Chaitin numbers of course can also be extended to the computational hierarchy (oracles, Turing jump iterations of the halting problem):
Super-Ω https://en.wikipedia.org/wiki/Chaitin%27s_constant#Super_Omega https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2011.0319 @philipthrift On Friday, August 16, 2019 at 6:31:31 PM UTC-5, Russell Standish 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]". > > Cheers > -- > > ---------------------------------------------------------------------------- > > Dr Russell Standish Phone 0425 253119 (mobile) > Principal, High Performance Coders > Visiting Senior Research Fellow [email protected] > <javascript:> > Economics, Kingston University http://www.hpcoders.com.au > ---------------------------------------------------------------------------- > > -- 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 [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/743f85fb-6da9-4cda-888d-a806e987d86a%40googlegroups.com.
