Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx>
- Date: Thu, 17 Jan 2008 06:13:07 -0600
On Thu, 17 Jan 2008 09:30:14 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:
lwalke3@xxxxxxxxx wrote:
OK, I think I see what the confusion is now.
Here tommy1729 is making an assumption which, while admittedly
intuitive, is false in ZFC. The assumption is that the composition of
infinitely many permutations must itself be a permutation. While it
is definitely true that the composition of finitely many permutations
must be a permutation, it's not true if there are infinitely many.
Plonk! Plonk! How can something be true for the composition of finitely
many permutations and not true if there are infinitely many? (Infinitely
many IMHO is just a limiting case of finitely many.)
As has been pointed out, we actually _are_ talking about a limit
here, not a literal composition of infintely many permutations.
The limit of a squence of permutations of N can indeed be a
non-permutation.
Consider the following sequence of permutations of N:
1,2,3,4,5,6,7,8,...
2,1,3,4,5,6,7,8,...
2,3,1,4,5,6,7,8,...
2,3,4,1,5,6,7,8,...
..
..
..
Each one of those is a permutation of the natural numbers.
But the limit is this:
2,3,4,5,6,7,8,...
which is not a permutation, because 1 does not appear.
But instead of IMH0
and insisting on it, I would like to learn how this can be in _standard_
mathematics.
I always like that, when you talk about "standard" mathematics...
Tell us, in _your_ mathematics what is the limit of that sequence
of permutations?
Especially because I was flabbergasted to learn that in the
universe of infinite permutations the cardinality of them is the one for
the continuum (c) and not Aleph_0. Did I got this right?
Han de Bruijn
************************
David C. Ullrich
.
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