Undecidable Statements and the Concept of Truth

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On Undecidable Statements in Enlarged Systems of Logic and the Concept of Truth Alfred Tarski The Journal of Symbolic Logic, Vol. 4, No. 3. (Sep., 1939), pp. 105-112. Stable URL: http://links.jstor.org/sici?sici=0022-4812%28193909%294%3A3%3C105%3AOUSIES%3E2.0.CO%3B2-C The Journal of Symbolic Logic is currently published by Association for Symbolic Logic.

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TEE JOURNAL OF SYMBOLIC Loarc Volume 4, Number 3, September 1939

ON UNDECIDABLE STATEMENTS IN ENLARGED SYSTEMS OF LOGIC AND THE CONCEPT OF TRUTH ALFRED TARSHI

I t is my intention in this paper to add somewhat to the observations already made in my earlier publications on the existence of undecidable statements in systems of logic possessing rules of inference of a "non-finitary" ("non-constructive") character ($$1-4).' I also wish to emphasize once more the part played by the concept of truth in relation to problen~sof this nature (§§5-8). At the end of this paper I shall give a result which was not touched upon in my earlier publications. It seems to be of interest for the reason (among others) that it is an example of a result obtained by a fruitful combination of the method of constructing undecidable statements (due to K. Godel) with the results obtained in the theory of truth.

1. Let us consider a formalized logical system L. Without giving a detailed description of the system we shall assume that it possesses the usual "finitary" ("constructive") rules of inference, such as the rule of substitution and the rule of detachment (modus ponens), and that among the laws of the system are included all the postulates of the calculus of statements, and finally that the laws of the system suffice for the construction of the arithmetic of natural numbers. Moreover, the system L may be based upon the theory of types and so be the result of some formalization of Prim'pia mat he ma tic^.^ I t may also be a system which is independent of any theory of types and resembles Zennelo's set theory.' In this paper we shall denote the class of all statements belonging to the Received February 6, 1939. Compare my earlier papers: Einige Betrachtungen iiber die Begrife der @-Widerspruchsjreiheit und der w-Vollstcindigkeit, Mombhejte fiir Mathematik und Physik, vol. 40 (1933), see p. 111; Pojfcie prawdy w jezykach nauk dedukcyjnych, Travaux de la Socibtb des Sciences et des Lettres de Varsovie, Classe 111, no. 34, Warsaw 1933, see p. 103; Der Wahrheitsbegriff in den formalisierten Sprachen, Studia philosophica, vol. 1 (1936) ; Ober den Begriff der logischen Folgerung, Actes du CongrPs International d e Philosophie ScientilFque 1935, VII h i q u e , 1936, see p. 4. The above papers will be further quoted as tar ski^, Tarski*, Tarskis, and Tarski, respectively. Compare for instance K. Ciidel, Uber formal unentscheidbare S&ze der Principia Mathematica und verwandter Sysleme I , Mowlshefte jiir Mathemafik und Physik, vol. 38 (1931), pp. 173-198; A. Tarski, S u r les ensembles d6jnissables de nombres reels I , Fundam e n k mathematicae, vol. 17 (1931), pp. 210-239; these two papers will be quoted below as Godell and Tarski6. Cf. also Tarskit, p. 96 ff., or Tarskis, p. 363 ff. a Compare, for example, Th. Skolem, Ober einige Grundlagenjragen der Mathemcrlik, Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo, I . Mat.-naturv. klasse, 1929, no. 4, see $1; P. Bernays, A system of aziomatic set theory-Part f, this JOURNAL, V O ~2 . (1937), pp. 65-77. 105

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system L by the symbol "S", and the class of all demonstrable statements belonging to L by "D".' The symbol "3" will denote the negation of the statement z, and the symbol "z-y" the implication which has the statement s as antecedent and the statement y as consequent. Finally, the symbol "z-y" will denote the equivalence whose terms are z and y. The forn~ulss"zsYJ' and "zaY" will, as usual, express respectively that an object z belongs and that it does not belong to the class Y. Similarly, the formula "XCY" expresses that the class X is contained. in the class Y. We assume it to be understood that metalogical statements about the system L can, at least in part, be formalized, or rather interpreted, in the system L itself.' 2. In what follows we shall also use the symbol "EJJ,which is assumed to fulfil the following conditions:

2.1. "E" is deflned in metalogic and denotes a class of statements belonging 20 the system L. 2.2. The metalogical definition of the symbol 'fE" can be jomuzlized in the sljstem L.

It follows from these assumptions that to every statement z there corresponds another statement y which is the result of formalising the metalogical statement "zeE". We shall denote this statement y by the symbol "z(E)J'.6 The following theorem can be proved on the basis of the conventions adopted: 2.3. There is a statement zeS such that Zc+zcn e D.

' It'is especially emphasized that the concept of a logical system, aa i t is used here, must not be confused with that of the class of all its demonstrable statements. A logical system is determinate when we know what signs occur in it, what series of its signs are to be regarded as statements, which among these statements are distinguished as demonstrable (i.e., as axioms and theorems), and, more generally, under what circumstances s statement of the system is said to follow from other statements of the system. 8 This was found by K. Godel and the present author independently of one another. Cf. Godel,, and Tarskill p. 35 ff., or Tarskia, p. 301 ff. 6 In order to make these remarks aa general as possible and, a t the same time, t o avoid complicated formulations, I have adopted a somewhat inaccurate mode of expression which may lead the reader to suppose that I do not always distinguish between the object and the symbol which denotes it. A typical instance of an inaccuracy of this kind is seen in the use of the symbol "z(E)"; i t should therefore be expressly noted that the meaning of this symbol is determined not by the clam E but by the symbol "E" (or by the definition of this symbol): t o one and the same class E correspond different statements z(r). I t is also quite clear that the conditions 2.1 and 2.2 concern the symbol " E and not the class E. In fact, all the theorems of the present paper in which "E" appears are, a t bottom, not metalogical theorems, but schemata from which whole series of particular theorems can be obtained by replacing "E" by any constant which satisfies conditions 2.1 and 2.2. These schemata can of course be transformed into general metalogical theorems in which, in the place of " E Ma variable "X" appears which denotes any sub-class of S. But in that case it is necessary to make w e of .the more powerful deductive devices spoken of in $05 and 8.

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The proof of this theorem (or, to be exact, of a closely related theorem) was outlined by me in one of my earlier I t depends on the application of the same idea with the help of which Godel succeeded in proving that, for the class D, undecidable statements (i.e., statements z such that ztD and Z a ) can be constructed.' 3. If some additional assumptions with regard to E are adopted it becomes. easy to deduce from Theorem 2.3 that undecidable'statements can also be constructed for the class E. These additional assumptions are the following: 3.1. D CE.

3.2. If zeE and z-vy c E, then ycE (in other words the class E is closed with respect to the rule of detachment). 3.3. If zcE, then .zE)@. 3.4. If ZEE, then zcB)aE.

For lack of a better term we shall call a class E which fulfills the conditions 3.3 and 3.4 a content-consistent class of statenzents.' If the class E satisfies the conditions 3.1 and 3.2 we shall say that it is a contenl-consistent enlargement of the class D. We then obtain the following theorems: 3.6. If a class E is a content-consistent enlargement of the class D and 2ttz(.] c Dl then z@ and Z@. In fact, if Z++Z(B) t D then, by the laws of the calculus of statements, we have z-+zi) ;D and and+z(.) E D. From these statements we derive, according to 3.1, z-+& E E and Z--+Z~B) c E. If, therefore, zcE, we should have, by reference to 3.2, z(.,cE, which is incompatible with 3.3; similarly, by 3.2 and 3.4, ZeE cannot occur. From the theorems 2.3 and 3.5 follows at once the theorem: 3.6. If the class E is a content-consistent enlargemend of the class D, then there is a statement which is undecidable in E, i.e., a zcS such that Z 4 E and Z4E. 4. Different examples of classes E are known of which it seems reasonable to assume that they are content-consistent enlargements of the class D. The simplest example of such a class is the class D itself. Other examples can be obtained in the following may: To the postulates and rules of inference of - -

-

Cf. Tarskit, pp. 96-99, or Tarskir, pp. 370-374 (Theorem I ( a ) ) . W f . Godel,, especially pp. 187-190. 9 The definition of content-consistency gives rise to difficulties which are analogous to those occasioned by the introduction of the symbol "z(r)"(cf. Footnote6). What ishereby defined is not a kind of class of statements but a kind of symbol (constant) denoting such classes. The concept of the content-consistent class E of statements is relative in character; it must be relativized to a definition of the class E. An exact definition of the concept in question would require just as pon,erful deductive devices as the definition of true statement, see below $05 and 8. 7

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the system L we add a finite or infinite number of new postulates, and we add new rules of a "finitary" or "non-finitary" character which can be formalized in the system L and which are believed always to lead from true statements to true statements. We then define E as the smallest class of statements which contains all the postulates (old and new) and which is closed with respect to all the rules of inference (old and new). The best known rule of a "non-finitary" character is the so-called rule of infinite induction, which provides that, if all statements of the form ii4(0)J1, "@(I)", "4(2)", etc. belong to El then the statement "for every natural number n +(n)" shall also belong to E." Let "Dn" denote the smallest class E which contains the class D and is closed with respect to the rule of infinite induction (and with respect to all the rules of inference which are valid in L). Theorems 3.5 and 3.6, when applied to the classes E= D and E =Dn, can be considerably simplified. Each of these classes obviously satisfies the first two of the conditions 3.1 to 3.4 (contained implicitly in the hypotheses of 3.5 and 3.6). It can also be proved that these classes satisfy the following condition: If x& then z c n e ~ . "I t is easily seen that for this reason condition 3.3 can be replaced by a weaker condition which expresses the consistency of the class E in the ordinary sense; but since this condition is a consequence of 3.4, the condition 3.3 can be omitted altogether. Of the four conditions assumed only 3.4 remains. This result when applied to D can be still further simplified: 3.4 can be replaced by a condition which ensures the so-called w-consistency or even the ordinary consistency of the class 0.12 6." We shall now make use, in our metalogical considerations, of those means which cannot be formalized in the system L itself. The class of all the true statements of L can then, as we know, be defined in metalogic. We shall denote this class by "Tr". Some known properties of the class Tr are stated in the following theorems: 6.1. The class Tr contains among its elements all the postulates of the system L and is closed with respect to all the rules of inference of this system; therefore D C T r . 6.2. The class Tr is closed with respect to the rule of infinite induction; therefore DpCTr. 10 I drew attention to this rule in 1926, and discussed it in a lecture before the Second Polish Philosophical Congress (Warsaw 1927-cf. the reference to this lecture in Ruch Jlozoficzny, vol. 10, 1926-7, p. 96). Compare Tarski,, pp. 97 and 111, as well as Tarski*, pp. 107-110, or Tarskia, pp. 383-387. Compare also D. Hilbert, Die Grundlegung der elementaren Zahlenlehre, Mathematische AnnoZen, vcl. 104 (1930-I), pp. 485494; R. Carnap, The logical syntax of language, New York and London 1937, pp. 38 and 173. The rule of infinite induction has recently been treated by B. Rosser in his paper Godel theorems for non-conslruclive logics, this JOURNAL, vol. 2 (1937), pp. 129-137, which will be cited below as Rosserl. I t may be mentioned that attention had already been called to certain other ("constructive") rules of inference, which Rosser describes (loc. cit., p. 134) as rules of Kleene's type, in Tarski,, pp. 3-4. 1' This result for the c!ass Dn was established in Rosser~, p. 134. l2 Compare Godel,, and B. Rosser, Eztensions of some theorems of Godel and Church, this JOURNAL, 101. 1, (1936), pp. 87-91.

18 In connection with 45 compare Tarskia, in particular pp. 316-318 and 393-405.

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6.3. If zcTr then ZdTr (in other words, the class Tr i s consistent).

6.4. If x t S and zdTr, then ZCTT(in other words, the class Tr i s complete). 6.6. xtE if and only if xca,tTr.

The property stated in 5.5 follows from a more general and very characteristic property of the class Tr, for it can be shown that, on the basis of the definition of Tr, every theorem which falls under the following schema can be proved: 6.6. p if and only if xrTr.

In this schema "p" can be replaced by any statement of the system L and

"x" by the metalogical designation of this statement." 6. From Theorems 5.1 to 5.5 some interesting conclusions regarding the problem of undecidable statements can be obtained. 6.1. If E C T r and Zttzcn a D, then zaTr, z@ and z ~ E . ' ~ In order to prove this statement let us assume that zaTr. Then by reference to 5.4 we have ZeTr. I t follows also from the hypothesis of the theorem that Z+z I . If there. ezist statements z c S of h e 2 v at all, then there exists a statement z e S of level v which is not equivalent to any statement of a lower level; or, more exactly, which satisfies the following condition: @ z e S is any statement of level 1, Tr, be the class of all true statements of level < v and E , the class of thsse statements x for which there exists a statement ytTr, such that y-tz t D. Ifi'contrast to the whole class Tr, the classes Tr, and E, can be defined by such means as can be formalized within the system L.21 Moreover from 5.1 we have D C E , C T r . Consequently we can apply Theorem 6.2 (in a stronger formulation2*)to the class E,. In this way Cf. Tarskit, p. 69 ff., or Tarskii, p. 338 ff. This follows from the considerations in Tarski2, pp. 71 ff., and 104, or Tarski,, pp. 340 ff. and 379 ff. (the proof of Theorem 11). a2 In order to obtain this sharper formulation of Theorem 6.2 the meaning of the expression, "the class X of statements is definable within the system L," is first established, and then that of theexpression, "the class X ha$a definition which can be formalized in the systemL, and in fact with the help only of signs of levcl