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A problem concerning the notion of definability

Published online by Cambridge University Press:  12 March 2014

Alfred Tarski
Alfred Tarski*
Affiliation:
University of California(Berkeley)
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Extract

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 definable 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 character.

Type
Articles
Information
The Journal of Symbolic Logic , Volume 13 , Issue 2 , June 1948 , pp. 107 - 111
Copyright
Copyright © Association for Symbolic Logic 1948

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References

1 Compare here Fraenkel, A., Einleitung in die Mengenlehre, 3rd edition, Berlin 1928CrossRefGoogle Scholar (reprinted 1946, U.S.A.), especially Chapter 4; further bibliographical 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 Sur les ensembles définissables de nombres réels I, Fundamenta mathematicae, vol. 17 (1931), pp. 210–239. Part II 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 discussed; compare also the formalism described in Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatsheƒte ƒür Mathematik und Physik, vol. 38 (1931), pp. 173–198.CrossRefGoogle Scholar

4 For a precise definition of the expression “a satisfies φ,” see the author's work Der Wahrheitsbegriff in den formalisierten Sprachen, Studia philosophica, vol. 1 (1936), pp. 261–405, especially Chapters 3 and 4.

5 See Lebesgue, H., Sur les fonctions représentables analytiquement, Journal de mathématique, series 6, vol. 1 (1905), pp. 139–216Google Scholar, specifically p. 213.

6 There is a close connection between the problem in question and the axiom of constructibility discussed by Gödel, K. in his monograph The consistency of the continuum hypothesis, Princeton 1940.Google Scholar This connection is of such a sort as to make it seem very unlikely that an 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 it gives at the same time a finitary proof of consistency and completeness of the system , 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.