Re: Two results of set geometry

On 25 Sep., 15:00, William Hughes <wpihug...@xxxxxxxxxxx> wrote:
On Sep 25, 8:42 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:





On 25 Sep., 13:36, William Hughes <wpihug...@xxxxxxxxxxx> wrote:

On Sep 25, 6:55 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:

On 24 Sep., 22:37, William Hughes <wpihug...@xxxxxxxxxxx> wrote:

On Sep 24, 4:09 pm, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:

On 24 Sep., 20:39, William Hughes <wpihug...@xxxxxxxxxxx> wrote:

<snip>

The only way to get the finite segments without the complete
column is to write each finite segment in its own column.

That can be done. And then these segments can be projected in one
column.

And note that you do not change the finite segments in any way.
So any bijection that exists when each finite segment is written
in its own column still exists when all finite segments are written
in the same column.

When each of the finite segments is written in its own column there
is a bijection from the finite segments to N.

Yes.

When each of the finite segments is written in the same column,
there is a bijection from the finite segments to N.

Yes.

Therefore there are as many elements in N as there
are finite segments. From

WM: There are omega finite segments 1, 11, 111, ...

we conclude there are omega elements in N.

Without the complete column. So the complete column is not more and
not less - it is simply not existing?

No, the complete column is the set of all finite segments.
It certainly exists. However, it is a *set* of finite
segments (or equivalently, a *set* of indices).
It is not a finite segment.

As you acknowledge, "there is a bijection from the finite
segments to N". As you point out
there are omega finite segments. So there
are omega elements in N.

Not without N.

Take the function f(x) = 1/x. The values are less than infinite for x
0. Only for x = 0 there is an undefined respectively infinite value.

The same holds for our matrix. No initial segment of the column is in
bijection with infinitely many rows. In order to have a bijection with
omega rows you need the complete column, i.e., you need to include x =
0 in the picture given above.

Regards, WM

.

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