Re: 1-1/2+1/3-1/4+1/5-1/6+1/7

David C. Ullrich wrote:

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 replied to Chas Brown, I didn't mean to put anyone in a killfile ..

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.

Yes. I've admitted that now in a couple of other replies.

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.

Nice example. But how about the following.

Suppose we call that last non-permutation an "Ullrich". Then we can
execute several of these Ullrich's, to successively obtain:

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,...
..
..
2,3,4,5,6,7,8,... : an Ullrich .

2,3,4,5,6,7,8,...
3,2,4,5,6,7,8,...
3,4,2,5,6,7,8,...
3,4,5,2,6,7,8,...
.
.
3,4,5,6,7,8,... : next Ullrich .

And so on:

1,2,3,4,5,6,7,8,...
2,3,4,5,6,7,8,...
3,4,5,6,7,8,...
4,5,6,7,8,...
5,6,7,8,...

Finally :
...

So, after an infinitude of Ullrich's, the _whole_ set of naturals has
just disappeared (or, depending upon your sense of humour: it has been
dropped down at the other side of the '...'). But, _each_ of the above
sets has the cardinality of the natural numbers. But, the last set ...
is the _empty_ set.

Consequently, the empty set has the cardinality of the natural numbers?

What goes wrong here?

Han de Bruijn

.

Relevant Pages

  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... infinitely many permutations must itself be a permutation. ... is definitely true that the composition of finitely many permutations ... So, after an infinitude of Ullrich's, the _whole_ set of naturals has ...
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  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... infinitely many permutations must itself be a permutation. ... Suppose we call that last non-permutation an "Ullrich". ... So, after an infinitude of Ullrich's, the _whole_ set of naturals has ...
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