David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Journal of Philosophical Logic 27 (6):569-586 (1998)
The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental.
|Keywords||algorithmic information theory incompleteness Kolmogorov complexity|
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References found in this work BETA
George Boolos, John Burgess, Richard P. & C. Jeffrey (2007). Computability and Logic. Cambridge University Press.
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Citations of this work BETA
Kojiro Higuchi, W. M. Phillip Hudelson, Stephen G. Simpson & Keita Yokoyama (2014). Propagation of Partial Randomness. Annals of Pure and Applied Logic 165 (2):742-758.
Jörgen Sjögren (2008). On Explicating the Concept the Power of an Arithmetical Theory. Journal of Philosophical Logic 37 (2):183 - 202.
Shingo Ibuka, Makoto Kikuchi & Hirotaka Kikyo (2011). Kolmogorov Complexity and Characteristic Constants of Formal Theories of Arithmetic. Mathematical Logic Quarterly 57 (5):470-473.
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