International Journal of Theoretical Physics

, Volume 46, Issue 8, pp 2013–2025 | Cite as

From Heisenberg to Gödel via Chaitin

  • Cristian S. Calude
  • Michael A. Stay


In 1927 Heisenberg discovered that the “more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.” Four years later Gödel showed that a finitely specified, consistent formal system which is large enough to include arithmetic is incomplete. As both results express some kind of impossibility it is natural to ask whether there is any relation between them, and, indeed, this question has been repeatedly asked for a long time. The main interest seems to have been in possible implications of incompleteness to physics. In this note we will take interest in the converse implication and will offer a positive answer to the question: Does uncertainty imply incompleteness? We will show that algorithmic randomness is equivalent to a “formal uncertainty principle” which implies Chaitin’s information-theoretic incompleteness. We also show that the derived uncertainty relation, for many computers, is physical. In fact, the formal uncertainty principle applies to all systems governed by the wave equation, not just quantum waves. This fact supports the conjecture that uncertainty implies algorithmic randomness not only in mathematics, but also in physics.

Key words

algorithmic randomness uncertainty incompleteness 


  1. Barrow, J. D. (1998). Impossibility: The Limits of Science and the Science of Limits, Oxford University Press, Oxford.Google Scholar
  2. Barrow, J. D. (2000). Mathematical jujitsu: Some informal thoughts about Gödel and physics.’ Complexity 5(5), 28–34.CrossRefMathSciNetGoogle Scholar
  3. Calude, C. S. (2002a). Information and Randomness: An Algorithmic Perspective, Revised and Extended, 2nd edn. Springer Verlag, Berlin.MATHGoogle Scholar
  4. Calude, C. S. (2002b). Chaitin Ω numbers, Solovay machines and incompleteness. Theoretical Computer Science 284, 269–277.MATHCrossRefMathSciNetGoogle Scholar
  5. Calude, C. S. (2002c). Incompleteness, complexity, randomness and beyond. Minds and Machines: Journal for Artificial Intelligence, Philosophy and Cognitive Science 12(4), 503–517.MATHGoogle Scholar
  6. Calude, C. S. and Pavlov, B. (2002). The Poincaré–Hardy inequality on the complement of a Cantor set. In lpay, D., Gohberg, I., and Vinnikov, V., eds., Interpolation Theory, Systems Theory and Related Topics, Operator Theory: Advances and Applications, Vol. 134, pp. 187–208. Birkhäuser Verlag, Basel.Google Scholar
  7. Casti, J. L. and Traub, J. F., eds. (1994). On Limits. Santa Fe Institute Report 94-10-056, Santa Fe, NM.Google Scholar
  8. Casti, J. L. and Karlquist, A., eds. (1996). Boundaries and Barriers. On the Limits to Scientific Knowledge. Addison-Wesley, Reading, MA.Google Scholar
  9. Chaitin, G. J. (1975a). A theory of program size formally identical to information theory. Journal of the Association for Computing Machinery 22, 329–340. (Received April 1974) (Reprinted in: Chaitin, 1990, pp. 113–128).Google Scholar
  10. Chaitin, G. J. (1975b). Randomness and mathematical proof. Scientific American 232(5), 47–52.CrossRefGoogle Scholar
  11. Chaitin, G. J. (1982). Gödel’s theorem & information. International Journal of Theoretical Physics 22, 941–954.CrossRefADSMathSciNetGoogle Scholar
  12. Chaitin, G. J. (1990). Information, Randomness and Incompleteness, Papers on Algorithmic Information Theory, 2nd edn. World Scientific, Singapore.Google Scholar
  13. Chaitin, G. J. (1992). Information-Theoretic Incompleteness. World Scientific, Singapore.Google Scholar
  14. Chaitin, G. J. (1999). The Unknowable. Springer Verlag, Singapore.MATHGoogle Scholar
  15. Chaitin, G. J. (2002). Computers, paradoxes and the foundations of mathematics. American Scientist 90, 164–171.CrossRefGoogle Scholar
  16. Delahaye, J.-P. (1994). Information, Complexité et Hasard. Hermes, Paris.Google Scholar
  17. Deutsch, D. (1997). The Fabric of Reality. Allen Lane, Penguin Press, New York.Google Scholar
  18. Feferman, S., Dawson, J. Jr., Kleene, S. C., Moore, G. H., Solovay, R. M., van Heijenoort, J., eds. (1986). Kurt Gödel Collected Works, Vol. I. Oxford University Press, New York.Google Scholar
  19. Feferman, S., Dawson, J. Jr., Kleene, S. C., Moore, G. H., Solovay, R. M., van Heijenoort, J., eds. (1990) Kurt Gödel Collected Works, Vol. II. Oxford University Press, New York.Google Scholar
  20. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter. Systeme Monatshefte für Mathematik und Physik 38, 173–198 (Received 17 November 1930).Google Scholar
  21. Geroch, R. and Hartle, J. B. (1986). Computability and physical theories. Foundations of Physics 16(6), 533–550.CrossRefMathSciNetADSGoogle Scholar
  22. Hawking, S. W. (2002). Gödel and the end of physics. Dirac Centennial Celebration, Cambridge, UK, July 2002,
  23. Heisenberg, W. (1926). Quantenmechanik. Die Naturwissenschaften 14, 899–894.CrossRefGoogle Scholar
  24. Heisenberg, W. (1927). Über den Anschaulichen Inhalt der Quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik 43, 172–198. (Received 23 March 1927) English translation. In Wheeler, J. A., and Zurek, H. eds., Quantum Theory and Measurement, pp. 62–84. Princeton University Press, Princeton, 1983.Google Scholar
  25. Kennard, E. H. (1927). Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik 44, 326–352.CrossRefADSGoogle Scholar
  26. Pauli, W. (1979). In Hermann, A., von Meyenn, K., and Weiskopf, V. F., eds., Wissentschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg u.a. Volume 1 (1919–1929). Springer-Verlag, Berlin.Google Scholar
  27. Peres, A. (1985). Einstein, Gödel, Bohr. Foundations of Physics 15(2), 201–205.CrossRefMathSciNetADSGoogle Scholar
  28. Peres, A. and Zurek, W. H. (1982). Is quantum theory universally valid? Am. J. Phys. 50(9), 807–810.CrossRefADSGoogle Scholar
  29. Solovay, R. M. (2000). A version of Ω for which ZFC can not predict a single bit. In Calude, C. S., and Păun, G., eds., Finite Versus Infinite. Contributions to an Eternal Dilemma, pp. 323–334. Springer-Verlag, London.Google Scholar
  30. Svozil, K. (2005). Computational universes, CDMTCS Research Report 216, May 2003; Chaos, Solitons & Fractals 25(4), 845–859.Google Scholar
  31. Svozil, K. (2004). Private communication to Calude, 8 February 2004.Google Scholar
  32. Tadaki, K. (2002). Upper bound by Kolmogorov complexity for the probability in computable POVM measurement, Los Alamos preprint archive, http://arXiv:quant-ph/0212071, 11 December 2002.
  33. Tarski, A. (1994). Introduction to Logic and to the Methodology of Deductive Sciences, 4th edn. Oxford University Press, New York.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Computer ScienceThe University of AucklandAucklandNew Zealand

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