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  • 1
    Electronic Resource
    Electronic Resource
    Dynamical system where proving chaos is equivalent to proving Fermat's conjecture (1993)
    Springer
    International journal of theoretical physics 32 (1993), S. 2187-2206 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We prove that we can explicitly construct the expression for a low-dimensional Hamiltonian system where proving the existence of a Smale horseshoe is equivalent to proving that Fermat's Conjecture is true. We then show that some sets of similar intractable problems are dense (in the usual topology) in the space of all dynamical systems over a finite-dimensional real manifold.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00675030
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Two questions on the geometry of gauge fields (1994)
    Springer
    Foundations of physics 24 (1994), S. 783-800 
    Details
    ISSN: 1572-9516
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for higher-degree similar relations. We then prove that there is no algorithmic procedure to determine, for a reasonable restricted algebra of functions on spacetime, whether a given connection form satisfies the preceding conditions. A parallel result gives a version of Gödel's first incompleteness theorem within an (axiomatized) theory of gauge fields.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02054673
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  • 3
    Electronic Resource
    Electronic Resource
    Classical physics and Penrose's thesis (1991)
    Springer
    Foundations of physics 4 (1991), S. 363-373 
    Details
    ISSN: 1572-9524
    Keywords: Halting problem ; computability ; classical mechanics ; R. Penrose
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We expose and discussPenrose's thesis: “Nature produces harnessable noncomputable processes, but none at the classical level.” We then suggest a partial counterexample to it, based on aGedanken experiment about an undecidable family of integrable Hamiltonian systems that could lead to a sort of idealized solution to the Halting problem for Turing machines.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00665895
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    A Suppes predicate for general relativity and set-theoretically generic spacetimes (1990)
    Springer
    International journal of theoretical physics 29 (1990), S. 935-961 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We summarize ideas from Zermelo-Fraenkel set theory up to an axiomatic treatment for general relativity based on a Suppes predicate. We then examine the meaning of set-theoretic genericity for manifolds that underlie the Einstein equations. A physical interpretation is finally offered for those set-theoretically generic manifolds in gravitational theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00673682
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Undecidable hopf bifurcation with undecidable fixed point (1994)
    Springer
    International journal of theoretical physics 33 (1994), S. 1885-1903 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We exhibit a polynomial dynamical system where one cannot decide whether a Hopf bifurcation occurs. Therefore one cannot decide whether there will be parameter values such that a stable fixed point becomes an unstable one. Related incompleteness results for previously described axiomatized versions of dynamical systems theory are also discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00671031
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  • 6
    Electronic Resource
    Electronic Resource
    Undecidability and incompleteness in classical mechanics (1991)
    Springer
    International journal of theoretical physics 30 (1991), S. 1041-1073 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00671484
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    Paper (German National Licenses)
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