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

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 George Greene Posts: 448 Registered: 12/12/04
Re: different sizes of denumerable sets?
Posted: Sep 12, 2007 10:59 AM

On Sep 8, 10:25 am, tc...@lsa.umich.edu wrote:
> 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?

Come on; that is a strawman; that is TOO easy!

> Did he need to be patiently told that the two sets are countably infinite
> and therefore must have the same size? Of course not.

Of COURSE, so why did you stoop so low?

> There is a real phenomenon here that begs for explanation.

Exactly, and the NATURAL place to start, the level to which you
SHOULD'VE
stooped, would've been NATURAL density, not simple denumerability
(that
was REALLY over-trivializing the question). Most people would think
that "more" would've meant exactly that -- more IN THE NATURAL
DENSITY
SENSE of "more".

> It turns out that natural density doesn't suffice in this case,

Well, there's two ways that could happen. One would be that natural
density could fail in the way it USUALLY fails, namely, the limit does
not
converge. But oddly, here, even that might NOT be a failure, because
while
you might not get absolute convergence, you might still get that as n
increases,
one of these forms always has a lot more than the other. The second
way,
which would be far more interesting, would be that despite oscillation
in who was
winning the "natural density" measurement, there was a more "subtle"
or appropriate
measure that resolves competition clearly on one side.

> and a more subtle analysis is needed.
> See the Granville-Martin paper for more details.

Thank you kindly; I shall.

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