On 12 Oct 2013, at 01:08, meekerdb wrote:

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On 10/11/2013 3:44 PM, Russell Standish wrote:On Fri, Oct 11, 2013 at 03:08:30PM -0700, meekerdb wrote:UD* (trace of the universal dovetailer) is a continuum, AFAICT.It hasthe cardinality of the reals, and a natural metric (d(x,y) = 2^{-n}, where n isthe number of leading bits in common between x and y).Hmm? So 1000 is the same distance from 10 and 111? What's themeasure 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.Maybe I'm not clear on what UD* means. I took it to be, at a givenstate of the UD, the last bit output by the 1st prog, the last bitoutput by the 2nd program,...up to the last prog that the UD hasstarted. Right?

`Imagine a universal (and thus finite) game of life pattern. Then you`

`can look at UD* as the infinite cone obtained by adding all the planes`

`describing its evolution. That gives a static view of UD* as a`

`discrete infinite 3D conic object.`

`It is, as Russell and Liz said, the trace of the programs which runs`

`all computations in that parallel -dovetailing manner.`

`But physics and theology are emerging from the internal relative`

`machines points of views, and that gives a richer structure, relying`

`on the continuum. The 1/3 distinction has a key role here.`

Bruno

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