Re: Two results of set geometry

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

omega is double valued.

No

Then it does not exist as a number which can be increased (see below).

omega is the number of finite elements (rows)
which is not completed,

By completed you mean there is an element
that completes the set.

No. By completed I mean a column of ordinal number omega. omega is the
first transfinite ordinal. So it is larger than every finite ordinal
(larger than every finite segment of the column). Therefore it must
have more 1's than every finite segment. In fact it has, because it is
not in the bijection with one of the finite rows.

And it is not in bijetion with the complete diagonal, at least with
the horizontal component, because this component is not infinite.

{f(x)| f(x) = 1/x, 1 < x < 2} does not include 1.
Therefore the horizontal component of the diagonal
{f(n)| f(n) = n, 0 < n < oo} does not include oo.

By this definition the set of rows is not completed
because there is no element in the set of rows
which completes the set.

and omega is the completed infinite set.

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? This would support the view of
potential infinity.

Regards, WM


.

Relevant Pages

  • Re: Two results of set geometry
    ... (larger than every finite segment of the column). ... the horizontal component, because this component is not infinite. ... No. omega is the *complete column*, ... potential infinity. ...
    (sci.math)
  • Re: Two results of set geometry
    ... *set* of all elements, the complete column, is adding one to omega. ... Appending 1 to the complete column produces a column ... Every finite segment is the set of smaller finite segments. ... William Hughes ...
    (sci.math)