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Undecidability in Rn: Riddled Basins, the KAM Tori, and the Stability of the Solar System

Parker, Matthew W. (2003) Undecidability in Rn: Riddled Basins, the KAM Tori, and the Stability of the Solar System. Philosophy of Science, 70 (2). pp. 359-382.

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Abstract

Some have suggested that certain classical physical systems have undecidable long-term behavior, without specifying an appropriate notion of decidability over the reals. We introduce such a notion, decidability in μ (or d-μ) for any measure μ, which is particularly appropriate for physics and in some ways more intuitive than Ko’s (1991) recursive approximability (r.a.). For Lebesgue measure λ, d-λ implies r.a. Sets with positive λ-measure that are sufficiently “riddled” with holes are never d-λ but are often r.a. This explicates Sommerer and Ott’s (1996) claim of uncomputable behavior in a system with riddled basins of attraction. Furthermore, it clarifies speculations that the stability of the solar system (and similar systems) may be undecidable, for the invariant tori established by KAM theory form sets that are not d-λ.


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Item Type: Published Article or Volume
Creators:
CreatorsEmailORCID
Parker, Matthew W.m.parker@lse.ac.uk0000-0002-7436-2149
Keywords: Decidable, stability, chaos, KAM, Ko, riddled basin, real numbers, continuous space, computable, planetary system, nearly integrable, n-body problem, 3-body problem, dynamical systems, computable analysis
Subjects: Specific Sciences > Computation/Information > Classical
Specific Sciences > Mathematics > Logic
Specific Sciences > Physics > Classical Physics
Specific Sciences > Computation/Information
Specific Sciences > Computer Science
Depositing User: Dr. Matthew Parker
Date Deposited: 03 Jul 2017 14:35
Last Modified: 03 Jul 2017 14:35
Item ID: 13175
Journal or Publication Title: Philosophy of Science
Publisher: University of Chicago Press
Official URL: http://www.journals.uchicago.edu/doi/abs/10.1086/3...
DOI or Unique Handle: https://doi.org/10.1086/375472
Subjects: Specific Sciences > Computation/Information > Classical
Specific Sciences > Mathematics > Logic
Specific Sciences > Physics > Classical Physics
Specific Sciences > Computation/Information
Specific Sciences > Computer Science
Date: 1 April 2003
Page Range: pp. 359-382
Volume: 70
Number: 2
URI: http://philsci-archive.pitt.edu/id/eprint/13175

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