International Journal of Theoretical Physics

, Volume 30, Issue 8, pp 1041–1073 | Cite as

Undecidability and incompleteness in classical mechanics

  • N. C. A. da Costa
  • F. A. Doria
Article

Abstract

We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary real-valued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamiltonian mechanics and dynamical systems theory. Our examples deal with the algorithmic impossibility of deciding whether a given Hamiltonian can be integrated by quadratures and related questions; they lead to a version of Gödel's incompleteness theorem within Hamiltonian mechanics. A similar application to the unsolvability of the decision problem for chaotic dynamical systems is also obtained.

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References

  1. Adler, A. (1969).Proceedings of the American Mathematical Society,22, 523.Google Scholar
  2. Arnold, V. (1976).Méthodes Mathématiques de la Mécanique Classique, Mir, Moscow.Google Scholar
  3. Blum, L., Shub, M., and Smale, S. (1989).Bulletin of the American Mathematical Society,21, 1.Google Scholar
  4. Caviness, B. F. (1970).Journal ACM,17, 385.Google Scholar
  5. Chaitin, G. J. (1982).International Journal of Theoretical Physics,22, 941.Google Scholar
  6. Chaitin, G. J. (1988).Algorithmic Information Theory, Cambridge.Google Scholar
  7. Cohen, P. J. (1966).Set Theory and the Continuum Hypothesis, Benjamin, New York.Google Scholar
  8. Da Costa, N. C. A., and Doria, F. A. (1990a). Suppes predicates for classical physics, inProceedings of the San Sebastián Congress on Scientific Structures.Google Scholar
  9. Da Costa, N. C. A., and Doria, F. A. (1990b). Non-computability and two conjectures by Penrose and Scarpellini, preprint.Google Scholar
  10. Da Costa, N. C. A., and Doria, F. A. (1991). Structures, Suppes predicates, and Booleanvalued models in physics, inFestschrift in Honor of V. I. Smirnov on His 60th Birthday, J. Hintikka, ed.Google Scholar
  11. Da Costa, N. C. A., Doria, F. A., and de Barros, J. A. (1990).International Journal of Theoretical Physics,29, 935.Google Scholar
  12. Dales, H. G., and Woodin, W. H. (1987).An Introduction to Independence for Analysts, Cambridge.Google Scholar
  13. Davenport, J. H. (1981).On the Integration of Algebraic Functions, Springer Lecture Notes in Computer Science #102.Google Scholar
  14. Davis, M. (1973).American Mathematics Monthly,80, 233.Google Scholar
  15. Davis, M., Matijaševič, Yu., and Robinson, J. (1976). Hilbert's Tenth Problem. Diophantine equations: Positive aspects of a negative solution, inMathematical Developments Arising from Hilbert Problems, F. E. Browder, ed., Proceedings of the Symposium on Pure Mathematics, Vol. 28.Google Scholar
  16. Davis, M., Putnam, H., and Robinson, J. (1961).Annals of Mathematics,74, 425.Google Scholar
  17. Ehrenfeucht, A., and Mycielski, J. (1971).Bulletin of the American Mathematical Society,17, 366.Google Scholar
  18. Gödel, K. (1931).Monatshefte für Mathematik and Physik,38, 173.Google Scholar
  19. Hirsch, M. (1985). The chaos of dynamical systems, inChaos, Fractals and Dynamics, P. Fischer and W. R. Smith, eds., Marcel Dekker.Google Scholar
  20. Holmes, P. J., and Marsden, J. (1982).Communications in Mathematical Physics,82, 523.Google Scholar
  21. Jantscher, L. (1971).Distributionen, W. de Gruyter.Google Scholar
  22. Jones, J. P. (1982).Journal of Symbolic Logic,47, 549.Google Scholar
  23. Jones, J. P., and Matijaševič, Yu. (1984).Journal of Symbolic Logic,49, 818.Google Scholar
  24. Kunen, K., and Vaught, J. E. (1984).Handbook of Set-theoretic Topology, North-Holland.Google Scholar
  25. Lichtenberg, A. J., and Lieberman, M. A. (1983).Regular and Stochastic Motion, Springer.Google Scholar
  26. Lorenz, E. (1963).Journal of Atmospheric Science,20, 130.Google Scholar
  27. Machtey, M., and Young, P. (1978).An Introduction to the General Theory of Algorithms, North-Holland.Google Scholar
  28. Manin, Yu. I. (1977).A Course in Mathematical Logic, Springer.Google Scholar
  29. Mendelson, E. (1987).Introduction to Mathematical Logic, 3rd ed., Wadsworth & Brooks.Google Scholar
  30. Moore, C. (1990).Physical Review Letters,64, 2354.Google Scholar
  31. Ornstein, D., and Weiss, B. (1973).Israel Journal of Mathematics,14, 184.Google Scholar
  32. Post, E. (1944).Bulletin of the American Mathematical Society,50, 284.Google Scholar
  33. Pour-El, M. B., and Caldwell, J. (1975).Zeitschrift für Mathematische Logik und Grundlagen der Mathematik,21, 1.Google Scholar
  34. Pour-El, M. B., and Richards, I. (1981).Advances in Mathematics,39, 215.Google Scholar
  35. Pour-El, M. B., and Richards, I. (1983a).Advances in Mathematics,48, 44.Google Scholar
  36. Pour-El, M. B., and Richards, I. (1983b).Transactions of the American Mathematical Society,275, 539.Google Scholar
  37. Pour-El, M. B., and Richards, I. (1989).Computability in Analysis and Physics, Springer.Google Scholar
  38. Richardson, D. (1968).Journal of Symbolic Logic,33, 514.Google Scholar
  39. Richardson, D. (1969).Zeitschrift für Mathematische Logik und Grundlagen der Mathematik,15, 333.Google Scholar
  40. Rogers, Jr., H. (1967).Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York.Google Scholar
  41. Scarpellini, B. (1963).Zeitschrift für Mathematische Logik und Grundlagen der Mathematik,9, 265.Google Scholar
  42. Şierpiński, W. (1956).Hypothèse du Continu, Chelsea Press, New York.Google Scholar
  43. Tabor, M. (1989).Chaos and Integrability in Nonlinear Dynamics, Wiley, New York.Google Scholar
  44. Uspensky, V. A. (1987).Gödel's Incompleteness Theorem, Mir, Moscow.Google Scholar
  45. Wang, P. (1974).Journal ACM,21, 586.Google Scholar
  46. Yosida, K. (1980).Functional Analysis, Springer.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • N. C. A. da Costa
    • 1
  • F. A. Doria
    • 2
  1. 1.Institute for Advanced StudiesUniversity of São PauloSão Paulo SPBrazil
  2. 2.IMSSS/Stanford UniversityStanford
  3. 3.Center for the Study of Mathematical Theories of Communication, IDEA/School of CommunicationsFederal University of Rio de JaneiroRio de Janeiro RJBrazil

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