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.
First I'll say a little something about infinite numbers. Many people are 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 the earlier 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, which the 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 until we 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 this pattern 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 infinite ordinals indefinitely. However all of the ordinals generated via this scheme share the property of being countable. That is, they can all be put into one-to-one correspondence with w. This leads to the definition of cardinal numbers; a cardinal number is the smallest one of a set of ordinals which can be put into one-to-one correspondence with each other. This distinction is irrelevant for finite numbers, but for transfinites, all of the numbers beyond w in the list above can be put into one-to-one correspondence, and all have the same cardinality, which is written aleph-zero and is equal to w. The first non-countable transfinite is aleph-one. You would have to put aleph-one w's together to reach aleph-one; you can't get there by any countable process involving w. Beyond aleph-one is of course aleph-two, and so on, and we can now use our ordinals very usefully to list 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 can also be considered aleph-aleph-aleph-... going on forever. And this is far from the end. Rucker describes vastly, vastly larger transfinites. So, what is the relevance of this for all-universe models? Rucker describes how numbers, in modern mathematics, can be considered to be special 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 universe could be a single large set U." Rucker then speculates on where this U would be in the framework of ordinals and other mathematical objects. He shows a diagram in the shape 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 are found. To the side of the line are other mathematical objects that have similar complexity. This diagram in principle holds all sets, meaning that it holds all mathematical objects. It is well known in set theory that "the set of all sets" is a contradictory concept, so instead the V as a whole is called "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 would > one expect to find U? > > ...How much information is in the universe? If the universe is > completely finite, then U is a set somewhere in V_w, perhaps in V_googol. > And even if it is infinite, we wouldn't expect it to be so very far out - > surely U must lie in V_(w+w). > > It seems strange to have our physical universe being just a little set > U floating around in the big universe V of all possible sets. Under > this viewpoint, all the possible universes would be sets in V. Is it > really reasonable to have an idea like V be so much bigger than the real > world? > > There is always the possibility... that the collection U coding up the > physical universe is very much larger than we had expected. If there were > many parallel universes to be included, if matter were transfinitely > divisible, if time were transfinitely long - in any of these cases, it might > actually be possible to have U too big to be a set, too big to fit inside > V.... > > Of course, our daily experience flies in the face of any suggestion that > U is so very big. But there is a traditional philosophical principle, > the Principle of Plenitude, which suggests that the physical universe should > be as rich as the set theoretic universe of pure Platonic forms. Insofar > as any physical structure can be coded up as a set, we already expect V > to be as large or larger than U. The Principle of Plenitude insists that > U must be as large or larger than V as well, leading to the conclusion > that U and V are equally large. > > A more extreme statement of this would be to insist that U and V are > identical, but this is really pretty hard to swallow. An argument in > this direction might be begun by remarking that we should understand U to > include all the alternate universes as well as our own perceived universe. > And one could then point out that a possibly existing alternate univese > is really an abstract form no different than a set. 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