William schrieb am Mittwoch, 4. Mai 2022 um 22:00:54 UTC+2:
> On Wednesday, May 4, 2022 at 4:16:32 PM UTC-3, WM wrote:
> >
> > Either there is something immediately before omega or there is nothing.
> There is no position "immediately before omega" so it makes no sense to ask if there is something there.

omega is a point. The next natnumber is in infinite distance. What is in between? If there is nothing in between, then the distance to every n cannot be infinite. It is deplorable how confusing matheology acts upon brains.

Regards, WM

Fritz Feldhase schrieb am Mittwoch, 4. Mai 2022 um 21:57:45 UTC+2:
> On Wednesday, May 4, 2022 at 9:16:32 PM UTC+2, WM wrote:
>
> > Either there is something immediately before omega or there is nothing immediately before omega.
> There is nothing immediately before omega.

That is called a gap.

Regards, WM

Jim Burns schrieb am Mittwoch, 4. Mai 2022 um 21:59:00 UTC+2:
> On 5/4/2022 3:14 PM, WM wrote:
> > Jim Burns schrieb
> > am Mittwoch, 4. Mai 2022 um 19:39:10 UTC+2:
> >> On 5/4/2022 9:53 AM, WM wrote:
>
> >>> Before omega there is either nothing, i.e., a gap,
> >>> or there is something. But this cannot be defined.
> >>
> >> For each thing before omega,
> >> there is room for at least one more before omega.
> >> Therefore, nothing before omega is
> >> immediately before omega.
> >
> > Then nothing is immediately before omega.
> > That is called a gap.
> In discussion A,
> each non-empty collection of things
> contains a first thing.

In discussion logic and mathematics, there is either something before the point omega or there is nothing. The latter is called a gap.
>
> Nothing in discussion A is in that "gap".

Then drop that kind of discussion.

Regards, WM

horand....@gmail.com schrieb am Mittwoch, 4. Mai 2022 um 21:59:06 UTC+2:
> On Wednesday, 4 May 2022 at 16:16:32 UTC-3, WM wrote:
> [...]
> > Either there is something immediately before omega or there is nothing. These are the two alternatives.
> You are going gangbusters today. Congratulations on this epiphany. The operative word, of course, is "immediately".

Say in finite distance.

Regards, WM

On 5/4/2022 4:16 PM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 4. Mai 2022 um 21:59:00 UTC+2:
>> On 5/4/2022 3:14 PM, WM wrote:
>>> Jim Burns schrieb
>>> am Mittwoch, 4. Mai 2022 um 19:39:10 UTC+2:
>>>> On 5/4/2022 9:53 AM, WM wrote:

>>>>> Before omega there is either nothing, i.e., a gap,
>>>>> or there is something. But this cannot be defined.
>>>>
>>>> For each thing before omega,
>>>> there is room for at least one more before omega.
>>>> Therefore, nothing before omega is
>>>> immediately before omega.
>>>
>>> Then nothing is immediately before omega.
>>> That is called a gap.
>>
>> In discussion A,
>> each non-empty collection of things
>> contains a first thing.
>
> In discussion logic and mathematics,
> there is either something before the point omega
> or there is nothing.

For each thing before omega,
something else is between it and omega.

There is nothing _immediately_ before omega.

> The latter is called a gap.

Whatever you call it,
there is nothing _immediately_ before omega.

"Nothing" includes dark numbers among
things which aren't immediately before omega.

>> Nothing in discussion A is in that "gap".
>
> Then drop that kind of discussion.

What does it say about your "argument"
that you (WM) need to _change what we say_
in order to "win"?

In discussion A,
each non-empty collection of things
contains a first thing.

j is _definable_ in discussion A iff
each non-empty sub-collection of things =< j
contains first and last things,

In discussion B,
each thing is definable in discussion A.

omega is defined to be
the first thing in discussion A which
is not in discussion B.

( Also,
( each thing in discussion A is followed by
( another thing in discussion A.

On 5/4/2022 11:00 AM, WM wrote:
> 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, ...

here you have assigned ω to be a natural number, just like n, fail.

>
>> 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]

Wrong. red herring, That statement does not support your math at all.

>
> Regards, WM
>

you are a Deceiver.

On 5/4/2022 1:59 PM, WM wrote:
> William schrieb am Mittwoch, 4. Mai 2022 um 19:55:02 UTC+2:
>> On Wednesday, May 4, 2022 at 2:08:16 PM UTC-3, WM wrote:
>>> William schrieb am Mittwoch, 4. Mai 2022 um 18:38:36 UTC+2:
>>>> 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 two ordinal numbers is a distance.
> omega lies on the ordinal line as zero does.
>
>> The difference between omega and any element of N_p is a set difference not a distance.
>
> That is your personal and inconsistent definition.
>
> Es fragt sich, in welchem Abstande von gamma dieser Gigant delta liegt
> (Abstand = Distance)
> 1886, 11. Oct. Cantor, letter to Goldscheider
>
>>> It is an infinite number, in geometry, an infinite distance.
>> Nope. Your calling it a distance does not make it so.
>
> An idea has no distance from a melody, but |a - b| is a distance for all ordinals a and b.

Phantasie

>
>> You can call a set difference an "infinite distance" but that does not make it a distance.
>
> That is your personal and inconsistent definition, by the way it is contradicting Cantor:
>
> Es fragt sich, in welchem Abstande von gamma dieser Gigant delta liegt
> (Abstand = Distance)
> 11. Oct 1886, Cantor, letter to Goldscheider

another quote from the Troll Book.

>
> Regards, WM