Undecidability in R n

Riddled Basins, the KAM Tori, and the Stability of the Solar System

Bibliographic Data

ID10705195
AuthorsMatthew W Parker (0000-0002-3772-3742, University of Chicago, corresponding author)
Year2003
Volume70
Issue2
Pages359-382
Publication date2003-04-01
Open AccessYes
TypeARTICLE
VenuePhilosophy of Science (JOURNAL)
Journal identifiersISSN: 0031-8248 • E-ISSN: 1539-767X
PublisherCambridge University Press (CUP) (PUBLISHER)
DOI10.1086/375472
OpenAlexW2015114054
LanguageEN
Citations received1
References cited13

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

Decidability · Discrete mathematics · Geometry · Invariant (physics) · Invariant measure · Kolmogorov–Arnold–Moser theorem · Lebesgue integration · Lebesgue measure · Measure (data warehouse) · Pure mathematics · Torus · Undecidable problem · Computability, Logic, AI Algorithms · Computer Science · Distributed and Parallel Computing Systems · Mathematical Dynamics and Fractals · Mathematical Physics · Mathematics

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