Abstract
Gödel's theorem may be demonstrated using arguments having an informationtheoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.
Similar content being viewed by others
References
Bell, E. T. (1951).Mathematics, Queen and Servant of Science, McGraw-Hill, New York.
Bennett, C. H. (1982). The thermodynamics of computation—a review,International Journal of Theoretical Physics,21, 905–940.
Chaitin, G. J. (1974a). Information-theoretic computational complexity,IEEE Transactions on Information Theory,IT-20, 10–15.
Chaitin, G. J. (1974b). Information-theoretic limitations of formal systems,Journal of the ACM,21, 403–424.
Chaitin, G. J. (1975a). Randomness and mathematical proof,Scientific American,232 (5) (May 1975), 47–52. (Also published in the French, Japanese, and Italian editions ofScientific American.)
Chaitin, G. J. (1975b). A theory of program size formally identical to information theory,Journal of the ACM,22, 329–340.
Chaitin, G. J. (1977). Algorithmic information theory,IBM Journal of Research and Development,21, 350–359, 496.
Chaitin, G. J., and Schwartz, J. T. (1978). A note on Monte Carlo primality tests and algorithmic information theory,Communications on Pure and Applied Mathematics,31, 521–527.
Chaitin, G. J. (1979). Toward a mathematical definition of ‘life’, inThe Maximum Entropy Formalism, R. D. Levine and M. Tribus (eds.), MIT Press, Cambridge, Massachusetts, pp. 477–498.
Chaitin, G. J. (1982). Algorithmic information theory,Encyclopedia of Statistical Sciences, Vol. 1, Wiley, New York, pp. 38–41.
Cole, C. A., Wolfram, S., et al. (1981).SMP: a symbolic manipulation program, California Institute of Technology, Pasadena, California.
Courant, R., and Robbins, H. (1941).What is Mathematics?, Oxford University Press, London.
Davis, M., Matijasevic, Y., and Robinson, J. (1976). Hilbert's tenth problem. Diophantine equations: positive aspects of a negative solution, inMathematical Developments Arising from Hilbert Problems, Proceedings of Symposia in Pure Mathematics, Vol. XXVII, American Mathematical Society, Providence, Rhode Island, pp. 323–378.
Davis, M. (1978). What is a computation?, inMathematics Today: Twelve Informal Essays, L. A. Steen (ed.), Springer-Verlag, New York, pp. 241–267.
Dewar, R. B. K., Schonberg, E., and Schwartz, J. T. (1981).Higher Level Programming: Introduction to the Use of the Set-Theoretic Programming Language SETL, Courant Institute of Mathematical Sciences, New York University, New York.
Eigen, M., and Winkler, R. (1981).Laws of the Game, Knopf, New York.
Einstein A. (1944). Remarks on Bertrand Russell's theory of knowledge, inThe Philosophy of Bertrand Russell, P. A. Schilpp (ed.), Northwestern University, Evanston, Illinois, pp. 277–291.
Einstein, A. (1954).Ideas and Opinions, Crown, New York, pp. 18–24.
Feynman, R. (1965).The Character of Physical Law, MIT Press, Cambridge, Massachusetts.
Gardner, M. (1979). The random number omega bids fair to hold the mysteries of the universe, Mathematical Games Dept.,Scientific American,241 (5) (November 1979), 20–34.
Gödel, K. (1964). Russell's mathematical logic, and What is Cantor's continuum problem?, inPhilosophy of Mathematics, P. Benacerraf and H. Putnam (eds.), Prentice-Hall, Englewood Cliffs, New Jersey, pp. 211–232, 258–273.
Hofstadter, D. R. (1979).Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, New York.
Levin, M. (1974).Mathematical Logic for Computer Scientists, MIT Project MAC report MAC TR-131, Cambridge, Massachusetts.
Polya, G. (1959). Heuristic reasoning in the theory of numbers,American Mathematical Monthly,66, 375–384.
Post, E. (1965). Recursively enumerable sets of positive integers and their decision problems, inThe Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, M. Davis (ed.), Raven Press, Hewlett, New York, pp. 305–337.
Russell, B. (1967). Mathematical logic as based on the theory of types, inFrom Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, J. van Heijenoort (ed.), Harvard University Press, Cambridge, Massachusetts, pp. 150–182.
Taub, A. H. (ed.) (1961).J. von Neumann — Collected Works, Vol. I, Pergamon Press, New York, pp. 1–9.
von Neumann, J. (1956). The mathematician, inThe World of Mathematics, Vol. 4, J. R. Newman (ed.), Simon and Schuster, New York, pp. 2053–2063.
von Neumann, J. (1963). The role of mathematics in the sciences and in society, and Method in the physical sciences, inJ. von Neumann — Collected Works, Vol. VI, A. H. Taub (ed), McMillan, New York, pp. 477–498.
von Neumann, J. (1966).Theory of Self-Reproducing Automata, A. W. Burks (ed.), University of Illinois Press, Urbana, Illinois.
Weyl, H. (1946). Mathematics and logic,American Mathematical Monthly,53, 1–13.
Weyl, H. (1949).Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, New Jersey.
Wilf, H. S. (1982). The disk with the college education,American Mathematical Monthly,89, 4–8.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chaitin, G.J. Gödel's theorem and information. Int J Theor Phys 21, 941–954 (1982). https://doi.org/10.1007/BF02084159
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02084159