In article <1189173869.254426.77510@19g2000hsx.googlegroups.com>, george <greeneg@cs.unc.edu> wrote: >I honestly think the more legitimate path toward a math dissertation >would be coming up with some kind of new numbers that would allow a >size/metric to be established for infinite subsets of natnums, >something like probability or density. Everyone wants to say that >there are twice as many natnums as even natnums.
As someone else mentioned, natural density is one such notion. Slightly more general (but complicated to define) is analytic density, which IIRC is needed to prove that Benford's law is still valid when restricted to prime numbers.
It's too optimistic to think that there is a *single* way of measuring the size of infinite sets that suffices for all applications. Mathematicians realize this and define different notions of size, depending on what phenomenon they're trying to understand. For a good example of this at work, see the article "Prime number races" by Andrew Granville and Greg Martin (easy to find with Google). No less a mathematician than Chebyshev once said, "There is a notable difference in the splitting of the prime numbers between the two forms 4n+3, 4n+1: the first form contains a lot more than the second." Did Chebyshev need a basic lesson in cardinality? Did he need to be patiently told that the two sets are countably infinite and therefore must have the same size? Of course not. There is a real phenomenon here that begs for explanation. It turns out that natural density doesn't suffice in this case, and a more subtle analysis is needed. See the Granville-Martin paper for more details. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
