Re: Two results of set geometry

On Sep 26, 1:30 pm, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
On 26 Sep., 18:08, William Hughes <wpihug...@xxxxxxxxxxx> wrote:

On Sep 26, 11:18 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:

On 26 Sep., 16:48, William Hughes <wpihug...@xxxxxxxxxxx> wrote:

<snip>

too difficult?



No. omega is the *complete column*, a set of elements.
There is no element in the complete column that
is not in the set of rows. Thus there is no element
in the complete column that completes the set of elements
of the column. The *complete column* is not *completed*.

Then we can say omega + 1 = omega?

No. If you append one 1 to the *complete column* you
get a different column of length omega +1.
The fact that *complete column* is not *completed*
does not change this fact. Even though the *complete
column* is not *completed* there is no hole where we
could stick the one.

If you append 1 element to each of the finite segments of the column,
then you get no infinite segment.

Correct. Appending 1 to each finite segment is not
the same thing as taking the *set* of finite segments and
appending 1 to the set.

If you append 1 element to the complete column,

Note the complete column is a *set* of finite elements,

then you get an
infinite segment, even of order type omega + 1.

Nevertheless the complete column is nothing but all its finite
segments?

The complete column is the *set* of all its finite segments.
Appending 1 to the *set* of all finite segments, is
not the same thing as appending 1 to each of the *elements*
of the set of all finite segments.

Appending one to each of the *elements* of the complete
column is not adding one to omega. Appending one to the
*set* of all elements, the complete column, is adding one to omega.
Appending 1 to the complete column produces a column
of length omega+1.

- William Hughes

.

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