Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T14:28:18.896Z Has data issue: false hasContentIssue false
  • English
  • Français

Gödel, Tarski, Church, and The Liar

Published online by Cambridge University Press:  15 January 2014

György Serény
György Serény*
Affiliation:
Department of Algebra, Budapest University of Technologyand Economics, 1111 Stoczek U.2. Hép.5. EM., Budapest, HungaryE-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

The fact that Gödel's famous incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of the community of logicians. Indeed, almost every more or less formal treatment of the theorem makes a reference to this connection. Gödel himself remarked in the paper announcing his celebrated result (cf. [7]):

The analogy between this result and Richard's antinomy leaps to the eye;

there is also a close relationship with the ‘liar’ antinomy, since … we are

… confronted with a proposition which asserts its own unprovability.

In the light of the fact that the existence of this connection is commonplace it is all the more surprising that very little can be learnt about its exact nature except perhaps that it is some kind of similarity or analogy. There is, however, a lot more to it than that. Indeed, as we shall try to show below, the general ideas underlying the three central theorems concerning internal limitations of formal deductive systems can be taken as different ways to resolve the Liar paradox. More precisely, it will turn out that an abstract formal variant of the Liar paradox, which can almost straightforwardly inferred from its original ordinary language version, is a possible common generalization of (both the syntactic and semantic versions of) Gödel's incompleteness theorem, the theorem of Tarski on the undefinability of truth, and that of Church concerning the undecidability of provability.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 9 , Issue 1 , March 2003 , pp. 3 - 25
Copyright
Copyright © Association for Symbolic Logic 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Baaz, M. and Pudlák, P., Kreisel's conjecture for L∃1 , Arithmetic, proof theory, and computational complexity (Clote, P. and Krajíˇek, J., editors), Claredon Press, Oxford, 1993, pp. 30–60.Google Scholar
[2] Barnes, D. W. and Mack, J. M., An algebraic introduction to mathematical logic, Springer-Verlag, New York, 1975.Google Scholar
[3] Boolos, G., The logic of provability, Cambridge University Press, Cambridge, 1995.Google Scholar
[4] Boolos, G. and Jeffrey, R. C., Computability and logic, Cambridge University Press, Cambridge, 1992.Google Scholar
[5] Chang, C. C. and Keisler, H. J., Model theory, North Holland, Amsterdam, 1973.Google Scholar
[6] Findlay, J. N., Goedelian sentences: A non-numerical approach, Mind, vol. 51 (1942), pp. 259–265.Google Scholar
[7] Gödel, K., On formally undecidable propositions of Principia Mathematica and related systems, Oliver and Boyd, Edinburgh, 1962, translated by Meltzer, B., introduction by R. B. Braithwaite.Google Scholar
[8] Grandy, R. E., Advanced logic for applications, D. Reidel, Dordrecht, 1979.Google Scholar
[9] Grelling, K. and Nelson, L., Bemerkungen zu den Paradoxien von Russell und Burali–Forti, Abhandlungen der Fries'schen Schule, vol. 2 (1907–1908), pp. 300–334.Google Scholar
[10] Mendelson, E., Introduction to mathematical logic, D. Van Nostrand Company, Inc., Princeton, 1965.Google Scholar
[11] Myhill, J., A system which can define its own truth, Fundamenta Mathematicae, vol. 37 (1950), pp. 190–192.Google Scholar
[12] Parikh, R. J., Some results on the length of proofs, Transactions of the America Mathematical Society, vol. 177 (1973), pp. 29–36.Google Scholar
[13] Quine, W. V., Mathematical logic, Harvard University Press, Cambridge, 1951.Google Scholar
[14] Quine, W. V., The Ways of Paradox and other essays, Harvard University Press, Cambridge, 1979.Google Scholar
[15] Ryle, G., Heterologicality, Analysis, vol. 11 (1951), no. 3.Google Scholar
[16] Shoenfield, J. R., Mathematical logic, Addison–Wesley, Reading, Massachusetts, 1967.Google Scholar
[17] Smorynski, C., The incompleteness theorems, Handbook of mathematical logic (Barwise, J., editor), North–Holland, Amsterdam, 1977, pp. 821–865.Google Scholar
[18] Smullyan, R. M., Languages in which self reference is possible, The Journal of Symbolic Logic, vol. 22 (1957), no. 1, pp. 55–67.Google Scholar
[19] Smullyan, R. M., Gödel's incompleteness theorems, Oxford University Press, New York, 1992.Google Scholar
[20] Tarski, A., The semantic conception of truth, Philosophy and Phenomenological Research, vol. 4 (1944).Google Scholar
[21] Tarski, A., Truth and proof, Scientific American, vol. 220 (1969), no. 6.Google Scholar