On Fri, Aug 16, 2019 at 6:31 PM Russell Standish <li...@hpcoders.com.au> wrote:
> On Fri, Aug 16, 2019 at 12:06:32PM -0500, Jason Resch wrote: > > > > Thanks for the background and explanation. Is it the case then that any > > undecidable (creative?) set is a compact description of universal > dovetailing? > > Would Chaitin's constant also qualify as a compact description of the > universal > > dovetailing (though being a single real number, rather than a set of > rational > > complex points)? > > > > Related to this, on page 218 of Li and Vitanyi's "Introduction to > Kolmogorov Complexity and it Applications", right under corollary 3.6.2 is > the statement: > > "Moreover, for all axiomatic mathematical theories that can be > extressed compactly enough to be conceivably interesting to human > beings, say in fewer than 10,000 bits, [the first 10,000 bits of the > Chatin probability Ω] can be used to decide for every statement in the > theory whether it is the true, false or independent. ... Thus Ω is > truly the number of Wisdom, and 'can be known of, but not known, > through human reason' [C.H Bennett and M. Gardner, Sci > Am. 241:11(1979),20-34]". > > What an incredible and fascinating discovery/insight. I couldn't understand it at first but did some searching and reading and came across a detailed explanation. The quote above seems to be based on an earlier work by Charles Bennett, in "On Random and Hard-to-Describe Numbers" ( http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.214.7866&rep=rep1&type=pdf ). In it I found another nice quote: Throughout history philosophers and mystics have sought a compact key to universal wisdom, a finite formula or text which, when known and understood, would provide the answer to every question. The Bible, the Koran, the mythical secret books of Hermes Trismegistus, and the medieval Jewish Cabala have been so regarded. Sources of universal wisdom are traditionally protected from casual use by being hard to find, hard to understand when found, and dangerous to use, tending to answer more and deeper questions than the user wishes to ask. Like God the esoteric book is simple yet undescribable, omniscient, and transforms all who know It. The use of classical texts to foretell mundane events is considered superstitious nowadays, yet, in another sense, science is in quest of its own Cabala, a concise set of natural laws which would explain all phenomena. In mathematics, where no set of axioms can hope to prove all true statements, the goal might be a concise axiomatization of all “interesting” true statements. Ω is in many senses a Cabalistic number. It can be known of, but not known, through human reason. To know it in detail, one would have to accept its uncomputable digit sequence on faith, like words of a sacred text. It embodies an enormous amount of wisdom in a very small space, inasmuch as its first few thousand digits, which could be written on a small piece of paper, contain the answers to more mathematical questions than could be written down in the entire universe, including all interesting finitely refutable conjectures. Its wisdom is useless precisely because it is universal: the only known way of extracting from Ω the solution to one halting problem, say the Fermat conjecture, is by embarking on a vast computation that would at the same time yield solutions to all other equally simply-stated halting problems, a computation far too large to be carried out in practice. Ironically, although Ω cannot be computed, it might accidentally be generated by a random process, e.g. a series of coin tosses, or an avalanche that left its digits spelled out in the pattern of boulders on a mountainside. The initial few digits of Ω are thus probably already recorded somewhere in the universe. Unfortunately, no mortal discoverer of this treasure could verify its authenticity or make practical use of it. Jason -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CA%2BBCJUi-2VXM6apKYCtnazGz53oSP65q6bJ1cU3WenowV9%3D05Q%40mail.gmail.com.
