Undecidable Statements and the Concept of Truth

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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ALFRED TIRSICI

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.

UNDECIDABLE STATEMENTS I N D THE CONCEPT O F TRUTH

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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<$) t D. Applying 5.1 twice we obtain Z+Z(E) r Tr and zc$)eTr. According to 5.5 we then have z&, apd since E C T r we obtain zrTr, which is contrary to our assumption. Hence we must have zrTr. I t follows at once that I@; for if the contrary were the case we should have ZeTr which would be incompatible with 5.3. Moreover we obtain from tbe hypothesis: z-+z7& E D. If zcTr, we therefore obtain, by 5.1: zz,aTr. I t follows from this by 5.3 that qE)dTr, and finally by 5.5 that z&. Thus the statement z has all the required properties. From Theorems 2.3 and 6.1 follows a t once:

6.2. If ECTr, then there i s a statement which i s undecidable in El i.e., a zcS such that zdF and z@." In this way undecidable statements can be constructed for every class E containing only true statements, provided the definition of this class can be formalized in L. By 5.1 and 5.2 this result applies in particular to the classes D and Do; if we are using sufficiently powerful deductive devices in metalogic we may, without any additional assumptions, assert that in both of these classes undecidable statements can be found." I t is to be noted that the sole method of constructing undecidable statements that we actually possess leads Cf. Tarskill p. 40, or Tarski& p. 305 f.

Tarski$, pp. 401-403.

16 This theorem can be derived immediately from Theorem I in Tarski2, p. 96, or Tarskil,

14

la Cf.

p.

370.

"

It was for this reason that I was able to state in my earlier papers, without any restriction, that the class Dn contains undecidable statements; this I did in those situations in which the question of the power of the. available deductive devices used in metalogic played no essential part. Cf. Tarski,, p. 111, and Tarski,, p. 4 (it seems to me that these places in my papers have escaped the attention of logicians who have discussed these problems subsequently; cf., e.g., Rosserl).

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to statements z which satisfy the formula " Z t t ~ ( ~e) D" (or to the negations of such statements). Now with regard to every such statement z it can be proved -as we may see from 6.1-not only that this statement is undecidable in El but also that it is true. From theorem 6.2 it follows a t once that it is impossible to construct such a definition of Tr as could be formalized in L. If the contrary were the case undecidable statements could be found in Tr, and this would be incompatible with 5.4.

7. In order to clarify the relation between Theorems 3.5 and 6.1, and between 3.6 and 6.2, we shall prove: 7.1. If E C T r , then E is a content-consistent class of statements. For, if xeB, then we have, on the basis of 5.5, zca)tTr;but by 5.3 we infer from this that x(B)aTr, and since E C T r we obtain xif)aE. Similarly, if Z& then ZeTr, whence by 5.3 we obtain xtTr and, a fortiori, x4E; and by 5.5 this last formula yields x(&)~tTr, hence, a fortio~i,z ( B ) ~ E . The class E therefore satisfies the conditions 3.3 and 3.4, Q.E.D. From 5.1, 5.2, and 7.1 it follows, in particular, that the classes D and DO are .content-consistent. The theorem 7.1 could be used for the derivation of 6.1 from 3.5, but this procedure would not be very convenient. For it might hsppen,that a class E, contained in Tr, failed to satisfy tlie conditions 3.1 and 3.2 and therefore was not a (content-consistent) enlargement of the class D. 8.18 Let us now assume that the system L has been enlarged to form a system

L1 in which the definition of the class Tr (for the statements of the original system L) can be formalized.19 If the system L is based upon the theory of types this enlargement will consist, in the first instance, in an enrichment of its linguistic forms by the introduction of variables of a higher level (cf. 99 below). The-postulates and the rules of inference must be adapted to the enlarged resources of the linguistic forms, but their content, roughly speaking, does not change. If the system L is constructed aft,er the manner of the system of Zermelo, neither the linguistic forms nor the rules of inference are changed, but the change consists entirely in strengthening the set of postulates. The essential point here is the introduction of a new postulate, the content of which may be described as follows: There is a set X the elenleilts of which. provide a model for the system of postulates of the original system L. Let us assume that we have succeeded in proving that a given class E is contained in Tr and that in accordance with theorem 2.3 an undecidable st,atement z for the class E has been constructed such that Z + q n c D. We shall then be able to prove on the basis of 6.1 that zeTr. Making use of the characteristic propkrty of the class Tr which was mentioned a t the end of $5 we shall also be able to prove the statement z itself in metalogic. Moreover, as the theory of u In connection with 58 compare Tarski,, pp. 393-405. See aleo Godel,, p. 191.

lo Compare

here Footnote 4.

UNDECIDABLE STATEMENTS AND THE CONCEPT O F TRUTH

111

the class Tr can be interpreted in LI,we shall be able finally to repeat this process in the system LIand obtain the proof of the statement z in L1itself. I n this way every undecidable statement constructed for the class E of statements of the system L becomes decidable in the enlarged system L1with the help of the method employed in the proof of 2.3, provided only that the theory of truth can be formalized in this enlarged system.

9. I n conclusion we shall give another interesting application of the theory of truth to the questions under consideration. For the sake of simplicity we assume in what follows that the system L is constructed in the manner of Princi@a mathematics and, in particular, is based on the theory of types. With every sign which appears in the statements of L and which belongs to a logical type there can be correlated a natural number which is called its level (two signs which belong to the same type also have the same level, but the converse does not necessarily hold). Thus level 1 is ascribed to the (variable and constant) designations of individuals; to level 2 will be reckoned those signs which denote classes of individuals as well as twoand more-termed relations between individuals, and so on." We,shall take rn the level of a statement the greatest number which occurs as the level of a sign in it. But this concept will be applied only to statements in which no defined signs occur. The following theorem then holds:

9.1. Let v be any natural number > 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
An exact proof of this theorem will be published on a later occasion; here it will suffice to give a short indication of such s proof. Let v be a given natural number > 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 1 and X Iias a dejinition which can be formalized within L 21

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ALFRED TARSKI

If now x is any statement of we obtain a statement z such that z@, and z@,. level < v then we have, by 5.4, zcTrv and therefore xeE, or Z&,. Hence it follows easily that zt+x 4 D. Finally we note that signs of level $ v suffice From this we conclude that z is a statement for the definition of Tr, and of level v (this result can be reached by a closer analysis of the proofs of 6.2 and 2.3 and comes to light in the stronger formulation of 6.2 mentioned above). The statement z thus has the required properties. From Theorem 9.1 it follows in particular that a statement from the ariihmetic of real numbers can be found which is not equivalent to any statement from the arithmetic of natural numbers (i.e., from number theory sensu stricto). Similarly a statement from the theory of real functions can be found which is not equivalent to any statement from the arithmetic of real numbers etc., etc. (It may be mentioned that an analogous result, which, however, concerned not the demonstrability of statements but the definability of concepts, was obtained by the author by another method some years ago.24) The proof of 9.1 depends upon 6.2 and therefore requires deductive devices which cannot be formalized within the system L. But, without having recourse to such devices, we are able to establish all the special cases of 9.1 which can be obtained in the following way. For the variable "v" we substitute any constant which denotes (e.g., in the decimal scale) a natural number, and we furnish the resulting statement with a suitable consistency-hypothesis (namely, with the hypothesis of the content-consistency of the corresponding class E, mentioned above). Apart from the question of the deductive devices available in metalogic, Theorem 9.1 can be strengthened in various ways. Thus in the formulation of this theorem the sign "D" may be replaced by "E", in which case, however, (1) it is assumed that the sign "EMsatisfies the conditions 2.1 and 2.2, and (2) the theorem must have the hypothesis ('ECTr". Finally theorem 9.1 can be ,related to systems of logic possessing signs of transfinite level (in this case the variable "v" in the formulation of the theorem will denote ordinal numbers finite or transfinite), as well as, mutatis mutandis, to systems of logic which are not based on the theory of types.*' WARSAW

with the help only of signs of level < v , then i t can be asserted that the undecidable statement z i s of level v. It should be pointed out that this is-in contrast to 6.2-not a schema but a correct metalogical theorem (cf. Footnote 6). Z a Cf. Footnote 21. 2' Cf. my report: uber definierbare Mengen reeller ZahEen, Annales de & Soci6t6 Polonaise de Mathbmatique, vol. 9 (1930), pp. 206-207. Part of the ideas and results there sketched have later been fully developed in Tarskib. Cf. remarks concerning such systems of logic in Tarskir, pp. 393-398.