On Wednesday, May 4, 2022 at 10:53:25 AM UTC-3, WM wrote:

> Before omega there is either nothing, i.e., a gap,

Nope, the word "gap" is only meaningful if there is a distance. The difference between omega and any element of N_p is a set difference, not a distance. The fact that this set difference has cardinality aleph_0, does not mean there is a distance which is infinite. There is no distance. The statement "omega -n = infinity" means "the cardinality of the set (omega\{1,2,3...,n}) is aleph_0". There is no largest ordinal that is smaller than omega. There is no "last" ordinal before omega'

==
William Hughes

Fritz Feldhase schrieb am Mittwoch, 4. Mai 2022 um 16:52:13 UTC+2:
> On Wednesday, May 4, 2022 at 3:59:46 PM UTC+2, WM wrote:
>
> > Either there is nothing immediately before omega, or it is dark.
> There is nothing "immediately before" omega.
>
> Hint: On the real number line there is nothing "immediately before" 1, still there is no "gap" between the real numbers < 1 and 1.

This is not so easy to understand for you. Therefore I have used the ordinals. There is either something immediately before omega or nothing is immediately befor omega. The detrimental influence of matheology forbids even this simple logic.

Of course the same prevails before any defined real number, but it is not so obvious, because you can get as close as you like. At omega it is clear that omega - n = omega.

Regards, WM

William schrieb am Mittwoch, 4. Mai 2022 um 17:45:14 UTC+2:
> On Wednesday, May 4, 2022 at 10:53:25 AM UTC-3, WM wrote:
>
>
> > Before omega there is either nothing, i.e., a gap,
> Nope,

Yes. Here we see the detrimental influence of matheology: denial of simplest logic.

If omega is a number larger than all natnumbers, then there is a difference which can be visualized as a distance.
This is the simplest obvious picture: 1, 2, 3, ..., ω, ω+1, ..., ω+k, 2ω, ...
It has a geometrical analogon.

> the word "gap" is only meaningful if there is a distance. The difference between omega and any element of N_p is a set difference, not a distance.

Wrong. "The system Omega in its natural order constitutes a 'sequence'." [Cantor, p. 445]

Regards, WM

On 5/4/2022 8:53 AM, WM wrote:
> William schrieb am Dienstag, 3. Mai 2022 um 23:01:47 UTC+2:
>> On Tuesday, May 3, 2022 at 5:43:02 PM UTC-3, WM wrote:
>>> William schrieb am Dienstag, 3. Mai 2022 um 19:16:34 UTC+2:
>>>> On Tuesday, May 3, 2022 at 10:47:56 AM UTC-3, WM wrote:
>>>>> William schrieb am Montag, 2. Mai 2022 um 23:38:48 UTC+2:
>>>>>> On Monday, May 2, 2022 at 5:15:01 PM UTC-3, WM wrote:
>>>>>>> William schrieb am Montag, 2. Mai 2022 um 19:29:53 UTC+2:
>>>>>>>> On Monday, May 2, 2022 at 9:11:30 AM UTC-3, WM wrote:
>>>>>>>>
>>>>>>>>> gap
>>>>>>>>
>>>>>>>> Nope, There is no "gap".
>>>>>>> If we have a complete set of natnumbers n all of which are smaller than the limit omega, then we can calculate the difference omega - n.
>>>>>>>
>>>>>> Piffle. The "difference" is a set difference not a distance (metric).
>>>>>>
>>>>> The difference either consists of natural numbers or it is a gap.
>>>> Nope, The fact that there is no natural number between the *set* N_p and ω does not mean there is a "gap" between the set N_p and ω.
>>> Then there is no gap. Then there exists an element of N_p touching ω.
>> Piffle. You keep using words like "gap" and "touching" that only make sense if there is a distance.
>
> There is a distance, if omega and omega + 1 and 2*omega etc. are existing.

A distance is a measure of length between two points.

what is your second point, and how are you measuring distance ?

>
>> There is no distance. There is a "difference" but it is a set difference not a distance.
>
> A difference in arithmetic is called a distance in geometry.

No. One is apples, the other is oranges. Google Vector Fields

> There is an ordinal number sequece

where ? which one ? or do you mean any sequence, {i,1,-0.3,17,n^n,} ?

> and thereferefor an ordinlal numbers axis.

Your words fail you.

> Before omega there is either nothing, i.e., a gap, or there is something. But this cannot be defined.

False Dichotomy

https://en.wikipedia.org/wiki/False_dilemma

>
> Regards, WM

keep looking in your gap, I am sure the dark ants will come out of it.

William schrieb am Mittwoch, 4. Mai 2022 um 17:45:14 UTC+2:
> On Wednesday, May 4, 2022 at 10:53:25 AM UTC-3, WM wrote:
>
>
> > Before omega there is either nothing, i.e., a gap,
> Nope, the word "gap" is only meaningful if there is a distance. The difference between omega and any element of N_p is a set difference, not a distance.

That would be the case if only the statement "omega is different from every natnumber" were valid. But there is another statement: omega is larger than every natnumber. That is expressed in geometry by the ordinal line.

> There is no largest ordinal that is smaller than omega.

The definable ordinals are potentially infinite. There is no largest.

> There is no "last" ordinal before omega'

The dark ordinals are not fixable.

Regards, WM

On Wednesday, May 4, 2022 at 1:00:22 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 4. Mai 2022 um 17:45:14 UTC+2:

> > the word "gap" is only meaningful if there is a distance. The difference between omega and any element of N_p is a set difference, not a distance.
> Wrong. "The system Omega in its natural order constitutes a 'sequence'." [Cantor, p. 445]

Indeed. So what? The difference between two elements of a sequence may or may not be a distance. The difference between omega and any element of N_p is a set difference not a distance.

--
William Hughes

On Wednesday, May 4, 2022 at 1:11:40 PM UTC-3, WM wrote:

> potentially infinite.

Piffle.

--
William Hughes

On 5/4/2022 9:58 AM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 4. Mai 2022 um 00:01:41 UTC+2:
>> On 5/3/2022 4:45 PM, WM wrote:

>>> Then explain please how the X's in
>>>
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> XOOO...
>>> ...
>>>
>>> can be shuffled to cover all positions of the matrix.
>>
>> If there was some β for which
>> each non-empty sub-collection of things =< β
>> contains first and last things,
>>
>> then that observation would lead to contradictions.
>
> The observation that the X canot cover all the matrix is

.... incorrect.

Consider only
first-column X's at j/1 such that
each non-empty sub-collection of things =< j
contains first and last things,

and only
whole-matrix X's and O's at p/q such that
each non-empty sub-collection of things =< p and
each non-empty sub-collection of things =< q
contains first and last things.

We know that,
for each j,
there is one and only one pair p,q such that
j = (p+q-1)*(p+q-2)/2 + p

It is the pair p,q such that
p+q = ceiling((sqrt(8*j+1)+1)/2)
p = j - (p+q-1)*(p+q-2)/2
q = (p+q) - p

We know that,
for each pair p,q,
there is one and only one j such that
j = (p+q-1)*(p+q-2)/2 + p

We've gone over how we know this, and
you've accepted that much.
You simply reject that
the X's covering all the matrix
counts as
the X's covering all the matrix.

> The observation that the X canot cover all the matrix is
> 1) simply a geometrical one and

You do not type all the X's.
Less than all the X's do not cover all the matrix.
No one has said otherwise.

All the X's cover all the matrix.

> 2) provable in case of Cantors enumeration.
>
> It is in contradiction with Cantors claim that
> the X could cover all the matrix.

j = (p+q-1)*(p+q-2)/2 + p