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


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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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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