On Fri, Oct 11, 2013 at 03:08:30PM -0700, meekerdb wrote: > >UD* (trace of the universal dovetailer) is a continuum, AFAICT. It has > >the cardinality of the reals, and a natural metric (d(x,y) = 2^{-n}, where n > >is > >the number of leading bits in common between x and y). > > Hmm? So 1000 is the same distance from 10 and 111? What's the measure on > this space? >

1000... and 101... are 0.25 apart. 1000.. and 111... are 0.5 apart. (the ... refers to an infinite number of bits that are not relevant to the computation). So the answer to your question is that these these three strings are not the same distance from each other. The measure over a set of these things would be something like the supremum over the distance between any two pairs drawn from the set. Of course, that assumes that only sets defined by finite length prefixes, and countable unions and intersections thereof are considered. My maths chops aren't quite up to generalising this for arbitrary sets of binary strings. -- ---------------------------------------------------------------------------- Prof Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Professor of Mathematics hpco...@hpcoders.com.au University of New South Wales 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.