Jim Burns schrieb am Freitag, 6. Mai 2022 um 15:19:37 UTC+2:
> On 5/6/2022 8:34 AM, WM wrote:
>
> > In the sequence of natural numbers there cannot exist
> > two consecutive actually infinite sets. Therefore
> ...invalidly swap quantifiers here and...

There is no swap and nothing invalid, except perhaps in matheology.

> > the set of elements followed by ℵo numbers cannot be
> > a set of ℵo numbers itself.

Why should the infinite distance omega - n = omega vanish for all n?

This is the most absurd statement of matheology:

Every n has infinite distance from omega.
The distance between all n and omega is zero.

Regards, WM

Jim Burns schrieb am Freitag, 6. Mai 2022 um 15:14:56 UTC+2:
> On 5/6/2022 8:36 AM, WM wrote:

> Ordinals are the things for which
> each non-empty collection contains a first.

Wrong for dark ordinals.

Every bright natnumber is infinitely far from omega.
All bright natnumbers together are infinitely far from omega.
If there is not an infinite gap, then the space is filled with dark numbers - without a first one.

Regards, WM

On 5/6/2022 8:45 AM, WM wrote:
> horand....@gmail.com schrieb am Freitag, 6. Mai 2022 um 15:30:06 UTC+2:
>> On Friday, 6 May 2022 at 09:39:31 UTC-3, WM wrote:
>>> horand....@gmail.com schrieb am Freitag, 6. Mai 2022 um 12:22:13 UTC+2:
>>>> |1/n - 1/m| is a perfectly good distance measure for natural numbers.
>>> It allows to calculate the distance, but it is not the distance. The distance between 4 and 7 is 3 because 4 + 3 = 7.
>> That is *ONE* way to compute and define a distance,
>
> No, that IS the distance.
>
> Regards, WM

No, distance has units. Your's does not.

You have subtraction.

On 5/6/2022 9:50 AM, WM wrote:
> Jim Burns schrieb
> am Freitag, 6. Mai 2022 um 15:19:37 UTC+2:
>> On 5/6/2022 8:34 AM, WM wrote:

>>> In the sequence of natural numbers there cannot exist
>>> two consecutive actually infinite sets. Therefore
>>
>> ...invalidly swap quantifiers here and...
>
> There is no swap and nothing invalid,
> except perhaps in matheology.

(i)
For each natural j of infinitely-many,
there are infinitely-many naturals which are after j.

....invalidly swap quantifiers here and...

(ii)
There are infinitely-many naturals which,
for each natural j of infinitely-many,
are after j.

"Valid" is a technical term which is *NOT*
analogous to "valid driver's license".
There is no authority responsible for declaring
quantifier shifts invalid.

A _quantifier shift_ does not *always* end at a true
claim when it begins at a true claim. That is what
makes a _quantifier shift_ invalid.

Your justification for using a _quantifier shift_
in this case is that it *sometimes always* ends at a
true claim when it begins at a true claim.
"Sometimes always" is not a thing.
Your dark numbers are not justified.

On 5/6/2022 7:34 AM, WM wrote:
> William schrieb am Donnerstag, 5. Mai 2022 um 22:13:10 UTC+2:
>
>> Nope. Cardinality is an equivalence class of sets is not a metric. "ω - n = ω" is shorthand for the set ({1,2,3,...} \ {1,2,3...,n}) has cardinality aleph_0.
>
> And that is shorthand for: the set of natural numbers which can be inserted at position n is finite (because the set of dark numbers which cannot be inserted at position n is actually infinite).

QUacK !! Wrong!

({1,2,3,...} \ {1,2,3...,n}) is called an *Endsegment*.

no dark insertions required

>
> In the sequence of natural numbers there cannot exit two consecutive actually infinite sets.

Wrong. ω = ω + ω

Go buy another set of natural numbers offa eBay, only $1.28

> Therefore the set of elements followed by ℵo numbers cannot be a set of ℵo numbers itself.

Wrong, QUACK !!
ω = ω + ω
>
> Regards, WM

On 5/6/2022 8:50 AM, WM wrote:
> Jim Burns schrieb am Freitag, 6. Mai 2022 um 15:19:37 UTC+2:
>> On 5/6/2022 8:34 AM, WM wrote:
>>
>>> In the sequence of natural numbers there cannot exist
>>> two consecutive actually infinite sets. Therefore
>> ...invalidly swap quantifiers here and...
>
> There is no swap and nothing invalid, except perhaps in matheology.
>
>>> the set of elements followed by ℵo numbers cannot be
>>> a set of ℵo numbers itself.
>
> Why should the infinite distance omega - n = omega vanish for all n?

please restate your question.

>
> This is the most absurd statement of matheology:
>
> Every n has infinite distance from omega.
> The distance between all n and omega is zero.

wrong. there are no numbers "next to" omega.

>
> Regards, WM
>

Why are you always wrong ?

you are stuck in your imagination, and cannot understand proofs.

On 5/6/2022 8:55 AM, WM wrote:
> Jim Burns schrieb am Freitag, 6. Mai 2022 um 15:14:56 UTC+2:
>> On 5/6/2022 8:36 AM, WM wrote:
>
>> Ordinals are the things for which
>> each non-empty collection contains a first.
>
> Wrong for dark ordinals.
>
> Every bright natnumber is infinitely far from omega.
> All bright natnumbers together are infinitely far from omega.
> If there is not an infinite gap, then the space is filled with dark numbers - without a first one.
>
> Regards, WM

you are out of math again.

in Math bright is meaningless, so is dark, dark numbers, gap, dark ordinals...

can you define these in terms of mathematics ?

sergio schrieb am Freitag, 6. Mai 2022 um 16:33:05 UTC+2:
> On 5/6/2022 8:45 AM, WM wrote:

> > No, that IS the distance.
> >
> No, distance has units.
>
Yes, in natnumbers the unit is 1, the unit.

Regards, WM