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Incompleteness, Complexity, Randomness and Beyond

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Abstract

Gödel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an information-theoretic approach to randomness and recent developments in quantum computing.

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References

  • Auburn, D. (2001), Proof.A Play, New York: Faber & Faber.

    Google Scholar 

  • Barrow, J.D. (1998), Impossibility: The Limits of Science and the Science of Limits, Oxford: Oxford University Press.

    Google Scholar 

  • Barrow, J.D. (2000), ‘Mathematical Jujitsu: Some Informal Thoughts About Gödel and Physics’, Complexity 5, pp. 28–34.

    Google Scholar 

  • Becher, V., Daicz, S. and Chaitin, G.J. (2001), ‘A Highly Random Number’, in C.S. Calude, M. J. Dinneen and S. Sburlan, eds., Combinatorics, Computability and Logic, Proceedings of DMTCS'01, London: Springer, pp. 55–68.

    Google Scholar 

  • Beltrami, E. (1999), What is Random? Chance and Order in Mathematics and Life, New York: Springer.

    Google Scholar 

  • Bennett, C.H. and Gardner, M. (1979), ‘The Random Number Omega Bids Fair to Hold the Mysteries of the Universe’, Scientific American 241, pp. 20–34.

    Google Scholar 

  • Brisson, L. and Meyerstein, L.F. (1995), Puissance et Limites de la Raison, Paris: Les Belles Lettres.

    Google Scholar 

  • Calude, C. S. (2002), Information and Randomness.An Algorithmic Perspective, Berlin: Springer.

    Google Scholar 

  • Calude, C.S. (2000), ‘A Glimpse into Algorithmic Information Theory’, in P. Blackburn, N. Braisby, L. Cavedon and A. Shimojima, eds., Logic, Language and Computation, Volume 3, CSLI Series, Cambridge: Cambridge University Press, pp. 65–81.

    Google Scholar 

  • Calude, C.S. (2002), ‘Chaitin Ω ΩNumbers, SolovayMachines and Incompleteness’, Theoret.Comput.Sci. 28, pp. 269–277.

    Google Scholar 

  • Calude, C.S. and Chaitin, G.J. (1999), ‘Randomness Everywhere’, Nature 400, pp. 319–320.

    Google Scholar 

  • Calude, C.S., Dinneen, M.J. and C.-K. Shu, C.-K. (2002), ‘Computing a Glimpse of Randomness’, Experimental Mathematics; see also CDMTCS Research Report 167, 2001, 12 pp.

  • Calude, C.S., Dinneen, M.J. and Svozil, K. (1999), ‘Counterfactual Effect, the Halting Problem, and the Busy Beaver Function’ (Preliminary Version), CDMTCS Research Report 107, 8 pp.

  • Calude, C.S., Dinneen, M.J. and Svozil, K. (2000), ‘Reflections on Quantum Computing’, Complexity 6, pp. 35–37.

    Google Scholar 

  • Calude, C.S., Hertling, P., Khoussainov, B. and Wang, Y. (2001), 'Recursively Enumerable Reals and Chaitin Ω ΩNumbers’, Theoret.Comput.Sci. 255 pp. 125–149.

    Google Scholar 

  • Calude, C. and Jürgensen, H. (1994), ‘Randomness as an Invariant for Number Representations’, in H. Maurer, J. Karhumäki and G. Rozenberg, eds. Results and Trends in Theoretical Computer Science, Berlin: Springer, pp. 44–66.

    Google Scholar 

  • Calude, C., Jürgensen, H. and Zimand, M. (1994), ‘Is Independence an Exception ?’, Appl.Math.Comput. 66, pp. 63–76.

    Google Scholar 

  • Calude, C.S. and Meyerstein, F.W. (1999), ‘Is the Universe Lawful ?’, Chaos, Solitons & Fractals 10, pp. 1075–1084.

    Google Scholar 

  • Calude, C. and Nies, A. (1997), ‘Chaitin Ω ΩNumbers and Strong Reducibilities’, J.Univ.Comput.Sci. 3, pp. 1161–1166.

    Google Scholar 

  • Calude, C.S. and Pavlov, B. (2002), ‘Coins, QuantumMeasurements, and Turing's Barrier’, Quantum Information Processing 1(1-2), pp. 107–127.

    Google Scholar 

  • Calude, C.S. and Păun, G. (2001), Computing with Cells and Atoms, London: Taylor & Francis Publishers.

    Google Scholar 

  • Calude, C. and Salomaa, A. (1994), ‘Algorithmically Coding the Universe’, in G. Rozenberg and A. Salomaa, eds. Developments in Language Theory, Singapore: World Scientific, pp. 472–492.

    Google Scholar 

  • Casti, J. (1997), ‘Computing the Uncomputable’, The New Scientist, 154/2082, p. 34.

    Google Scholar 

  • Casti, J. (2000), Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20th-Century Mathematics, New York: Wiley.

    Google Scholar 

  • Casti, J. and DePauli, W. (2000), Gödel.A Life in Logic, Cambridge: Perseus.

    Google Scholar 

  • Chaitin, G.J. (1966), ‘On the Length of Programs for Computing Finite Binary Sequences’, J.Assoc.Comput.Mach. 13, pp. 547–569. (Reprinted in: Chaitin (1990), pp. 219–244.)

    Google Scholar 

  • Chaitin, G.J. (1975), ‘A Theory of Program Size Formally Identical to Information Theory’, J.Assoc.Comput.Mach. 22, pp. 329–340. (Reprinted in: Chaitin (1990), pp. 113–128)

    Google Scholar 

  • Chaitin, G.J. (1982), ‘Gödel's Theorem & Information’, International Journal of Theoretical Physics 22, pp. 941–954.

    Google Scholar 

  • Chaitin, G.J. (1990), Algorithmic Information Theory, Cambridge: Cambridge University Press (Third printing).

    Google Scholar 

  • Chaitin, G.J. (1990), Information, Randomness and Incompleteness, Papers on Algorithmic Information Theory, Singapore: World Scientific, Singapore (Second edition).

    Google Scholar 

  • Chaitin, G.J. (1992), Information-Theoretic Incompleteness, Singapore: World Scientific, Singapore.

    Google Scholar 

  • Chaitin, G.J. (1997), The Limits of Mathematics, Singapore: Springer.

    Google Scholar 

  • Chaitin, G.J. (1999), The Unknowable, Singapore: Springer.

    Google Scholar 

  • Chaitin, G.J. (2000), Exploring Randomness, London: Springer.

    Google Scholar 

  • Chaitin, G.J. (2001), Conversations with a Mathematician, London: Springer.

    Google Scholar 

  • Chown, M. (2001), ‘The Omega Man’, New Scientist 10 March, pp. 29–31.

  • Chown, M. (2002), ‘Smash and Grab’, New Scientist 6 April, pp. 24–28.

  • Collins, G. P. (2001), ‘Computing with Light’, Scientific American, Aug. p. 12.

  • Copeland, J. (1999), ‘The Modern History of Computing’, in E.N. Zalta, ed. The Stanford Encyclopedia of Philosophy http://plato.stanford.edu/entries/ computing-history/.

  • Copeland, J. (2000), ‘Narrow Versus Wide Mechanism: Including a Re-examination of Turing's Views on the Mind-machine Issue’, Journal of Philosophy XCVI1, pp. 5–32.

    Google Scholar 

  • Dawson, J.W. Jr. (1984), ‘Kurt Gödel in Sharper Focus’, The Mathematical Intelligencer6, pp. 9–17.

    Google Scholar 

  • Dawson, J.W. Jr. (1997), Logical Dilemmas.The Life and Work of Kurt Gödel, Massachusetts: A K Peters.

    Google Scholar 

  • Dembski, W. A. (1998), ‘Randomness’, in E. Craig, ed., Routledge Encyclopedia of Philosophy, Routledge, London, Vol. 8, pp. 56–59.

    Google Scholar 

  • Denker, M., Woyczyński, M. W. and Ycart, B. (1998), Introductory Statistics and Random Phenomena: Uncertainty, Complexity, and Chaotic Behavior in Engineering and Science, Boston: Birkhäuser.

    Google Scholar 

  • Detlefsen, M. (1998), ‘Gödel's Theorems’, in E. Craig, ed., Routledge Encyclopedia of Philosophy, Routledge, London, Vol. 4, pp. 106–119.

    Google Scholar 

  • Deutsch, D. (1985), ‘Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer’, Proceedings of the Royal Society London A 400, pp. 97–119.

    Google Scholar 

  • Doxiadis, A. (2000), Uncle Petros & Goldback's Conjecture.A Novel about Mathematical Obsession, New York: Bloomsbury.

    Google Scholar 

  • Etesi, G. and Németi, I. (2002)., ‘Non-Turing Computations via Malament-Hogarth Space-times’, International Journal of Theoretical Physics 41, pp. 341–370.

    Google Scholar 

  • Feferman, S. (1984), ‘Kurt Gödel: Conviction and Caution’, Philos.Natur. 21, pp. 546–562.

    Google Scholar 

  • Feferman, S., Dawson, J., Jr., Kleene S.C., Moore, G.H., Solovay, R.M. and van Heijenoort, J., eds. (1990), Kurt Gödel Collected Works, Volume II, Oxford: Oxford University Press.

    Google Scholar 

  • Feynman, R.P. (1985), ‘Simulating Physics with Computers’, International Journal of Theoretical Physics 11, pp. 11–20.

    Google Scholar 

  • Hey, J.G., ed. (1999), Feynman and Computation.Exploring the Limits of Computers, Reading: Perseus Books.

  • Gödel, K. (1964), ‘Russell's Mathematical Logic’, in P. Benacerref and H. Putnam, eds. Philosophy of Mathematics, Englewood Cliffs, NJ: Prentice-Hall, pp. 211–232.

    Google Scholar 

  • Gruska, J. (1999), Quantum Computing, London: McGraw-Hill.

    Google Scholar 

  • Hayes, B. (2001), ‘Randomness as a Resource’, American Scientist 89, 4 July–August, pp. 300-304.

    Google Scholar 

  • Hertling, P. and Weihrauch, K. (1998), ‘Randomness Spaces’, in K.G. Larsen, S. Skyum, and G. Winskel, eds. Automata, Languages and Programming, Proceedings of the 25th International Colloquium, ICALP'98 (Aalborg, Denmark), Berlin: Springer, pp. 796–807.

    Google Scholar 

  • Kac, M. (1983), ‘What is Random ?’, American Scientist 71, pp. 405–406.

    Google Scholar 

  • Kieu, T.D. (2001a), ‘Hilbert's Incompleteness, Chaitin's ΩΩ Number and Quantum Physics’, Los Alamos preprint archive http://arXiv:quant-ph/0111062, v1, 10 November.

  • Kieu, T. D. (2001b), ‘Quantum Algorithm for the Hilbert's Tenth Problem’, Los Alamos preprint archive http://arXiv:quant-ph/0110136, v2, 9 November.

  • Kleene, S.C. (1976), ‘The Work of Kurt Gödel’, J.Symbolic Logic 41, pp, 761–778; addendum J.Symbolic Logic 43, p. 613.

    Google Scholar 

  • Kolata, G. (1986), ‘What Does it Mean to be Random ?’, Science 7, pp. 1068–1070.

    Google Scholar 

  • Kolmogorov, A. N. (1965), ‘Three Approaches for Defining the Concept of “Information Quantity”’, Problems Inform.Transmission 1, pp. 3–11.

    Google Scholar 

  • Kreisel, G. (1980), ‘Kurt Gödel’, Biographical Memoirs of Fellows of the Royal Society of London 26, pp. 149–224; corrigenda 27, p. 697, 28, p. 718.

    Google Scholar 

  • Kučera, A. and Slaman, T.A. (2001), ‘Randomness and Recursive Enumerability’, SIAM J.Comput. 31, pp. 199–211.

    Google Scholar 

  • van Lambalgen, M. (1989), ‘Algorithmic Information Theory’, J.Symbolic Logic 54, pp. 1389–1400.

    Google Scholar 

  • Landauer, R. (1987), ‘Computation: A Fundamental Physical View’, Physica Scripta 35, pp. 88–95.

    Google Scholar 

  • Li, M. and Vitányi, P. M. (1997), An Introduction to Kolmogorov Complexity and Its Applications, Berlin: Springer (Second edition).

    Google Scholar 

  • Martin-Löf, P. (1966), Algorithms and Random Sequences, Nürnberg: Erlangen University.

    Google Scholar 

  • Martin-Löf, P. (1966), ‘The Definition of Random Sequences’, Inform.and Control 9, pp. 602–619.

    Google Scholar 

  • Marxen, H. and Buntrock, J. (1990), ‘Attaching the Busy Beaver 5’, Bull EATCS 40, pp. 247–251.

    Google Scholar 

  • Nagel, E. and Newman, J. R. (1986), Gödel's Proof, New York: University Press (Second printing).

    Google Scholar 

  • Post, E. ( 1965), ‘Absolutely Unsolvable Problems and Relatively Undecidable Propositions: Account of an Anticipation’, in M. Davis, ed., The Undecidable, New York: Raven Press, pp. 340–433.

    Google Scholar 

  • Raatikainen, P. (1998), ‘On Interpreting Chaitin's Incompleteness Theorem’, J.Philos.Logic 27, pp. 569–586.

    Google Scholar 

  • Rice, H. (1954), ‘Recursive Reals’, Proc.Amer.Math.Soc. 5, pp. 784–791.

    Google Scholar 

  • Rozenberg, G. and Salomaa, A. (1994), Cornerstones of Undecidability, Englewood Cliffs, NJ: Prentice Hall.

    Google Scholar 

  • Rucker, R. (1982), Infinity and the Mind, New York: Bantam.

    Google Scholar 

  • Siegelmann, H. (1995), ‘Computation Beyond the Turing Limit’, Science 268, pp. 545–548.

    Google Scholar 

  • Soare, R. I. (1969), ‘Recursion Theory and Dedekind Cuts’, Trans.Amer.Math.Soc. 140, pp. 271–294.

    Google Scholar 

  • Solomonoff, R. J. (1964), ‘A Formal Theory of Inductive Inference’, Part 1 and Part 2, Inform.and Control 7, pp. 1–22, 224–254.

    Google Scholar 

  • Solovay, R. M. (1975), Draft of a paper (or series of papers) on Chaitin's work...done for the most part during the period of Sept.–Dec.1974, New York: IBM Thomas J. Watson Research Center, 215, pp.

    Google Scholar 

  • Solovay, R. M. (2000), ‘A Version of Ω for Which Z F C Cannot Predict a Single Bit’, in C.S. Calude and G. Păaun, eds., Finite Versus Infinite.Contributions to an Eternal Dilemma, London: Springer, pp. 323–334.

    Google Scholar 

  • Specker, E. (1949), Nicht konstruktiv beweisbare 'sätze der Analysis’, J.Symbolic Logic 14, pp. 145–158.

    Google Scholar 

  • Svozil, K. (1993), Randomness & Undecidability in Physics, Singapore: World Scientific.

    Google Scholar 

  • Svozil, K. (1995), ‘Halting Probability Amplitude of Quantum Computers’, J.UCS 1, pp. 201–203.

    Google Scholar 

  • Staiger, L. (1999), ‘The Kolmogorov Complexity of Real Numbers’, in G. Ciobanu and Gh. Păun, eds. Proc.Fundamentals of Computation Theory, Lecture Notes in Comput. Sci. No. 1684, Berlin: Springer, pp. 536-546.

    Google Scholar 

  • Stewart, I. (1991), ‘Deciding the Undecidable’, Nature 352, pp. 664–665.

    Google Scholar 

  • Turing, A. M. (1936/7), ‘On Computable Numbers with an Application to the Entscheidungsproblem’, Proc.Amer.Math.Soc. 42, pp. 230–265; a correction, 43, pp. 544–546.

    Google Scholar 

  • Zwirn, H. (2000), Les Limites de la Connaissance, Paris: Odile Jacob.

    Google Scholar 

  • Uspensky, V.A., Semenov, A. L. and Shen, A. Kh. (1990), ‘Can an Individual Sequence of Zeros and Ones be Random ?’, Russian Math.Surveys 45, pp. 121–189.

    Google Scholar 

  • Vitányi, P.M. (2001), ‘Quantum Kolmogorov Complexity Based on Classical Descriptions’, IEEE Trans.Inform.Theory 47, pp. 2464–2479.

    Google Scholar 

  • Wang, H. (1996), A Logical Journey: From Gödel to Philosophy, Cambridge: MIT Press.

    Google Scholar 

  • Williams, C. P. and Clearwater, S. H. (2000), Ultimate Zero and One, New York: Copernicus.

    Google Scholar 

  • Wittgenstein, L. (1964), ‘Selections from “Remarks on the Foundations of Mathematics”’, in P. Benacerref and H. Putnam (eds). Philosophy of Mathematics: Selected Readings, Princeton, NJ: Prentice-Hall, pp. 421–480.

    Google Scholar 

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Calude, C.S. Incompleteness, Complexity, Randomness and Beyond. Minds and Machines 12, 503–517 (2002). https://doi.org/10.1023/A:1021132605507

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