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How Much Information Can There Be in a Real Number?

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Computation, Physics and Beyond (WTCS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7160))

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Abstract

This note gives some information about the magical number Ω and why it is of interest. Our purpose is to explain the significance of recent work by Calude and Dinneen attempting to compute Ω. Furthermore, we propose measuring human intellectual progress (not scientific progress) via the number of bits of Ω that can be determined at any given moment in time using the current mathematical theories.

This paper has originally appeared in International Journal of Bifurcation and Chaos 17(6), 1933–1935 (©2007, WSPC). It is reprinted with kind permission of World Scientific Publishing Company.

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References

  1. Calude, C.S., Dinneen, M.J.: Exact approximations of omega numbers. Int. Journal of Bifurcation & Chaos 17(6), 1937–1954 (2007)

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  2. Calude, C.S., Calude, E., Dinneen, M.J.: A new measure of the difficulty of problems. Journal of Multiple-Valued Logic and Soft Computing 12, 285–307 (2006)

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  3. Chaitin, G.: The limits of reason. Scientific American 294(3), 74–81 (2006)

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  4. Chaitin, G.: Meta Math! Pantheon, New York (2005); Meta Maths. Atlantic Books, London (2006)

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  5. Chaitin, G.: The halting probability Ω: Irreducible complexity in pure mathematics. Milan Journal of Mathematics 75, 291–304 (2007)

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Author information

Authors and Affiliations

  1. IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY, 10598, USA

    Gregory Chaitin

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  1. Gregory Chaitin
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Auckland, Private Bag 92019, 1142, Auckland, New Zealand

    Michael J. Dinneen , Bakhadyr Khoussainov  & André Nies ,  & 

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Chaitin, G. (2012). How Much Information Can There Be in a Real Number?. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds) Computation, Physics and Beyond. WTCS 2012. Lecture Notes in Computer Science, vol 7160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27654-5_19

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  • DOI: https://doi.org/10.1007/978-3-642-27654-5_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27653-8

  • Online ISBN: 978-3-642-27654-5

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