Re: Two results of set geometry


William Hughes schrieb:
On Sep 24, 9:18 am, Albrecht <albst...@xxxxxx> wrote:
WM schrieb:



On 19 Sep., 13:23, William Hughes <wpihug...@xxxxxxxxxxx> wrote:
On Sep 19, 2:52 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:

1
11
111
...

Omega is an ordinal. Like any ordinal, it is also a *set* of
ordinals.
The complete first column represents the *set* Omega. No shorter
segment of the column represents the *set* Omega. A bijection
involving Omega involves the *elements* of the *set* Omega.
The complete first column is not an *element* of Omega.

That is not in question.
1) The initial segments of the diagonal are projections of the initial
segments of the first column including the whole infinite column. (No
problem.)
2) The initial segments of the diagonal are projections of rows. There
is no infinite row. But there is an infinite initial segment of the
diagonal. (A problem, in particular with respect to the fact that all
diagonal elements belong to a strict subset of the elements in the
rows.)

I think we are forced to accept that mathematicians don't have a
problem with the fact that the first column is infinite since it ever
runs on and on, but no row is infinite since any row starts with an
element of the first column which has a finite index.
And in spite of the fact that the first column - and any column -
doesn't come to an end they like to handle this structure as a
complete one.


You have not yet definied "complete".


To talk about ordinal numbers should be the same as to talk about
strings of symbols like 111....
So let's talk about the strings of the symbol "1" in the system

1
11
111
1111
....


So we can say correctly:
"The strings in the horizontal rows _grow_ infinitely. But all of them
_are_ finite."

Now, in modern math we can say:
"The string in the first (and any) vertical column _is_ infinite, and
the string on the diagonal _is_ infinite."

But we can't say:
"If there is a (complete) infinitude of growing strings, there must be
at least one which is infinite," since this leads to the well-known
contradiction.

In my opinion we have to say right:
"As the horizontal strings grow infinitely, the vertical strings (and
the diagonal string) _grow_ infinitely too."

So there is no infinitude of strings. There is only an infinitude of
_growing_ of strings.
To talk about an infinitude of objects is just a sloppy way of
talking.
In fact, the infinitude isn't tamed by Cantor.

(So, dear William, a complete structure is a finite structure.
(Perhaps one might say: a complete structure is a structure with at
least one finite extension? (Yes, I know, I should define
"extension".)))


Best regards
Albrecht S. Storz

.

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  • Re: Two results of set geometry
    ... The complete first column is not an *element* of Omega. ... The initial segments of the diagonal are projections of the initial ... segments of the first column including the whole infinite column. ... So let's talk about the strings of the symbol "1" in the system ...
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  • Re: Two results of set geometry
    ... The complete first column represents the *set* Omega. ... The complete first column is not an *element* of Omega. ... The initial segments of the diagonal are projections of the initial ... segments of the first column including the whole infinite column. ...
    (sci.math)
  • Re: Two results of set geometry
    ... The complete first column is not an *element* of Omega. ... The initial segments of the diagonal are projections of the initial ... segments of the first column including the whole infinite column. ... So let's talk about the strings of the symbol "1" in the system ...
    (sci.math)