International Journal of Theoretical Physics

, Volume 21, Issue 12, pp 941–954 | Cite as

Gödel's theorem and information

  • Gregory J. Chaitin
Physical Models of Computation


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.


Field Theory Elementary Particle Quantum Field Theory Traditional Proof Incompleteness Phenomenon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bell, E. T. (1951).Mathematics, Queen and Servant of Science, McGraw-Hill, New York.Google Scholar
  2. Bennett, C. H. (1982). The thermodynamics of computation—a review,International Journal of Theoretical Physics,21, 905–940.Google Scholar
  3. Chaitin, G. J. (1974a). Information-theoretic computational complexity,IEEE Transactions on Information Theory,IT-20, 10–15.Google Scholar
  4. Chaitin, G. J. (1974b). Information-theoretic limitations of formal systems,Journal of the ACM,21, 403–424.Google Scholar
  5. 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.)Google Scholar
  6. Chaitin, G. J. (1975b). A theory of program size formally identical to information theory,Journal of the ACM,22, 329–340.Google Scholar
  7. Chaitin, G. J. (1977). Algorithmic information theory,IBM Journal of Research and Development,21, 350–359, 496.Google Scholar
  8. 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.Google Scholar
  9. 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.Google Scholar
  10. Chaitin, G. J. (1982). Algorithmic information theory,Encyclopedia of Statistical Sciences, Vol. 1, Wiley, New York, pp. 38–41.Google Scholar
  11. Cole, C. A., Wolfram, S., et al. (1981).SMP: a symbolic manipulation program, California Institute of Technology, Pasadena, California.Google Scholar
  12. Courant, R., and Robbins, H. (1941).What is Mathematics?, Oxford University Press, London.Google Scholar
  13. 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.Google Scholar
  14. Davis, M. (1978). What is a computation?, inMathematics Today: Twelve Informal Essays, L. A. Steen (ed.), Springer-Verlag, New York, pp. 241–267.Google Scholar
  15. 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.Google Scholar
  16. Eigen, M., and Winkler, R. (1981).Laws of the Game, Knopf, New York.Google Scholar
  17. 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.Google Scholar
  18. Einstein, A. (1954).Ideas and Opinions, Crown, New York, pp. 18–24.Google Scholar
  19. Feynman, R. (1965).The Character of Physical Law, MIT Press, Cambridge, Massachusetts.Google Scholar
  20. 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.Google Scholar
  21. 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.Google Scholar
  22. Hofstadter, D. R. (1979).Gödel, Escher, Bach: an Eternal Golden Braid, Basic Books, New York.Google Scholar
  23. Levin, M. (1974).Mathematical Logic for Computer Scientists, MIT Project MAC report MAC TR-131, Cambridge, Massachusetts.Google Scholar
  24. Polya, G. (1959). Heuristic reasoning in the theory of numbers,American Mathematical Monthly,66, 375–384.Google Scholar
  25. 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.Google Scholar
  26. 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.Google Scholar
  27. Taub, A. H. (ed.) (1961).J. von Neumann — Collected Works, Vol. I, Pergamon Press, New York, pp. 1–9.Google Scholar
  28. von Neumann, J. (1956). The mathematician, inThe World of Mathematics, Vol. 4, J. R. Newman (ed.), Simon and Schuster, New York, pp. 2053–2063.Google Scholar
  29. 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.Google Scholar
  30. von Neumann, J. (1966).Theory of Self-Reproducing Automata, A. W. Burks (ed.), University of Illinois Press, Urbana, Illinois.Google Scholar
  31. Weyl, H. (1946). Mathematics and logic,American Mathematical Monthly,53, 1–13.Google Scholar
  32. Weyl, H. (1949).Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, New Jersey.Google Scholar
  33. Wilf, H. S. (1982). The disk with the college education,American Mathematical Monthly,89, 4–8.Google Scholar

Copyright information

© Plenum Publishing Corporation 1982

Authors and Affiliations

  • Gregory J. Chaitin
    • 1
  1. 1.IBM ResearchYorktown Heights

Personalised recommendations