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
Adler, A. (1969).Proceedings of the American Mathematical Society,22, 523.
Arnold, V. (1976).Méthodes Mathématiques de la Mécanique Classique, Mir, Moscow.
Blum, L., Shub, M., and Smale, S. (1989).Bulletin of the American Mathematical Society,21, 1.
Caviness, B. F. (1970).Journal ACM,17, 385.
Chaitin, G. J. (1982).International Journal of Theoretical Physics,22, 941.
Chaitin, G. J. (1988).Algorithmic Information Theory, Cambridge.
Cohen, P. J. (1966).Set Theory and the Continuum Hypothesis, Benjamin, New York.
Da Costa, N. C. A., and Doria, F. A. (1990a). Suppes predicates for classical physics, inProceedings of the San Sebastián Congress on Scientific Structures.
Da Costa, N. C. A., and Doria, F. A. (1990b). Non-computability and two conjectures by Penrose and Scarpellini, preprint.
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.
Da Costa, N. C. A., Doria, F. A., and de Barros, J. A. (1990).International Journal of Theoretical Physics,29, 935.
Dales, H. G., and Woodin, W. H. (1987).An Introduction to Independence for Analysts, Cambridge.
Davenport, J. H. (1981).On the Integration of Algebraic Functions, Springer Lecture Notes in Computer Science #102.
Davis, M. (1973).American Mathematics Monthly,80, 233.
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.
Davis, M., Putnam, H., and Robinson, J. (1961).Annals of Mathematics,74, 425.
Ehrenfeucht, A., and Mycielski, J. (1971).Bulletin of the American Mathematical Society,17, 366.
Gödel, K. (1931).Monatshefte für Mathematik and Physik,38, 173.
Hirsch, M. (1985). The chaos of dynamical systems, inChaos, Fractals and Dynamics, P. Fischer and W. R. Smith, eds., Marcel Dekker.
Holmes, P. J., and Marsden, J. (1982).Communications in Mathematical Physics,82, 523.
Jantscher, L. (1971).Distributionen, W. de Gruyter.
Jones, J. P. (1982).Journal of Symbolic Logic,47, 549.
Jones, J. P., and Matijaševič, Yu. (1984).Journal of Symbolic Logic,49, 818.
Kunen, K., and Vaught, J. E. (1984).Handbook of Set-theoretic Topology, North-Holland.
Lichtenberg, A. J., and Lieberman, M. A. (1983).Regular and Stochastic Motion, Springer.
Lorenz, E. (1963).Journal of Atmospheric Science,20, 130.
Machtey, M., and Young, P. (1978).An Introduction to the General Theory of Algorithms, North-Holland.
Manin, Yu. I. (1977).A Course in Mathematical Logic, Springer.
Mendelson, E. (1987).Introduction to Mathematical Logic, 3rd ed., Wadsworth & Brooks.
Moore, C. (1990).Physical Review Letters,64, 2354.
Ornstein, D., and Weiss, B. (1973).Israel Journal of Mathematics,14, 184.
Post, E. (1944).Bulletin of the American Mathematical Society,50, 284.
Pour-El, M. B., and Caldwell, J. (1975).Zeitschrift für Mathematische Logik und Grundlagen der Mathematik,21, 1.
Pour-El, M. B., and Richards, I. (1981).Advances in Mathematics,39, 215.
Pour-El, M. B., and Richards, I. (1983a).Advances in Mathematics,48, 44.
Pour-El, M. B., and Richards, I. (1983b).Transactions of the American Mathematical Society,275, 539.
Pour-El, M. B., and Richards, I. (1989).Computability in Analysis and Physics, Springer.
Richardson, D. (1968).Journal of Symbolic Logic,33, 514.
Richardson, D. (1969).Zeitschrift für Mathematische Logik und Grundlagen der Mathematik,15, 333.
Rogers, Jr., H. (1967).Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York.
Scarpellini, B. (1963).Zeitschrift für Mathematische Logik und Grundlagen der Mathematik,9, 265.
Şierpiński, W. (1956).Hypothèse du Continu, Chelsea Press, New York.
Tabor, M. (1989).Chaos and Integrability in Nonlinear Dynamics, Wiley, New York.
Uspensky, V. A. (1987).Gödel's Incompleteness Theorem, Mir, Moscow.
Wang, P. (1974).Journal ACM,21, 586.
Yosida, K. (1980).Functional Analysis, Springer.
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da Costa, N.C.A., Doria, F.A. Undecidability and incompleteness in classical mechanics. Int J Theor Phys 30, 1041–1073 (1991). https://doi.org/10.1007/BF00671484
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DOI: https://doi.org/10.1007/BF00671484