The Superstring Theorists have the idea (which I am highly skeptical of, but never mind) that string theory is the "final theory" of physics and will predict the values of all fundamental physical constants, uniquely. However, the uniqueness parade got rained on when the notion of the "landscape" came along. That is, in string theory, the universe is not 3+1 dimensional like you thought, but actually has 6 (or 7 for "M theory") extra dimensions, all "microscopic" and having the topology of a "Calabi Yau manifold." The question then is WHICH Calabi Yau manifold? Also, there are various integer valued "fluxes" or "winding numbers" trapped in this topology. So, to specify the Final Laws Of The Universe, you do not actually get unique laws -- you get a large set of possible laws depending on which CY manifold and which integer vectors you choose. Oops. But, supposedly, once you'd specified that stuff, from then on everything was going to be unique and all physical constants were going to get predicted exactly from the power of pure thought (if only anybody was smart enough to do the computation), etc. So: How many Calabi-Yau manifolds and acceptable integer vectors are there? One paper guesstimated the number as 10^500. This suggests that quite possibly, specifying that stuff will require more bits than it takes to specify all known physical constants, which would be kind of a philosophical failure. But anyhow, nobody has even proven the CY-manifold count is even BOUNDED. It might be "infinite." And apparently, it is now conjectured that the CY-manifolds come in a finite number of families, each containing a countably infinite number of topologically inequivalent members. Which leads me to my QUESTION -- can we even prove it is at most COUNTABLY infinite? That, at least, might be easy. Maybe even trivial. Or already known. The inequivalent 2-dimensional-manifold compact topologies were classified by Mobius and plainly are countably infinite in number. For 3D, the Thurston Geometrization Conjecture proven by G.Perelman classifies all possible compact topologies, and I believe makes it clear that they too are countable. More generally, in any fixed dimension, if we regard a topology as defined by a simplicial complex having that same topology ("piecewise linear" manifolds) -- well, it seems clear the number of PL topologies is countable. Proof: specify the number V of D-simplices you want. Then their connection pattern is defined by a (D+1)-regular graph with V vertices and each edge labeled with up to D! permutations. All that information happens in only a countable set of possible ways. In fact we could easily program a Turing machine to print them all out (runs forever, and probably generates a superset, but does print them all) which proves countability kind of "constructively." QED I suspect/hope that "wild" topologies (not describable in piecewise linear manner) are forbidden in stuperstring theory (and physics generally) in which case this would prove countability. Do the string theorists claim that, once you specify the topology of the CY, then its geometry (metric) is forced ("rigidity")? Anyhow the hope I here am trying to hope is: that at least, if superstringers are right, then the laws of physics can be specified using a FINITE number of bits, because of countability. If it had been uncountable, then that would not be obvious. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)