On 5/5/2022 6:36 AM, WM wrote:
> zelos...@gmail.com schrieb am Donnerstag, 5. Mai 2022 um 07:07:29 UTC+2:
>> onsdag 4 maj 2022 kl. 15:59:46 UTC+2 skrev WM:
>>> zelos...@gmail.com schrieb am Mittwoch, 4. Mai 2022 um 06:54:32 UTC+2:
>>>> tisdag 3 maj 2022 kl. 15:50:05 UTC+2 skrev WM:
>>>
>>>>> The difference ω - n = ω for every definablePeano n. Therefore the dark numbers are required.
>>>> There is no such implication because YOU CANNOT EVEN FUCKING DEFINE WHAT A DARK NUMBER IS!
>>> Either there is nothing immediately before omega, or it is dark.
>>
>> Omega is an ordinal . THere is no "Before" it, Omega-1 is meaningless.
>
> omega is a point on the extended real axis.

Not on a linear scale, try Riemann Sphere,

Oh, you dont believe in imaginary numbers, or mapping fields...

The points there have distances. In arithmetic we have "a is smaller than b, the difference is b - a", in geometry the same is expressed by "a lies left
of b, the distance is b - a. There is no chance to keep the first and to reject the second.
>
> "The question is which distance from gamma this giant delta has." [11. Oct 1886, Cantor, letter to Goldscheider] delta and gamma are very large transfinite ordinals. If you defend Cantor's theory, then accept his rules pleeze. And don't delete the clear evidence against your opinion.

that quote does not apply to any of this topic. are you that desperate ?

>
>> You sitll have failed to define "dark"
>
> Dark is all between omega and the definable natnumbers. That is an infinite amount of numbers since "omega - n = omega" [Cantor].

Liar, Cantor did not say that.

>
> Regards, WM
>

WM wrote:

>> For each thing before omega,
>> something else is between it and omega.
>
> Right. Each definable thing has many successors because of potential
> infinity.
> Each undefinable thing cannot be put in order.

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On 5/5/2022 6:31 AM, WM wrote:
> Chris M. Thomasson schrieb am Donnerstag, 5. Mai 2022 um 01:09:47 UTC+2:
>
>> A dark number is one he has not thought of yet?
>
> Many dark numbers can be made definable as individuals. But most dark numbers cannot be thought of individually.
>
> Regards, WM

How would you know ?

horand....@gmail.com schrieb am Donnerstag, 5. Mai 2022 um 13:44:14 UTC+2:
> On Thursday, 5 May 2022 at 08:29:49 UTC-3, WM wrote:
> [...]
> > omega is a point on the extended real axis. The points there have distances. In arithmetic we have "a is smaller than b, the difference is b - a", in geometry the same is expressed by "a lies left of b, the distance is b - a. There is no chance to keep the first and to reject the second.
> Not true in this generality. Look up "taxicab norm".

True for one-dimensional problems.

Regards, WM

On 5/5/2022 11:45 AM, WM wrote:
> horand....@gmail.com schrieb am Donnerstag, 5. Mai 2022 um 13:44:14 UTC+2:
>> On Thursday, 5 May 2022 at 08:29:49 UTC-3, WM wrote:
>> [...]
>>> omega is a point on the extended real axis. The points there have distances. In arithmetic we have "a is smaller than b, the difference is b - a", in geometry the same is expressed by "a lies left of b, the distance is b - a. There is no chance to keep the first and to reject the second.
>> Not true in this generality. Look up "taxicab norm".
>
> True for one-dimensional problems.
>
> Regards, WM

Use your Cursor function, have it go along the real line from left to right, defining all numbers, until it arrives at omega.

On Thursday, May 5, 2022 at 8:29:49 AM UTC-3, WM wrote:
> William schrieb am Donnerstag, 5. Mai 2022 um 00:55:42 UTC+2:
> > On Wednesday, May 4, 2022 at 5:11:43 PM UTC-3, WM wrote:
> > > 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.
> > Nope. There is no "distance".
> omega is a point on the extended real axis.

There is no "distance" (metric) on the extended real axis for which d(n,m) = |n - m| for n,m elements of N_p. That is because "infinity"
is not a "distance"

-- William Hughes

On 5/5/2022 7:25 AM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 4. Mai 2022 um 22:59:41 UTC+2:
>> On 5/4/2022 4:16 PM, WM wrote:

>>> 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.
>
> Right.
> Each definable thing has many successors
> because of potential infinity.

Each definable thing has many definable successors
because
adding one more to a set with two ends
produces a set with two ends.

Thus,
for each definable j
(each non-empty collection =< j has two ends),

there is definable j+1
(each non-empty collection =< j+1 has two ends).

> Each undefinable thing cannot be put in order.

You've changed what "definable" means.

Some things α in discussion A are _definable_
Each non-empty collection =< α has two ends.

Some things β in discussion A are _not definable_
Some non-empty collection =< β doesn't have two ends.

For each non-empty collection of things in discussion A,
that collection contains a first thing.

Let α and β be two things in discussion A.
α ≠ β
The non-empty collection {α,β} contains a first thing.
α ≠ β -> α < β or β < α

The things in discussion A are in a total order.

>> There is nothing _immediately_ before omega.
>
> How far from omega ist the first definable point
> before omega?

There is no last definable point
because
adding one more to a set with two ends
produces a set with two ends.

> According to Cantor it has distance omega.

I think that it's very unlikely you have Cantor right.
You show us how you change what people say to
what you would prefer arguing against.

Hypothetically, if Cantor wrote of a last definable
point, Cantor would be wrong.

>>> The latter is called a gap.
>>
>> Whatever you call it,
>> there is nothing _immediately_ before omega.
>
> How would you know?

By knowing that omega is defined to be
the first thing in discussion A which
is not in discussion B,

and knowing that each thing in discussion B
is followed by something in discussion B,

which I know because
adding one more to a set with two ends
produces a set with two ends.

>> omega is defined to be
>> the first thing in discussion A which
>> is not in discussion B.
>
> omega is a point on the extended real axis.

omega is the first infinite ordinal.

You have gotten the idea from somewhere that,
if you call _something else_ 'omega',
that something else and the first infinite ordinal
are combined into a single entity.

It doesn't work like that.

We call the first infinite ordinal 'omega', but
even if everyone stopped calling the first infinite
ordinal 'omega', everything true of the first
infinite ordinal remains true of the first infinite
ordinal, only not stated so concisely.