Saturday, July 21, 2007

Of Mice and Mengen

As mentioned in my last post, the Burali-Forti paradox seems to be, from the standard perspective (see the aforementioned paper), the most fundamental and paradoxical of the set-theoretical paradoxes, although from my own perspective it is the least fundamental, which is why I’ve already looked at Russell’s and Cantor’s paradoxes.
......I’ll just briefly describe the standard paradox, which arises from the standard presupposition that the natural numbers are a standard set, that they form an actual (or finitesque) totality—in more picturesque terms, one that can, when they (or rather, their tokens) are envisaged spatially, be thought of as being all together, in the same space; or one that can, when their endless sequence is thought of more temporally (in the intuitive sense, i.e. not as part of a quasi-spatial space-time), be completely run through. (For an independent, scientific reason why that standard presumption is likely to be false, see my forthcoming paper.)
......Beginning with standard set theory and its von Neumann ordinals (0 = the empty set, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, and so forth), we have that each ordinal is the set of the preceding ordinals. The first transfinite ordinal is omega = the set of all the natural numbers, and it is followed by omega + 1 = {0, 1, 2, …, omega}, and so on. There is no set of all such ordinals because if there were it would similarly be an ordinal, and it would be greater than (since arising higher in the set-theoretic hierarchy than) all its members, whereas they were presumed to be all such ordinals. Paradox arises because our intuition that there should be a set of all the natural numbers (and hence our first transfinite ordinal) seems also to tell us that there should be such a set of all those ordinals (and hence some last ordinal, usually denoted by capital-omega).
......Cantor originally needed transfinite inductions, and hence his set (Menge) theory, because he was investigating arbitrary functions of the real number line, as part of the development of Fourier analysis. Although the functions usually met with in physics are well behaved (smooth, differentiable etc.) almost everywhere, one generally expects there to be points (i.e. real numbers) where they are discontinuous etc. Analysis is simplest when there are not infinitely many such points in any finite interval, but if there are then they will have at least one accumulation point, and there might be infinitely many such accumulation points, with accumulation points of their own, etc.
......Cantor was therefore considering sequences of point-sets, each set of points obtained from the preceding one by keeping only the accumulation points. Often, for a given function, the Nth point-set in that sequence would, for some natural number N, contain only a finite number of points, with the (N + 1)th point-set being empty, but the particularly troubling (and therefore mathematically interesting) cases were the functions for which the Nth point-set contained infinitely many points for every natural number N (e.g. the function whose value is 1 at each rational point, and 0 elsewhere). So Cantor was considering what was left after all the naturally numbered stages of that process had been gone through, i.e. at the omegath stage, and (since that might also contain infinitely many points) beyond.
......And one reason why the Burali-Forti paradox remains so paradoxical (see the aforementioned paper) is that today’s mathematicians seem to find a use for inductions that range over ordinals that are (in some mysterious way) even bigger than capital-omega, e.g. in mouse theory. So, what got us using standard ordinals in the first place does seem to take us beyond them, paradoxically. And so perhaps the epistemic possibility that the natural numbers do not actually form a set (which does not presuppose a constructivistic view of numbers, but only that they are an objective structure, whose properties are to be discovered rather than stipulated) should not be too casually dismissed.
......If the natural numbers happen not to form a set, but are rather only potentially infinite, then although we could still consider ordinals insofar as they are the order-types of well-orderings (i.e. as reifications of certain structures) we would not have the same use for them. Cantor’s naturally numbered stages could not have been completely run through, to get us to an omegath stage, but that would not matter because geometrical lines would not then be isomorphic to real number lines, so there would not be such arbitrary functions to investigate in the first place.
......That is not to say that investigating them was not useful, that the standard approach has not given us a nice formal model of geometry, adequate for most applications, but only that it would then be (as it seems to be) no more than that. Note that Cantor did dismiss that possibility rather casually—remarking that a potential infinity (thought of as a variable) presupposes a logically prior actual infinity (the domain of the values that the variable could take)—perhaps too casually (or Hegelian?) given that he was aware of these paradoxes.

1 comment:

SteveG said...

Best. Title. Ever.