Re: Two results of set geometry

On Sep 26, 8:53 am, Albrecht <albst...@xxxxxx> wrote:
William Hughes schrieb:



On Sep 25, 7:48 am, Albrecht <albst...@xxxxxx> wrote:
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.

Again you use "complete" before you define it.

I thought, if you answer, first you like to read a posting. But you
are right: the definition I suggested is at the end of my posting.
Since I used the word "complete" in this connection only to clarify my
point, I think my carelessness isn't very bad.
But I hoped for you would support the sentence you find fault.

Best regards
Albrecht S. Storz

I took the mention of complete at the end
as a conclusion, not a definition.

So your definition of complete is

"a complete structure is a finite structure."

You uae a new term "structure".

Do you mean?

"a complete set is a finite set"

In any case, using this definition,
the term "complete infinitude"
means nothing more or less than
"finite infinitude".
So we know that there is no
finite infinitude of growing
strings. Not a very useful definition.

- William Hughes


.

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