Hal Finney wrote: > > I was re-reading Rudy Rucker's 1982 book Infinity and the Mind last week. > This is a popular introduction to various notions of infinity. Rucker > includes some speculations about the possibility that the multiverse > could be identified with the class of all possible sets, similar to the > idea that Tegmark later developed in greater detail.

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... > > So here Rucker is advancing the notion that U, the universe, is identical > to the class of all sets, which is itself the same as the class of all > mathematical structures. This is the same idea which Tegmark championed, > where he brought in the anthropic principle to explain why the visible > universe has the lawful and orderly structure that we observe. > > I had some very enlightening discussions with Wei Dai at the Crypto > conference last week, and he mentioned that this view of the multiverse, > which we associate with Tegmark, implies a very much larger multiverse > than the computational view advanced by Schmidhuber, at least if we > restrict the notion of computation to Turing machines and simple > extensions. Most of the objects treated by modern set theorists > are vastly larger than even the transfinite theta I mentioned above, > putting them far outside the reach of a Turing machine. A computer is > fundamentally a sequential object with a finite, or at most countably > infinite, complexity, and these infinite objects are far more complex. > > When we do mathematics, we are no more than a Turing machine, but we > should not confuse our limited understanding of these mathematical objects > with the objects themselves. Godel teaches us that axiomatic reasoning > is a very limited tool for approaching mathematical truth, but it is > unfortunately the only tool we have (modulo claims of extra-algorithmic > "mathematical intuition"). A multiverse built on computational engines > would be far more limited than one which includes all the endless richness > of mathematical set theory. > > Hal Finney > Tegmark is suitably obscure as to whether he is referring to some grander collection of mathematical objects, or just the axiomatisable ones. Rucker is obviously talking about the former, but I'm inclined to think that the idea is just plain incoherent. Therefore, I've always chosen to interpret Tegmark as referring to the axiomatisable stuff. This is wholly contained with the set of all descriptions, which is a set, and has cardinality "c" (can be placed in one-to-one correspondence with the reals, modulo a small set of measure zero). This set of all descriptions is the Schmidhuber approach, although he later muddies the water a bit by postulating that this set is generated by a machine with resource constraints (we could call this Schmidhuber II :). This latter postulate has implications for the prior measure over descriptions, that are potentially measurable, however I'm not sure how one can separate these effects from the observer selection efects due to resource constraints of the observer. One can consider this complete set of descriptions to be generated by a machine running a dovetailer algorithm, however the machine would need to run for c clock cycles, so it would be a very unusual machine indeed (not your typical Turing machine). Personally, I don't think this view is all that productive. The advantage of the set of all descriptions is that it does contain anything accessible by an observer, and it has precisely zero information content. I find it hard to see what is gained by adding in other mythematical beasts such as powersets of the reals - somehow they must have zero measure, or be otherwise irrelevant to observers (although a neat proof of this would be nice!). ---------------------------------------------------------------------------- A/Prof Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile) UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 (") Australia [EMAIL PROTECTED] Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 ----------------------------------------------------------------------------