On Characteristic Constants of Theories Defined by Kolmogorov Complexity

Ibuka, Shingo; Kikuchi, Makoto; Kikyo, Hirotaka

doi:10.1007/978-3-540-69937-8_19

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The Characteristic Constants1

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2011

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“…In 2, we proved that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$r_\mathbf {S}< r_\mathbf {T}$\end{document} for some universal Turing machine if and only if there exists some Π 1 ‐sentence which is provable in T but not in S , which is obtained as a corollary of the following theorem.…”

Section: The Characteristic Constantsmentioning

confidence: 99%

Kolmogorov complexity and characteristic constants of formal theories of arithmetic

Ibuka¹,

Kikuchi²,

Kikyo³

2011

Mathematical Logic Quarterly

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MSC (2010) 03A99, 03B25We investigate two constants c T and r T , introduced by Chaitin and Raatikainen respectively, defined for each recursively axiomatizable consistent theory T and universal Turing machine used to determine Kolmogorov complexity. Raatikainen argued that c T does not represent the complexity of T and found that for two theories S and T, one can always find a universal Turing machine such that c S = c T . We prove the following are equivalent: c S = c T for some universal Turing machine, r S = r T for some universal Turing machine, and T proves some Π1 -sentence which S cannnot prove.

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Section: The Characteristic Constantsmentioning

confidence: 99%