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Topic: Matheology § 170
Replies: 41   Last Post: Dec 8, 2012 5:35 PM

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 netzweltler Posts: 473 From: Germany Registered: 8/6/10
Re: Matheology § 170
Posted: Dec 6, 2012 5:21 PM

On 4 Dez., 07:48, WM <mueck...@rz.fh-augsburg.de> wrote:
> Matheology § 170
>
> The infinite triangle formed by the sequence
>
> 0.1
> 0.11
> 0.111
> ...
>
> has height aleph_0 but width less than aleph_0 (because the limit 1/9,
> the first line with aleph_0 digits, does not belong to the triangle).
> This lack of symmetry is disturbing for a physicist.
>
> But it would be completely unclear, what side of the infinite triangle
> is the first one to complete infinity aleph_0, when constructed in the
> following manner:
>
> a
>
> a
> bb
>
> c
> ac
> bbc
>
> d
> dc
> dac
> dbbc
>
> d
> dc
> dac
> dbbc
> eeeee
>
> ...
>
> Regards, WM

There is a similar problem. If we construct this infinite list in
countably many steps:

00
01

000
001
010
011
100
101
110
111

0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111

...

we cannot tell how the infinite list looks like. It could be the list
of natural numbers { ...0000, ...0001, ...0010, ... }, or it could be
the list of the binary complement of the natural numbers { ...1111, ...
1110, ...1101, ... }. Even if the resulting infinite lists are
completely different - the finite lists during construction are
exactly the same for each step in both cases.

--
netzweltler