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Topic: No Identity Bijection for Omega
Replies: 116   Last Post: Sep 22, 2007 5:58 PM

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 tchow@lsa.umich.edu Posts: 1,133 Registered: 12/6/04
Re: different sizes of denumerable sets?
Posted: Sep 8, 2007 10:25 AM

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

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