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.

...

> 
> 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!).

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