Abstract
In this paper I discuss possible ways of measuring the power of arithmetical theories, and the possiblity of making an explication in Carnap’s sense of this concept. Chaitin formulates several suggestions how to construct measures, and these suggestions are reviewed together with some new and old critical arguments. I also briefly review a measure I have designed together with some shortcomings of this measure. The conclusion of the paper is that it is not possible to formulate an explication of the concept.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boolos, G.: 1982, Extremely undecidable sentences, The Journal of Symbolic Logic 47, 191–196.
Chaitin, G.J.: 1971, Computational complexity and Gödel’s Incompleteness Theorem, ACM SIGACT News 9, 11–12.
Chaitin, G.J.: 1972, Information-theoretic aspects of post’s construction of a simple set, AMS Notices 19, A–712.
Chaitin, G.J.: 1974, Information-theoretic limitations of formal systems, Journal of the ACM 21, 403–424.
Chaitin, G.J.: 1975a, Randomness and mathematical proof, Scientific American 232(5), 47–52.
Chaitin, G.J.: 1975b, A theory of program size formally identical to information theory, Journal of the ACM 22, 329–340.
Chaitin, G.J.: 1977, Algorithmic information theory, IBM Journal of Research and Development 21, 403–424.
Chaitin, G.J.: 1982, Gödel’s theorem and information, International Journal of Theoretical Physics 22, 941–954.
Chaitin, G.J.: 2000, A century of controversy over the foundations of mathematics, Complexity 5(5), 12–21.
Chaitin, G.J.: 2001, Exploring Randomness, Springer, Berlin.
Cohen, P.J.: 1966, Set Theory and the Continuum Hypothesis, W. A. Benjamin, New York.
Franzén, T.: 2005, Gödel’s Theorem – An Incomplete Guide to its Use and Abuse, A K Peters, Wellesley, MA.
Hofstadter, D.R.: 1979, Gödel’s, Escher, Bach: An Eternal Golden Braid, Basic Books, New York.
Jagers, P.: 2001, Den Osannolika Slumpen, Forskning och Framsteg 36(3), 34–37.
Lindström, P.: 2003, Aspects of incompleteness, 2 edition, A K Peters, Wellesley, MA.
Li, M. and Vitányi, P.: 1993, An Introduction to Kolmogorov Complexity and its Applications, Springer, Berlin.
Odifreddi, P.: 1993, Classical Recursion Theory, Springer, Berlin.
Raatikainen, P.: 1997, The problem of the simplest Diophantine representation, Nordic Journal of Philosophical Logic 2(2), 47–54.
Raatikainen, P.: 1998, On interpreting Chaitin’s incompleteness theorem, Journal of Philosophical Logic 27, 569–586.
Raatikainen, P.: 2000, Algorithmic information theory and undecidability, Synthese 123, 217–225.
Raatikainen, P.: 2001, Exploring randomness and the unknowable, review article, Notices of the AMS 48, 992–996.
Schwichtenberg, H.: 1997, Proof theory: Some applications of cut-elimination, in J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, pp. 867–895.
Sjögren, J.: 2004, Measuring the power of arithmetical theories, Dept. of Philosophy, University of Göteborg, Thesis for the Licentiate Degree.
van Lambalgen, M.: 1989, Algorithmic information theory, Journal of Symbolic Logic 54, 1389–1400.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sjögren, J. ON EXPLICATING THE CONCEPT THE POWER OF AN ARITHMETICAL THEORY . J Philos Logic 37, 183–202 (2008). https://doi.org/10.1007/s10992-007-9077-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-007-9077-8