for equations of the seventh degree, i.e., to be precise, he suggested that this cannot be done by means of continuous functions of one or two variables. This conjecture turned out to be mistaken when Kolmogorov and Arnold showed that all continuous functions, of any number of variables, can be obtained by the composition of continuous functions of one or two variables. However, even before that, Segre and Brauer had amended Hilbert's problem by stipulating that the solutions in question be obtained by the composition of algebraic functions of one or two variables. One may wonder why Hilbert himself did not make this assumption, which seems so much more natural. One possible explanation comes to mind when one looks at Hilbert's twelfth problem, which deals with the representability of algebraic numbers by values of transcendental functions, more particularly the elliptic modular function (Kronecker's Jugendtraum).-Following Segre and Brauer, let 4, be the smallest number of variables, such that the general equation of degree n can be obtained by the concatenation of algebraic functions of not more than I, arguments. It turns out that lim(n - In) = Go. However, it is still unknown if 4, is bounded.

Let us now consider this question from the metamathematical point of view. Let K, be a set of axioms for the notion of a commutative field of characteristic p formulated in terms of addition, multiplication and equality and in terms of the constants 0 and 1. For p > 0, K, may be taken to be finite while, for p = 0, the assertion that this is the case requires the introduction of an infinite sequence of existential sentences, e.g., in abbreviated notation, (3x)[x + x # 0], (3x)[x + x + x # 0], etc. In order to obtain a set of axioms for the notion of an algebraically closed field we have to add another sequence of sentences Xn, n = 2, 3,.*., where Xn states that every monic polynomial of degree n has a root. It is not difficult to see that Xn can be written as an V3-sentence (a sentence in prenex normal form in which no existential quantifier precedes a universal quantifier) with just n universal quantifiers, e.g., for X2, (Vx1)(Vx2)(3y)[y2 + x1y + x2 = 0]. Let H = {Xn}, n = 2, 3, 4,. -. Some simple field theory shows that H cannot be replaced by a finite subset. Let K1* = K, u H. We now ask the following question, for p = 0 or for prime p:

Can K,* be replaced by a sequence of V3-sentences Kp such that there is a uniform bound on the number of universal quantifiers in each sentence ? That is to say, is there a Kp of this kind such that K,* is logically equivalent to K; ?

This problem is actually equivalent to the problem concerning the boundedness of ln (for p = 0, for which the former question was asked originally). For Suppose the answer to the last question is affirmative. Then every solution of an algebraic equation can be represented by Skolem functions which are associated with the elements of Kp. But it is known that every such Skolem functionfcan be represented by a finite number of algebraic functions whose domains jointly exhaust the domain off. Accordingly, the existence of a Kp as described would imply the boundedness of 1ns Conversely, suppose that 1n is bounded. Then the sentences which assert the existence of the corresponding algebraic functions together with K, make up a set K, of the required type.

Although the algebraic problem and the metamathematical problem are thus equivalent the latter presents the matter in a new light. It also suggests analogous




questions in other areas (e.g., for p-adic fields) some of which, while nontrivial might turn out to be simpler than the original question. The torsion-free divisible abelian group provide an example of a complete theory which is axiomatizable by an infinite number of V3-axioms such that there is a common bound on the number of universal quantifiers in each. On the other hand, for arithmetic it can be shown in several ways that the "axiom of induction" (which is an infinite set of instances of a scheme) is not equivalent to any set of sentences with a bounded number of universal quantifiers. But what has really been proved here is that the number of alternations of quantifiers must be unbounded and this has no immediate bearing on the original problem. However, it shows that the answer is still negative if we drop the assumption that the elements of Kp are in V3-form, retaining only the condition that they are in prenex normalform and that there is a uniform bound on the number of universal quantifiers in each. We may modify the general problem in the same way. (This makes no essential difference for the case of algebraically closed fields.) An equivalent formulation then is that there is a uniform bound on the number of arguments of the Skolem functions arising out of the sentences of K, and this is closer to Hilbert's own version.

It is easy to produce more sophisticated questions in this area. For example, defining A(X) as the number of universal quantifiers in a given sentence which is in prenex normal form, let K be a specified set of axioms in that form. Define K& as the set {Xe K I A(X) = n}. Let sun be the smallest number for which there is a set of sentences Hn in prenex form such that K F H1, HIn F Kn, and A(X) < It, for all Xe H,. Clearly sun, < n. Determine sun, or the rate of growth of sun The result of Brauer and Segre quoted earlier shows that if K is a set of axioms for the notion of an algebraically closed field of characteristic zero as detailed above, then 1imnoD(n - 1un) = x0.

There are at present no apparent tools in logic for tackling this kind of problem. Only, perhaps, in polyadic algebra one notices arguments which might be relevant here.

?7. Nonstandard analysis. Nonstandard analysis makes use of nonstandard models of the integers and of the real and complex numbers in order to develop analysis in terms of infinitely small and infinitely large numbers. It also includes analogous notions for other types of structures, e.g., topological spaces and function spaces. Since its inception in 1960 it has been applied to many areas, for example complex function theory, measure theory and probability, functional analysis, nonstandard economics. So at this point I only wish to draw your attention to three areas in which, in my view, further work by nonstandard methods would be desirable and enjoyable. They are the following:

(i) The theory offunctions of several (perhaps infinitely many) complex variables. Here the initial work involving both germs and sheaves has been done, and the next important step would be to frame the cohomological notions involved in nonstandard terms.

(ii) Lie groups. Here, the direct use of infinitesimals is altogether natural, so it may come as a surprise that only the beginnings of a nonstandard theory of the subject exist at this time.




(iii) Algebraic number theory and algebraic function theory. The effectiveness of nonstandard analysis depends on the combination of internal

notions, to which the transfer principle applies, with external ones. Thus, let *M be a nonstandard extension of a structure M in one of the several accepted senses of the terms. Clearly, we should get nothing new for M if we considered only internal notions in *M since these are, by definition, the notions which are transferred from M and they have the same properties in *M as in M. So we have to introduce external notions. These are, in the first place, precisely the notions of the infinitely large and infinitely small (infinitesimal) or their counterparts for nonmetric structures. The literature of the subject shows that one can go quite a long way using only these external notions, or notions derived from them (such as S-con- tinuity in function theory). However further progress may be expected to come through the introduction of additional external notions. Thus, the theorem of Ax and Kochen, when interpreted in terms of nonstandard analysis hinges on the fact that for a prime field Rp, where p is an infinite prime, the field of formal power series R,((t)) is externally isomorphic (in certain nonstandard models) to the field of p-adic numbers Q,. Again, let I be an irreducible algebraic curve over an algebraic number field k, such that F has an infinite number of points with coordinates in k. Then the function field of F, k(x, y) can be injected into *k, where the injection is necessarily external, and this fact has been used to throw light on the standard theory.

?8. A heuristic principle in complex function theory. Soon after Weierstrass stated the theorem known under his and Casorati's name, Picard (1879) published his famous result that an entire function can omit not more than one value. Related theorems state that in any neighborhood of an isolated essential singularity an analytic function omits at most one value (Picard's "great theorem") and that a meromorphic function omits at most two values. In the first quarter of this century, Montel developed his theory of normal families, which showed that Picard's theorems and many related results can be proved with the aid of compactness arguments. Let F be a family of complex functions which are defined and holomorphic on a domain D of the complex plane. F is called normal if from any sequence of elements of F one can select a subsequence which converges either to a holomorphic function or to A0, uniformly on compact subsets of D.

The following heuristic connection between entire functions and normal families was put forward after having been verified in several particular cases (see [14]). It appears to go back to common lore current during the heyday of the subject between 1915 and 1935.

" Let P be a property which cannot be possessed by any nonconstant entire function. If all functions which belong to a family F of functions holomorphic on a joint domain D possesses this property then F is (apt to be) normal."

To make this or any other heuristic principle precise is a natural task for a logician who is interested in mathematics, but one may not necessarily need logic for this purpose. A first step is to try and eliminate cases which are obvious excep- tions to the validity of the principle. For example, if P is the property not to be an entire function then the principle fails. On the other hand, it is in the nature of




things that a precise wordixig may also exclude some special cases which can still be fitted into the informal assertion. At any rate, the following formulation includes many of the cases to which the heuristic principle is known to apply.

We consider properties (i.e., sets) P of holomorphic functions which are specified with their (open) domains of definition. Thus <1/(1 - z), jzj < 1> is different from <1/(1 - z), IzI < i>. Suppose that P satisfies the following conditions:

(i) If <f(z), D> e P and D' c D then <f(z), D'> e P. (ii) If D1 c D2 c D3 . are domains on the complex plane and <f(z), Dn>

eP, n = 1, 2,. , then <f(z), D> eP where UDn = D. (iii) If <fn(z), D> E P, n = 1, 2, *-, and limf,(z) = f(z) in the sense of uniform

convergence on compact subsets of D then <f(z), D> e P. (iv) If <f(z), D> E P and D is mapped conformally one-to-one on a domain D' by

means of a function g(w): D -- D' then <f(g(w)), D'> also belongs to P. The principle now asserts that if P does not contain any nonconstant entire function,

then for any domain D the family off(z) such that <f(z), D> E P is normal on D. For example, let r be a positive real number and let <f(z), D> E P iff(z) is holo-

morphic on D and If(z)l < r for all z E D. Then <f(z), C> E P where C is the entire plane, only iff(z) is a constant, by Liouville's theorem. Accordingly the hypotheses of the proposed principle apply and so, indeed, does the conclusion. By contrast, the class P of <f(z), D> which is defined by the property that f(z) is holomorphic and bounded on D (without a specified bound) does not fall under the principle. Another example where both the hypotheses and the conclusion of the principle apply is provided by the condition that f(z) omit both 0 and 1 on its domain D.

If we take for F the family of all functions holomorphic in the unit circle and belonging to a set P that satisfies (i)-(iv) above then it is a consequence of (iv) that F is closed under one-to-one conformal maps z -? z' of the unit circle on itself. If F is normal it is then called uniformly normal.

Although the principle under consideration does not require the use of a formal language (beyond the amount of formalization common in mathematics) it seems to me that the matter is properly our concern from the point of view of the present talk. Beyond that I now wish to show that the principle as formulated is correct if the answer to the following question in nonstandard analysis is positive.

Let F be a family of functions holomorphic on a domain D. Nonstandard analysis shows that F is not normal if and only if there exist an f(z) E *F and a point zo E D such thatf(z) is irregular on the monad of zo, ,u(zo). That is to sayf(z) takes values in at least two distinct monads as z ranges over ~t(zo).

Suppose that an internal function f(z) is holomorphic but irregular on a monad tt(zo). Does there always exist an internal conformal map z = g(w) wl-ichi maps the external set D' = {vI I it, Ifinite} one-to-one into p(zo) such thatf(g(w)) is finite on D' but takes values in at least tiv'o different monads as it, ranges orer D'?

Suppose that the answer to this question is affirmative, and let F be a family of functions holomorphic on a domain D, such that, for all f(z) E F, <f(z), D> E P for a set P which satisfies (i)-(iv). If F is not normal, then somef(z) E *F is irregular on the monad tz(zo) at a point zo E D. Sincef(z) E *F, we also havef(z) E *P. Let g(w) be a map as described. Then there exists an infinite R > 0 such that g(w) maps the




domain I wI < R one-to-one on a subset of ju(zo), Do say. Then <f(z), Do> belongs to *P. by (i) and so <f(g(w)), I wI < R>, belongs to *P by (iv). It follows, again by (i), that <f(g(w)), I wl < n> E *P for all standard positive integers n. Also, since f(g(w)) is finite on D' it possesses a standard part h(w) = 0(f(g(w))), which is then a standard entire function of w. Moreover, h(w) cannot be a constant sincef(g(w)) takes values in different monads. We now claim that for any standard positive integer n, <h(w), I wI < n> belongs to P. Indeed, h(w) - f(g(w)) is infinitesimal for all I wl < n. Accordingly, for any standard integer k > 0 it is true in the nonstandard universe that there exists an element of *P, i.e., hk(w) = f(g(w)) such that

Ih(w) - hk(w) 1/k for all IwJ < n. But then, by transfer, h(w) can be approxi- mated uniformly for I w I < n, by elements of P, and so <h(w), l vi < n> e P by (iii). Hence, by (ii), <h(w), C> e P where C is the complex plane. This shows that P contains nonconstant entire functions, and proves our principle, conditionally.

The following example shows, in a particular case, how an entire function h(w) can be obtained in the manner postulated above. Consider the family F of functions <zn, IZI < 2>, n = 1, 2, 3, . Fis not normal and, in accordance with the general test mentioned earlierf(z) = zn e *F where n is an infinite positive integer, is irregular on the monad of the point zo = 1. Now let g(w) = 1 + w/n. This maps the finite w-plane into the monad of 1. Also f(g(w)) = (1 + w/n)n and the standard part of this function, for finite w is h(w) = '((1 + w/n)n) = ew, a nonconstant entire function.

?9. Baire's theorem. We recall the following definitions. Let T be a topological space. A set A c T is nowhere dense if the interior of the closure A of A is empty. That is to say, every point of A is a limit point of X - A and this is the case if and only if X - A is dense in X. A set is meager or of the first category if it is the union of a countable number of nowhere dense sets; a set is co-meager or residual if it is the complement of a set of the first category; and a set is of the second category if it is not of the first category.

Baire's argument leads to the THEOREM. Let X be a complete metric space or a complete pseudo-metric space or

a locally compact regular space. Then the intersection of a countable family of open dense subsets of X is itself dense in X.

For the metric case, the proof involves the selection of a nested sequence of closed sets with diameter tending to zero. From these sets one selects a Cauchy sequence which must have a limit because the space is complete. The argument is analogous in the nonmetric case.

Sometimes the name "Baire's theorem" is given to the following: COROLLARY. With any one of the three assumptions of the preceding theorem, the

space X is of the second category. That is to say, X cannot be the union of a countable family of nowhere dense sets

An. For if this were the case then nn(X - An) would be dense in X, hence not empty.

A major application is the Banach-Steinhaus theorem. Let Tn: B -+ B' be a sequence of bounded linear maps from the Banach space B to the Banach space B'




such that IITnxl, n = 1, 2,. ., is bounded for each x E B. Then IIT.II, n = 1, 2,. , is bounded.

The assignment of this section is-to produce a metamathematicalframework for Baire's theorem, more particularly for the metric case, and its applications.

In order to appreciate what is required here, one may look at the many strands that tie the mathematical notion of compactness to metamathematics. On one hand, the compactness theorem of the lower predicate calculus is the most powerful single tool of model theory. On the other hand, model theory, particularly through nonstandard analysis has thrown new light on the notion, in many ways, as in the theory of compact or polynomially compact linear operators, or in the theory of normal families. Moreover, if one regards compactness only as a tool, as one may choose to do with regard to normal families, then it can actually be replaced by metamathematical arguments and constructs, e.g., in the case just mentioned by nonstandard internal analytic functions.

There is no corresponding metamathematical edifice related to Baire's theorem or, more generally, to the notion of completeness. In the opposite direction, Morley has used completeness in infinitary logic to replace compactness, and Takeuti and Bowen have given versions of set theoretic and model theoretic forcing in the sense of Baire. Perhaps these efforts offer a hint how one should go about the complementary task formulated here.

?10. Numbers as functions. Here we shall be concerned with a pervasive analogy that can be drawn between arithmetic or, more generally, algebraic number theory on one hand and algebraic function theory on the other hand. The problem of formulating such an analogy in precise terms falls naturally within the domain of a logician although it is of course conceivable that this can be done in traditional mathematical terms as far as it can be done at all. Moreover inasmuch as the analogy is concerned with the "deep structure" of arithmetic it seems to me that it may be regarded as part of the foundations of mathematics no less, or perhaps more, than many questions concerning infinite cardinals which, though interesting in their own right, have no bearing on the general scheme of mathematics.

I am thinking of the interpretation of rational and, more generally, of algebraic numbers as functions on a "Riemann surface" whose points are just the prime divisors of the field. For the rational numbers, these may be identified with the rational primes p, or equivalently, with the p-adic valuations, together with the archimedean valuation by ordinary absolute value. For an algebraic number field, the "points" of the surface are given by the prime ideals of the field together with a finite number of archimedean valuations. The " values " taken by these " functions" at a "point" that corresponds to a prime number or prime ideal, lies in the corresponding residue class field while the counterpart of the Laurent or Puiseux expansions for functions at any point is the injection of the given number into the corresponding p-adic or archimedean completion.

Ever since the end of the nineteenth century, the analogy between number fields and algebraic function fields has been a guiding principle for the development of both theories. As far as one can tell it inspired or helped to inspire Hensel to invent or discover the theory of p-adic numbers. It showed Dedekind and Weber how to




formulate a purely algebraic theory of algebraic functions. And in his twelfth problem, which is entitled Extension of Kronecker's Theorem (that every abelian field over the rationals can be obtained by the adjunction of roots- of unity), Hilbert gives a long list of parallel examples from the two areas. Since then, these have been followed by many more, for example in class field theory and in the theory of rational points on curves. It is in fact common practice nowadays to transfer a conjecture which has not been proved for number fields to function fields, particularly but not necessarily to function fields over a finite ground field and to prove it there. So the possibility of going beyond a mere formal analogy and offinding a deeper justification for it, perhaps in a metamathematical framework, should be considered seriously. However, the difficulties that stand in the way of this task should not be underrated.

Perhaps the most conspicuous difference between function fields and number fields is the absence of a nontrivial transformation group for the latter. Thus, while a choice of the independent variable in a function field singles out the point or points at infinity, they can be transformed into finite points by substitution. This is not possible in a number field where the infinite points ("infinite prime spots") that correspond to the archimedean valuation retain a distinct character throughout. Accordingly, it would be of great interest to produce a (mathematical or metamathematical) framework in which the infinite divisors of a number field can be considered more uniformly on a par with the finite divisors. However, contrary to what one might expect by recalling other results of model theory it is not true that the field of real numbers, R (i.e., the completion of Q at the infinite prime spot), can be regarded as a limiting case of the p-adic number fields in the sense that for every sentence X of first order field theory there exist a natural po such that X is true for R only if it is true also for Q, when p > po.

Before leaving this .subject let me draw your attention to one very special cir- cumstance which is perhaps not accidental in the sense that a deeper reason might yet be found for it. This is the importance of the Chinese remainder theorem both in number theory-where it leads to use of the approximation theorems-and in G6del's famous incompleteness paper, where it is used to prove the representability of primitive recursive functions. Roughly speaking, the use of the theorem in both cases is due to the fact that it expresses the great flexibility of the rational number system.

?11. Theorems and proofs. Except for the several examples of decision problems (where a negative solution would involve the concepts and notions of recursion theory or of one of its equivalents) all our problems up to this point have been based on, or have called for, the use of model theory as a framework for a specific mathematical situation. This is, in a sense, surprising if we remember that logic is often regarded as the science of deductive reasoning. The fact that, up till now, I have not mentioned proof theory may to some extent be due to my own predilections but it also reflects an objective situation. There can, indeed, be no doubt that proof theory should be regarded as one of the major branches of logic, irrespective of concrete applications. And one of the outstanding experts in the field, Kreisel, gave, in the fifties, several important examples which showed that proof theoretic




ideas are applicable to concrete mathematics. But while the subject as such has expanded greatly in recent years, to a large extent under Kreisel's lead, its development has been for the most part inward looking. Nevertheless there still seems to be considerable scope for the growth of the subject in closer connection with concrete mathematical theories and I wish to indicate several possibilities in this direction.

As Kreisel has emphasized, a proof may be regarded as an interesting object in its own right, so the detailed comparison of the deductive steps involved in two different proofs of the same mathematical result may be a worthwhile occupation. Thus, one may try to devise a purely syntactical transformation which correlates standard and nonstandard proofs of the same theorems in a large area, e.g., complex function theory. It is in fact evident that nonstandard analysis achieves its greater syntactical simplicity because it can eliminate certain quantifiers by means of the external predicates "infinitely small" or "infinitely large." For example we express the fact that limn_0. sn, = 0 in standard language by: " For every e > 0 there exists a natural u such that, for all n > ju, ISnI < e"; while the corresponding nonstandard expression is: " For all infinite natural n, Sn is infinitesimal" replacing the quantifiers in two of the three instances by the use of the new predicates.

Going on to another area there are several theorems which were proved originally for the real numbers with the aid of topological methods, but which (by Tarski's theorem on the completeness of the notion of a real-closed field) must be equally true for all other real-closed fields. The most conspicuous among them at the present time is the theorem of Bott and Milnor which asserts that all finite-dimensional (associative or nonassociative) division algebras over the real numbers must be of dimension 1, 2, 4, 8. Now the assertion " there is no division algebra of dimension n over the reals" can be formulated as a first order statement Xn about the real numbers; so for each n other than 1, 2, 4, 8, it must be deducible from the axioms for a real-closed field. In a comparable case-a theorem which had been proved by Eckmann for the complex numbers by topological methods but is nevertheless true for all algebraically closed fields by transfer-Habicht showed how the required topological notions can be recreated step by step for an arbitrary real closed field. But clearly, there are some topological notions, such as local compactness which, at least in the absolute sense, are lost as one passes from the field of real numbers to an arbitrary real closed field. It would be interesting to obtain a general class of theorems, perhaps including the theorem of Bott-Milnor, for which Habicht's program can be carried through.

It is conceivable that an analytic or topological proof of an algebraic theorem may be very much shorter than any algebraic proof. A classical result of Godel's establishes the existence of an analogous phenomenon for arithmetic. Rabin has begun a detailed investigation of the lengths of some well-known decision procedures for particular algebraic theories and has already obtained several substantial results.

Adapting these results to the case of particular theorems considered here, one might proceed as follows. Let K be a recursively enumerable set of axioms. Intro- duce a reasonable measure for the length A(X) of a shortest proof of a sentence X from K, e.g., by some form of godelization. Let {Xn} be a sequence of sentences de-




ducible from K. Find the order of growth of A(Xn) as a function of n, e.g., in the two concrete cases mentioned above. Perhaps A(Xn) may increase so rapidly that the explicit formulation of first order proofs for Xn, n = 0, 1, 2, , is not feasible even in cases when K is decidable.

Apart from the intrinsic interest of the subject, a detailed understanding of mathematical proof procedures might infuse some badly needed new ideas into the effort to develop methods for proving theorems by computer when no decision procedure is available. While the use of Herbrand's theorem has been of some help in this area it still has a long way to go to reach maturity.

?12. Foundations of mathematics. For a period of fifty years, from Frege to Godel, the most distinguished work in mathematical logic was concerned with the foundations of mathematics. To make these foundations secure by anchoring them firmly in logic, to make their consequences unshakable by means of a well delimited deductive system-these things were regarded by many as the preeminent reason for the development of our subject. That this is no longer the case is well illustrated by the contents of my talk up to this point. However, it seems to me that even nowadays it behooves the logician to maintain an active interest in the philosophical foundations of mathematics and to adopt an attitude towards them. At the present stage, even the formulation of an absolutely meaningful problem in this area is fraught with difficulties. All the same, here is my question.

Is there a well-formedfirst order assertion about the natural numbers which can be neither proved nor refuted by a formal or by a generally acceptable informal argument?

Clearly, there is a difficulty in the interpretation of the term "generally acceptable informal argument," and I cannot see how to overcome it in a universally acceptable way. In the present context, we may choose to disregard the intuitionists and other kinds of constructivists, their substantial merits in some respects notwithstanding. But even so, one cannot predict for certain that some argument in arithmetic will not one day split the classical mathematicians down the middle, perhaps precisely because it leads to the kind of result adumbrated in my question. On the other hand, the reasoning that establishes the truth of the Godel sentence ("I am not provable") which is the outstanding example of a theorem of arithmetic arrived at by going beyond the specified formal framework has indeed met with general acceptance. Godel's work also shows that a positive answer to my question would have to be given by means that are not strictly finitistic since it would imply the consistency of Peano arithmetic. Post attempted to give such an answer as far back, at least, as 1941.

At any rate, the question shows where I stand. While others are still trying to buttress the shaky edifice of set theory, the cracks that have opened up in it have strengthened my disbelief in the reality, categoricity or objectivity, not only of set theory but also of all other infinite mathematical structures, including arithmetic. I am thus taking sides in an ancient controversy that has appeared and reappeared in different forms over thousands of years. In our time no less a man than Paul J. Cohen has indicated his agreement with my point of view. However, it is equally true that the array ranged against it is formidable. Among the platonic realists, it




includes the greatest of all mathematical logicians, Godel, as well as Bernays, whom we revere as the Nestor of our subject, and the dynamic and influential Professor Kreisel. Not long ago, Bernays, in an article published in Dialectica, criticized my attitude in his usual gentle manner, while Kreisel had stated his disagreement with me previously, also in his usual manner. From another direction comes the voice of the constructivists who deplore the fact that the formalists, whose philosophical attitude towards the actual infinite in mathematics is akin to their own, do not join with them also in their mathematical practice. However, I must confess that neither invective nor moral indignation have induced me to change my opinion in this matter. The relevance of mathematics to the investigation of the physical world is an empirical fact. It is neither affected by, nor does it determine the outcome of the controversies on the philosophy of mathematics. And, regardless of their attitude towards its foundations, all mathematicians have a strong feeling for the beauty and fascination of their subject. I hope that I have conveyed some of this feeling in my talk.

REFERENCES

[1] V. I. ARNOL'D, Some questions of approximation and representation offunctions, Proceedings of the International Congress of Mathematicians (Edinburgh, 1958), Cambridge University Press, New York, 1960, pp. 339-398. (Russian)

[2] J. Ax and S. KOCHEN, Diophantine problems over local fields. I, II, American Journal of Mathematics, vol. 87 (1965), pp. 605-630, 631-648; Diophantine problems over local fields. III. Decidable fields, Annals of Mathematics, vol. 83 (1966), pp. 437-456.

[3] P. BERNAYS, Zum Symposium fiber die Grundlagen der Mathematik, Dialectica, vol. 25 (1971), pp. 171-195.

[4] K. A. BOWEN, Forcing in a general setting, 1972 (unpublished). [5] R. BRAUER, A note on systems of homogeneous algebraic systems, Bulletin of the American

Mathematical Society, vol. 51 (1945), pp. 749-755. [6] P. J. COHEN, Comments on the foundations of set theory, Proceedings of Symposia in Pure

Mathematics, vol. 13, Part I, American Mathematical Society, Providence, R.I., 1971, pp. 9-15. [7] R. DEDEKIND and H. WEBER, Theorie der algebraischen Functionen einer Veranderlichen,

Journal fur reine und angewandte Mathematik, vol. 92 (1882), pp. 181-290. [8] Y. L. ERSHOV, On the elementary theory of maximal normal fields, Algebra i Logika, vol. 4

(1965), pp. 31-70. [9] K. GODEL, Oiber formal unentscheidbare Satze der Principia Mathematica und verwandter

Systeme. I, Monatshefte fur Mathematik und Physik, vol. 38 (1931), pp. 173-198. [10] , Ober die Lange von Beweisen, Ergebnisse eines mathematischen Kolloquiums,

Fasc. 7, 1936, pp. 23-24. [11] W. HABICHT, Ober Polynomabbildungen, Mathematische Annalen, vol. 126 (1953), pp.

149-176. [12] G. HERRMANN, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale,

Mathematische Annalen, vol. 95 (1926), pp. 736-788. [13] D. HILBERT, Mathematische Probleme, Archiv fur Mathematik und Physik (3), vol. 1

(1901), pp. 44-63, 213-237 (Collected works, vol. 3 (1930), pp. 290-329). [14] L. HILLE, Analytic function theory, Vol. II, Ginn, Boston, 1962, p. 250. [15] J. M. HIRSCHFELD, Existentially complete and generic structures in arithmetic, Doctoral

Dissertation, Yale University, New Haven, Conn., 1972. [16] R. JOHNSON, The construction of elementary extensions and the Stone-Cech compactifica-

tion, Doctoral Dissertation, Yale University, New Haven, Conn., 1971. [17] E. R. KOLCHIN, Some problems in differential algebra, Proceedings of the International

Congress of Mathematicians (Moscow 1966), "Mir", Moscow, 1968, pp. 269-276.




[18] G. KREISEL, Mathematical significance of consistency proofs, this JOURNAL, vol. 23 (1958), pp. 155-182.

[19] , Observations on popular discussion of foundations, Proceedings of Symposia in Pure Mathematics, vol. 13, Part I, American Mathematical Society, Providence, R.I., 1971, pp. 189-198.

[20] W. M. LAMBERT, A notion of effectiveness in arbitrary structures, this JOURNAL, vol. 33 (1968), pp. 577-602.

[21] P. S. LAPLACE, Memoire sur les solutions particulieres des equations diffierentfelles et sur les inegalites scculaires des planetes, (Collected works), vol. 8 (1772), pp. 326-365.

[22] A. H. LIGHTSTONE and ABRAHAM ROBINSON, On the representation of Herbrandfunctions in algebraically closed fields, this JOURNAL, vol. 22 (1957), pp. 187-204.

[23] P. LORENZEN, Constructive mathematics as a philosophical problem, Compositio Mathematica, vol. 20 (1968), pp. 133-142.

[24] W. A. J. LUXEMBURG (Editor), Applications of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart and Winston, New York, 1969.

[25] A. MACINTYRE, On algebraically closed groups, Annals ofMathematical Logic (to appear). [26] M. MCCORD, Nonstandard analysis and homology, Fundamenta Mathematicae, vol. 74

(1972), pp. 21-28. [27] M. MORLEY, Applications of topology to L.,., mimeographed, not dated. [28] E. L. POST, Absolutely unsolvable problems and relatively undecidable propositions, 1941,

The Undecidable (Martin Davis, Editor), Raven Press, New York, 1965, pp. 338-433. [29] J. F. Rirr, Differential algebra, American Mathematical Society Colloquium Publications,

vol. 33, American Mathematical Society, Providence, R.I., 1950. [30] A. ROBINSON, Proving a theorem (as done by man, logician, or machine), Summer Institute

for Symbolic Logic (Cornell, 1957), Summaries, pp. 350-352. [31] , Introduction to model theory and to the metamathematics of algebra, Studies in

Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1963. [32] , Formalism '64, Proceedings of the International Congressfor Logic, Methodology,

and Philosophy of Science, North-Holland, Jerusalem 1964, pp. 228-246. [33] , Nonstandard analysis, Studies in Logic and the Foundations of Mathematics,

North-Holland, Amsterdam 1966. [34] , Forcing in model theory, Symposia Mathematica, vol. 1, Academic Press, Lon-

don, 1971, pp. 245-250. [35] , Nonstandard arithmetic and generic arithmetic, Proceedings of the International

Congress for Logic, Philosophy and Methodology of Science (Bucharest 1972) (to appear). [36] J. ROBINSON, The decision problem for fields, Studies in Logic and the Foundations of

Mathematics (J. W. Addison, L. Henkin, A. Tarski, Editors), North-Holland, Amsterdam 1965, pp. 299-311.

[37] B. SEGRE, Mathematical questions on algebraic varieties, The Athlone Press, London, 1957.

[38] G. TAKEUTI, Topological space and forcing, mimeographed, 1966. [39] A. TARSKI and J. C. C. MCKINSEY, A decision method for elementary algebra and

geometry, 1st edition, Rand Corp., Santa Monica, Calif., 1948; 2nd edition, University of California Press, Berkeley and Los Angeles, 1951.

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