Re: Rucker's Infinity and the Mind
Title: Re: Rucker's Infinity and the Mind George Levy wrote in part: Beautiful post, Hal. I have read and reread Rudy Rucker's "Infinity and the Mind" four or five times. This is such a rich book that I enjoy it everytime. His explanation of the infinite always leaves me in awe. I share the joy. Rudy Rucker's "Infinity and the Mind" is formidable. Please note that: Its "excursion II" on Godel's incompleteness theorem is very good. The excursion ends on a technical note on man-machine equivalence quite genuine to introduce yourself to ... the consistent machine's elaborations ... Hal Finney wrote in part: A multiverse built on computational engines would be far more limited than one which includes all the endless richness of mathematical set theory. I disagree. To disentangle the machine dreams, not *one* grain dust of Cantor paradise will be too much ... Mathematics is really like wonderland (or yellow submarine if you prefer) full of spaces which looks little (countable or even finite) from the outside and are undefinissably big from inside. Godel's theorems limits what machines can say and prove about ... machines. Even machine will need the non mechanical "notions" to speed up their learning about machines and anticipated their behaviors. Bruno
Re: Rucker's Infinity and the Mind
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!).
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 Centrehttp://parallel.hpc.unsw.edu.au/rks
International prefix +612, Interstate prefix 02
Re: Rucker's Infinity and the Mind
Beautiful post, Hal. I have read and reread Rudy Rucker's "Infinity and the Mind" four or five times. This is such a rich book that I enjoy it everytime. His explanation of the infinite always leaves me in awe. I agree with you that our brains and our bit-based digital computers are limited to countable sets. Quantum computers, on the other hand may not be so limited. Who knows, there might be as many versions and models of quantum computers as there are of infinite cardinals. One day we may even easily "simulate" on a desk top, using an Aleph1 computer model, an infinite Aleph0 universe. That Aleph1 computer running faster than real time could then become a perfect oracle for that universe. George Hal Finney wrote: [EMAIL PROTECTED]"> 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. Ruckerincludes some speculations about the possibility that the multiversecould be identified with the class of all possible sets, similar to theidea that Tegmark later developed in greater detail.First I'll say a little something about infinite numbers. Many peopleare familiar with the transfinite cardinals: aleph-zero (or aleph-null),the cardinality of the integers; aleph-one, aleph-two, and so on; C,the cardinality of the continuum, which may or may not be one of theearlier alephs. However there is a whole other system of transfinites,the ordinals.The transfinite ordinals are generalizations of counting numbers.The first infinite ordinal is omega, which I will write here as w, whichthe greek lower-case omega resembles. We can create a number series like:0,1,2,3,... w,w+1,w+2,... w*2,w*2+1,... w*3,... w*4, ..., w^2, ... w^3, ...,w^w, ..., w^w^w, ...The idea here is that w has no predecessor, but it has a successor, w+1.And that has a successor, w+2. We can keep on with this process untilwe get to the next number which has no predecessor, w+w, written as w*2.>From here we get to w*3, w*4 and so on, and we can then generalize thispattern to get to w*w which we write as w^2.There is no end to this type of pattern; we can go on finding new infiniteordinals indefinitely. However all of the ordinals generated via thisscheme share the property of being countable. That is, they can all beput into one-to-one correspondence with w.This leads to the definition of cardinal numbers; a cardinal number isthe smallest one of a set of ordinals which can be put into one-to-onecorrespondence with each other. This distinction is irrelevant forfinite numbers, but for tran sfinites, all of the numbers beyond w inthe list above can be put into one-to-one correspondence, and all havethe same cardinality, which is written aleph-zero and is equal to w.The first non-countable transfinite is aleph-one. You would have toput aleph-one w's together to reach aleph-one; you can't get thereby any countable process involving w. Beyond aleph-one is of coursealeph-two, and so on, and we can now use our ordinals very usefully tolist the alephs:aleph-0, aleph-1, aleph-2, ..., aleph-w, aleph-(w+1), ..., aleph-(w*2), ...aleph-(w^w), ... aleph-(aleph-one), ...Theta is the smallest transfinite for which theta = aleph-theta, which canalso be considered aleph-aleph-aleph-... going on forever. And this isfar from the end. Rucker describes vastly, vastly larger transfinites.So, what is the relevance of this for all-universe models? Ruckerdescribes how numbers, in modern mathematics, can be con sidered to bespecial kinds of sets; specifically sets of sets in certain combinations.Similarly, all mathematical objects can be built ultimately on set theory.And this raises the possibility that physical objects are sets as well.Rucker writes on page 200:"And consider this: If reality is physics, if physics is mathematics,and if mathematics is set theory, then everything is a set. I am a set,my thoughts are sets, my emotions are sets If everything is a set,then only pure form exists, which is nice. The whole physical universecould be a single large set U."Rucker then speculates on where this U would be in the framework ofordinals and other mathematical objects. He shows a diagram in theshape of a V, with the empty set at the bottom, at the point of the V.Going straight up above it is a vertical line where the ordinals arefound. To the side of the line are other mathematical objects that havesimilar complexity.This diagram in principle holds all sets, meaning that it holds allmathematical objects. It is well known in set theory that "the set ofall sets" is a contradictory concept, so instead the V as a whole iscalled "the class of all sets" and Rucker uses V to denote this concept.He then asks how U, the set which is the universe, compares with V.>From page 201-202: Say that U is the set coding up our physical universe. How far up wouldone expect to find U?...How much information is in the universe? If the universe iscompletely finite, then U is a set som
