A Problem Concerning the Notion of Definability Alfred Tarski The Journal of Symbolic Logic, Vol. 13, No. 2. (Jun., 1948), pp. 107-111. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28194806%2913%3A2%3C107%3AAPCTNO%3E2.0.CO%3B2-Y The Journal of Symbolic Logic is currently published by Association for Symbolic Logic.
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TEEJoarwar, oa SYMBOLIC LOGIC
Volume 13, Number 2, June 1948
A PROBLEM CONCERNING THE NOTION OF DEFINABILITY ALFRED TARSKI
We are inclined to believe that, by means of an argument entirely analogous to that which leads to the Richard antinomy,' the notion of definability as applied to entities discussed in a formal system can easily be shown not to be itself d e h able in this system. It will be seen from this discussion that actually the situation is not quite so simple as it would appear at first glance. Our discussion will have a rather sketchy and informal ~haracter.~ To fix the ideas we consider a formal system 6 which is assumed to be consistent and to provide an adequate basis for the formalization of "classical" mathematics. 6 may be thought of as a system close to that of PrincQia mathernatica, but based upon the simple theory of types (though our discussion applies as well to various axiomatic systems of set theory known from the literature). Thus, each variable occurring in 6 has a definite order. Variables of the zeroth order are interpreted as representing natural numbers (in other words, the range of these variables is the set of all natural numbers) ; variables of the first order represent real numbers-which are identified here, as it is often done, with sets of natural numbers; variables of the second order represent sets of real numbers, and so on. In general we shall refer to entities represented by variables of the nth order as elements of the nth order. Among constants occurring in (3we find logical constants such as sentential connectives "+", "v", " A " , and "-", the universal quantifier "A", the identity symbol "=", and the membership symbol "e"; in addition, G contains some specifically mathematical constants, for instance, the " and " ." of operations on natural numbers. We shall not describe symbols the formalism of G in detail.3 Let us now consider the notion of definability in a. Given an element a of the nth order and a formula (sentential function) 4 in 6 which contains a certain variable of the nth order as the only free variable, we can ask the question whether or not a satisfies +.4 If a proves to satisfy 4 and if nothing else satisfies this formula, we say that defines a. Thus, for instance, if "x" and "y" are variables
"+
+
Received October 31,1947. 1 Compare here A. Fraenkel, Einleiiung in die Mengenlehre, 3rd edition, Berlin 1928 (reprinted 1946, U.S.A.), especially Chapter 4; further biblidgraphical references can be found there. 2 The notion of definability, in its application to sets of real numbers, was discussed in the author's paper S u r les ensembles dk$nissables de nombres rdels I , Fundarnenia rnaihernaticae, vol. 17 (1931), pp. 210-239. Part I1 of this paper, which did not appear in print, was intended to include a more detailed discussion of the ideas sketched in the present article. 3 Cf. the author's paper cited in the preceding footnote where a closely related system is r unentscheidbare discussed; compare also the formalism described in K. Godel, ~ b e formal Satze der Principia Mathernatica und verwandter Systeme I , Monaishejte jiir Maihematik und Physik, vol. 38 (1931), pp. 173-198. 4 For a precise definition of the expression " a satisfies 4," see the author's work Der Wahrheitsbegrijtf in den formalisierten Sprachen, Studia philosophica, vol. 1 (1936), pp. 261405, especially Chapters 3 and 4.
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ALFRED TARSKI
of the zeroth order, the formula A, x + y = y d e h e s the number 0, while the formula
A, x . Y = Y defines the number 1. The element a is said to be definable in 6 if there is a formula 4 in 6 which defines a. In case a is a set, i. e., in case the order n of a is different from 0, this definition can be replaced by another, equivalent one, which proves to be very convenient in some applications. We agree to say that a formula $ which contains a certain variable of the (n - 1)st order as the only free variable defines a in a new sense if a is the set of all elements which satisfy $; and a is definable in 6 if there is a formula $ in 6 which defines it in this new sense. Similarly, a binary relation R (between elements of the same order or of two different orders) is definable if there is a formula which contains just two different free variables and is satisfied by those and only those ordered couples of elements between which R holds. This new definition of definability enables us to discuss the definability of a set a (or a relation R) in a formal system Z even though a (or R) does not belong to the range of any variable occurring in 2 ; it suffices to assume that all elements of a belong to the range of some variable in 2 (or that all elements of the domain of R belong to the range of a variable in 5 and all elements of the counter-domain r the same-variable in P). of R belong to the range of a n o t h e r ~ possibly We shall concentrate our attention on the problem whether the n o t k of an element of the nth order definable in 6 is itself definable in G ; or, in other words, whether, for any given natural number n, there is a formula $ in 6 which contains a variable of the nth order as the only free variable and is satisfied by all those and only those elements of the nth order that are definable in 6. If we agree to denote by "D," the set of all elements of the nth order definable in EJ, the problem assumes a concise form: is it true that the set D, (which is clearly an 1)st order) is a member of the set D,+I? element of the (n A discussion of the problem just formulated leads to the following conclusions: the solution of the problem is (trivially) positive if n = 0; the solution is negative if n 2 2; in the (perhaps most interesting) case n = 1 the problem remains open. I n fact, it is easily seen that, like the numbers 0 and 1 for which defining formulas were explicitly given above, each natural number is definable in G. Hence the set Doof all natural numbers definable in G is simply the set of all natural numbers. We can easily construct a formula $ in 6 which contains a variable of t h e zeroth order, say "x", as the only free variable and such that Do is the set of all elements satisfying $; for instance, we can take "x = x" for $. Thus the solution of our problem is positive for n = 0. We now take up the case n = 2; our argument, however, will apply practically without changes to any natural number n 2 2. We can argue as follows. It was pointed out above that the elements of the second order are sets of real numbers. I t is known that a well-ordering relation R can be defined whose field consists of sets of real numbers and is nondenumer-
+
A PROBLEM CONCERNING THE NOTION OF DEFINABILITY
109
able.' In fact, we can establish a one-to-one correspondence between all real numbers and all sets of rational numbers. We agree to call a real number a "well-ordered" if the correlated set 6 of rational numbers is well-ordered by the ordinary 5 relation, and to call two real numbers a and b "similar" if the correlated sets 6 and 6 are similarly ordered. We consider sets of real numbers each of which consists of all real numbers similar to a certain well-ordered real number. Let R be the relation which holds between two such sets x and y if, and only if, there are two real numbers a in x and b in y such that the correlated set 6 of rationals is set-theoretically included in the correlated set 6 of rationals. I t is not hard to show that the relation R thus defined is indeed a well-ordering relation between sets of real numbers and that the field of R is non-denumerable. Since the system 6 provides us with adequate devices for a formalization of the whole of "classical" mathematics, the relation R is definable in 6. In other words, a formula in 6 can be constructed which contains two different variables of the second order, say "2" and "y", as the only free variables and which is satisfied by all those and only those couples of sets x and y of real numbers between which R holds; we can even construct this formula without using any variable of an order higher than 1 as a bound variable. The formula in question may be symbolically represented by "p(x, y)". The set of all formulas in G being denumerable, the set Dz of all sets x of real numbers which are definable in 6 is also denumerable. Since the field of the relation R is non-denumerable, there are sets x of real numbers which belong to the field of R, but do not belong to Dz ; and since R is a well-ordering relation, there is a uniquely determined set xo of real numbers which belongs to the field of R but not to Dz,and which is in the relation R to any other set y of real numbers belonging to the field of R but not to Dz . Assume now that the set Dz is definable in 6. Thus, there is a formula in (3which contains a variable of the second order, say "x", as the only free variable and which is satisfied by all sets belonging to D zand by nothing else; we may symbolize this formula by "6(x)". Consider the following formula 4 :
here "6(y)" clearly denotes the formula obtained from "6(x)" by changing everywhere "x" to "y" ("x" and "y" may be assumed not to occur in "6(x)" as bound variables), and analogously for "p(x, x)" and "p(y, y)". I t is easily seen that 4 is a formula in 6 containing "x" as the only free variable and that the set xo previously determined is the only set which satisfies 4; hence 4 defines s , and xo is a set of real numbers which is definable in G. We have thus arrived a t a contradiction: xoproves both to belong and not to belong to Dz ;and therefore we must reject the assumption that Dzis definable in G. I t may be noticed that our argument even leads to a somewhat stronger conclusion: every set which i s definable in 6 and contains all definable sets of real numbers as elements i s non-denumerable (and in fact has a t least the power of the field of the relation R). At any rate See H . Lebesgue, S u r les fonctions reprtsentables analytiquement, Journal d e math6rnatique, series 6, vol. 1 (1905), pp. 139-216, specifically p. 213.
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ALFRED TARSKI
the solution of our main problem proves to be negative for n = 2 and, more generally, for every natural number n 1 2. m e n we now turn to the case n = 1, the method of reasoning just applied is seen to fail. In fact, we have been unable so far to construct a well-ordering relation of real numbers which has a non-denumerable field, and it seems doubtful that such a relation will ever be constructed. This can also be put in another way: the argument outlined above for n = 2 reduces essentially to the fact that we can construct a function f which correlates, with every denumerable set a of sets of real numbers, a set f(a) of real numbers which does not belong to a ; we do not know, however, whether a function with an analogous property can be defined for denumerable sets of real numbers. Also, no other method of attacking the problem has proved successful so far. Thus, the problem remains open whether the notion of a real number definable in a formal system 6 (in which the theory of real numbers can adequately be formalized) is itself definable in G . ~ It may be observed that the problem can be solved for certain fragmentary systems in which more or less comprehensive portions of the theory of real numbers can be formalized. This applies, for instance, to the system 6' in which all variables represent real numbers and the only constants-besides the sentential oonnectives, the quantifier, and the identity sign-are the symbols of arithmetical addition and multiplication. In fact, it can be shown that a real number is definable in 6' if, and only if, it is algebraic; and that a set of real numbers is definable in G' if, and only if, it is the set-theoretical union of a finite number of intervals (closed or open, bounded or unbounded) with algebraic endpoints;7 hence the set of all real numbers which are definable in G' is not definable in it. Some special cases of the notion of definability in our original system G are of considerable interest. Let, for instance, D,,, be the set of all those elements of the nth order which can be defined in G by means of formulas containing no variables of an order higher than p; we use here the notion of a defining formula in the second sense mentioned above (except of course the case n = O), and we assume that n - 1 5 p, for otherwise the set D,,, is empty. All the sets D,,, can be shown to be definable in (3;in fact, each set D,,, proves to be a member of ,+,+ . If, however, we discuss the problem whether the set D,,, is a member of Dn+l,p, we arrive at conclusions analogous to those obtained for the sets D, : the solution of the problem is positive for n = 0, is negative for n 2 2, and the problem is open for n = 1 (except the trivial case n = 1, p = 0 when the set D,+I , is empty). On the other hand, the situation changes essentially if we concern ourselves with the sequences, and not with the sets, of all definable elements of a given There is a close connection between the problem in question and the axiom of constructibility discussed by K. Gijdel in his monograph The consistency of the continuum hypothesis, Princeton 1940. This connection is of such a sort as to make i t seem very unlikely that a n affirmative solution of the problem is possible. We shall not elaborate on this point. 7 The result just mentioned was stated without proof on p. 234 in the author's paper cited in footnote 2. The argument which leads to this result is rather involved, but i t gives a t the eame time a finitary proof of consistency and completeness of the system G', and permits the establishing of a decision procedure in this system ; and these results can in turn be extended to the formalized system of elementary geometry.
A PROBLEM CONCERNING THE NOTION OF DEFINABILITY
111
order. As we remember, a definable element of the nth order can be regarded as defined by a formula which contains a certain variable "s" as the only free variable. (It is irrelevant in which sense the notion of a defining formula is used here, and hence the variable "z" may be assumed to be either of the nth or of the (n - 1)st order.) All such formulas can easily be arranged in an infinite sequence; we can thus speak of the mth formula in 6 containing "s" as the only free variable. Consequently, we can also speak of the mth element of the nth order definable in G ; and therefore, for each natural number n, we can consider the relation R, which holds between an element x of the nth order and a natural number m if, and only if, z is the mth element of the nth order which is definable in G. If we now ask the question whether these relations R, are themselves definable in 6 , the answer is again trivially affirmative for n = 0, but for every positive value of n, n = 1 included, a straight application of the diagonal procedure leads to a negative answer.
