¹ Compare my earlier papers: Einige Betrachtungen über die Begriffe der ω-Wider-spruchsfreiheit und der ω-Vollständigheit, Monatshefte für Mathematik und Physik, vol. 40 (1933), see p. 111Google Scholar; Pojęcie prawdy w językach nauk dedukcyjnych, Travaux de la Société des Sciences et des Lettres de Varsovie, Classe III, no. 34, Warsaw 1933, see p. 103Google Scholar; Der Wahrheitsbegriff in den formalisierten Sprachen, Studio philosophica, vol. 1 (1936)Google Scholar; Über den Begriff der logischen Folgerung, Actes du Congrès International de Philosophie Scientifique 1935, VII Logique, 1936, see p. 4Google Scholar. The above papers will be further quoted as Tarski₁, Tarsk₂, Tarski₃, and Tarski₄ respectively.

² Compare for instance Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198CrossRefGoogle Scholar; Tarski, A., Sur les ensembles définissables de nombres reels I, Fundamenta mathematicae, vol. 17 (1931), pp. 210–239CrossRefGoogle Scholar; these two papers will be quoted below as Göde₁ and Tarski₅. Cf. also Tarski₂, p. 96 ff., or Tarski₃,, p. 363 ff.

³ Compare, for example, Skolem, Th., Über einige Grundlagenfragen der Mathematik, Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo, I. Mat.-naturv. klasse, 1929, no. 4, see §1Google Scholar; Bernays, P., A system of axiomatic set theory—Part I, this Journal, vol. 2 (1937), pp. 65–77Google Scholar.

⁴ It is especially emphasized that the concept of a logical system, as it 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 a statement of the system is said to follow from other statements of the system.

⁵ This was found by K. Gödel and the present author independently of one another. Cf. Gödel₁, and Tarski₂, p. 35 ff., or Tarski₃, p. 301 ff.

⁶ In order to make these remarks as general as possible and, at the same time, to 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 “x(E)”; it should therefore be expressly noted that the meaning of this symbol is determined not by the class E but by the symbol “E” (or by the definition of this symbol): to one and the same class E correspond different statements x(E). It 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, at 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” a variable “X” appears which denotes any sub-class of S. But in that case it is necessary to make use of the more powerful deductive devices spoken of in §§5 and 8.

⁷ Cf. Tarski₁, pp. 96–99, or Tarski₂, pp. 370–374 (Theorem 1(α)).

⁸ Cf. Gödel,, 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 “^x(E)” (cf. Footnote 6). What is hereby 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 powerful deductive devices as the definition of true statement, see below §§5 and 8.

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 filozoficzny, vol. 10, 1926–1927, p. 96)Google Scholar. Compare Tarski₁( pp. 97 and 111, as well as Tarski₂, pp. 107–110, or Tarski₃, pp. 383–387. Compare also Hilbert, D., Die Grundlegung der elementaren Zahlenlehre, Mathematische Annalen, vol. 104 (1930–1931), pp. 485–494CrossRefGoogle Scholar; Carnap, R., The logical syntax of language, New York and London 1937, pp. 38 and 173Google Scholar. The rule of infinite induction has recently been treated by Rosser, B. in his paper Gödel theorems for non-constructive logics, this Journal, vol. 2 (1937), pp. 129–137Google Scholar, which will be cited below as Rosser₁. It 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 Tarski₄, pp. 3–4.

¹¹ This result for the class D _Ω was established in Rosser₁, p. 134.

¹² Compare Gödel, ₁, and Rosser, B., Extensions of some theorems of Gödel and Church, this Journal, vol. 1, (1936), pp. 87–91Google Scholar.

¹³ In connection with §5 compare Tarski₃, in particular pp. 316–318 and 393–405.

¹⁴ Cf. Tarski₂, p. 40, or Tarski₃, p. 305 f.

¹⁵ Cf. Tarski₃, pp. 401–403.

¹⁶ This theorem can be derived immediately from Theorem I in Tarski₂, p. 96, or Tarski₃, p. 370.

¹⁷ It was for this reason that I was able to state in my earlier papers, without any restriction, that the class D _Ω 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. Tarski₁, p. 111, and Tarski₄, 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., Rosser₁).

¹⁸ In connection with §8 compare Tarski₃,, pp. 393–405. See also Gödel₁, p. 191.

¹⁹ Compare here Footnote 4.

²⁰ Cf. Tarski₂, p. 69 ff., or Tarski₃, p. 338 ff.

²¹ This follows from the considerations in Tarski₂, pp. 71 ff., and 104, or Tarski₃, pp. 340 ff. and 379 ff. (the proof of Theorem II).

²² 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 the expression, “the class X has a definition which can be formalized in the system L, and in fact with the help only of signs of level <ν.” The paper Tarski₅, will give an idea how the definition of these expressions is to be set up; the same deductive devices are here required as in the case of the definition of truth. Theorem 6.2 can now be given the following stronger form: If X is any subclass of Tr which is definable within the system L, then there is a statement which is undecidable in X, i.e., a zϵS such that z∉X and z¯macr;∉X. If, moreover, ν is a natural number > 1 and X lias a definition which can be formalized within L with the help only of signs of level < ν, then it can be asserted that the undecidable statement z s of level ν. It should be pointed out that this is—in contrast to 6.2—not a schema but a correct metalogical theorem (cf. Footnote 6).

²³ Cf. Footnote 21.

²⁴ Cf. my report: Über definierbare Mengen reeller Zahlen, Annales de la Société Polonaise de Mathématique, vol. 9 (1930), pp. 206–207Google Scholar. Part of the ideas and results there sketched have later been fully developed in Tarski₅.

²⁵ Cf. remarks concerning such systems of logic in Tarski₃, pp. 393–398.