[1. I regret that the obvious size limitations of this paper do not allow me to tell about hypercomputation as a fascinating case of information processing which is not utmcomputational. To give the taste of the problem, let me refer to Hector Zenil’s blog “Anima ex Machina”, the post: http://www.mathrix.org/liquid/category/recreation entitled “Hypercomputation in A Computable Universe”.]Search in Google Scholar
[2. See: Alan Turing, “Systems of Logic Based on Ordinals”, Proc. London Math. Soc., ser. 2, 45 (1939).]Search in Google Scholar
[3. See: G. J. Chaitin, Algorithmic Information Theory, Cambridge University Press, 1990 (2nd ed.), p. 62.]Search in Google Scholar
[4. An extensive account of Chatin’s theory and its applications to the progress of sciences can be found in the book by Douglas S. Robertson Phase Change: The Computer Revolution in Science and Mathematics, Oxford University Press 2003. As for G. J. Chaitin’s original texts, for present purposes his Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory (World Scientific, Singapore 1990) would be very useful.]Search in Google Scholar
[5. This term appears in most recent discussions to take advantage of the explanatory merits of the idea of information with respect to the nature of the universe. See, e.g., Hector Zenil’s polemics with Seth Lloyd in the former’s blog “Anima ex Machina”: http://www.mathrix.org/liquid/archives/tag/quantum-computer.]Search in Google Scholar
[6. A recent approach to the exponential growth of information is found in discussions inspired by Ray Kurzweil’s bold predictions. See, e.g., the blog discussion entitled “Why so slow” at the page http://sciencehouse.wordpress.com/2008/06/10/why-so-slow/. Also “Big and Small” by R. D. Ekers at http://arxiv.org/pdf/1004.4279.pdf.]Search in Google Scholar
[7. Still in the first decades of the 20th century it was projected in the Vienna Circle to establish a logic of induction, able to grant such certainty to the natural sciences, as the logic of deduction does with respect to mathematics.]Search in Google Scholar
[8. See http://pl.wikipedia.org/wiki/Max Planck, and http://en.wikipedia.org/wiki/ Special relativity.]Search in Google Scholar
[9. See http://en.wikipedia.org/wiki/Initial singularity.]Search in Google Scholar
[10. Nicholas Rescher, Satisfying Reason: Studies in the Theory of Knowledge (Kluwer, Dordrecht 1995). See chapter 3. Reason and Reality, section 6. The Burdens of Complexity, p. 38.10.1007/978-94-011-0483-8]Search in Google Scholar
[11. See the paper by Gordana Dodig-Crnkovic “Significance of Models of Computation, from Turing Model to Natural Computation”, Minds and Machines, May 2011, volume 21, issue 2, pp. 301-32. Available with Springer if addressed: http://link.springer.com/article/10.1007/s11023-011-9235-1.10.1007/s11023-011-9235-1]Search in Google Scholar
[12. Cp. http://www.mathrix.org/liquid/category/recreation - H. Zenil’s post: “Meaningful Math Proofs and ‘Math is not Calculation’”.]Search in Google Scholar
[13. Available at https://mises.org/journals/jls/121/1219.pdf. Published in: Journal for Libertarian Studies, 12(1) (Spring 1996), pp. 179-192. Center for Libertarian Studies.]Search in Google Scholar
[14. See http://dl.acm.org/citation.cfm?doid=2580723.2591012. Published in Communications of the ACM, April 2014, volume 57, issue 4, pp. 66-75. John Harrison belongs among the most renowned computer scientists in the field of automated theorem proving. Jeremy Avigad is a professor in the departments of philosophy and of mathematics at Carnegie Mellon University.10.1145/2591012]Search in Google Scholar
[15. More on this subject, see chapter 25 in George Boolos’ book Logic, Logic, and Logic, Harvard University Press 1998. The proof of Gödel’s 1936 theorem is given in: Samuel R. Buss, “On Gödel’s Theorems on Lengths of Proofs I: Number of Lines and Speedups for Arithmetic”, Journal of Symbolic Logic, 39, 1994, pp. 737-756.]Search in Google Scholar
[16. See http://logika.uwb.edu.pl/studies/index.php?page=search&vol=22, sections 1.1-1.5. ]Search in Google Scholar
