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

Citation data:

Philosophy of Science, ISSN: 0031-8248, Vol: 70, Issue: 2, Page: 359-382

Publication Year:
2003
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Repository URL:
http://philsci-archive.pitt.edu/id/eprint/13175
DOI:
10.1086/375472
Author(s):
Matthew W. Parker
Publisher(s):
University of Chicago Press
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article description
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 mu (or d-mu) for any measure mu, which is particularly appropriate for physics and in some ways more intuitive than Ko's recursive approximability (r.a.). For Lebesgue measure lambda, d-lambda implies r.a. Sets with positive lambda-measure that are sufficiently "riddled" with holes are never d-lambda 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-lambda. (edited)

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