On Wed, Sep 12, 2007 at 07:32:32AM -0700, Youness Ayaita wrote: > > The two concerns, how to give a precise notion of the Everything, and > how to deduce predictions from a chosen notion, lie at the very heart > of our common efforts. Though, I did not go into them for the simple > reason that I wanted to avoid discussions that are not directly linked > to the topic. > > When I first wanted to capture mathematically the Everything, I tried > several mathematicalist approaches. But later, I prefered the > Everything ensemble that is also known here as the Schmidhuber > ensemble. I assume that the no-justification naturally leads to this > ensemble. This comes from the development of the (degenerate) property > of existence which is then assigned to all imaginable things. I don't > think that a metaphysical discussion of the term "imaginable thing" is > necessary now, I'm satisfied with the idea that an imaginable thing > can be completely described by means of language. For further > research, it is then natural to identify imaginable things with their > descriptions and to choose a simple alphabet for expressing the > descriptions (e.g. strings of 0 and 1). In the past I assumed these > strings to be of finite length. I read that Russell Standish also > permits infinite strings. >

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These sorts of discussions "No-justification", "Zero-information principle", "All of mathematics" and Hal Ruhl's dualling All and Nothing (or should that be "duelling") are really just motivators for getting at the ensemble, which turns out remarkably to be the same in each case - the set of 2^\aleph_0 infinite strings or histories. Where differences lie is in the measure attached to these strings. I take each string to be of equal weight to any other, so that there are twice the measure of strings satisfying 01* as 011*. This leads naturally to a universal prior. Schmidhuber has a different measure, assuming that the strings are generated in real time from a machine with bounded resources. This is his "speed prior", and leads to a quite different measure on the strings. Neither Bruno's nor Max's theories give a measure, but remarkably the Occam's razor theorem and White Rabbit result is fairly insensitive to the measure chosen (so long as it's not too pathological!). On your comment on permitting infinite strings - the ensemble I describe in my book has only infinite strings, which belong to syntactic space. A finite string corresponds to a set of infinite strings all having the same finite prefix, and as such belongs to semantic space. It would be possible to construct an ensemble of purely finite strings (all strings of length googol bits, say). This wouldn't satisfy the zero information principle, or your no-justification, as you still have the finite string size to justify (why googol and not googol+1, for instance). I suspect the observable results would be indistinguishable from the infinite string ensembles for large enough string string size, however. -- ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---