Jim Burns schrieb am Donnerstag, 1. Dezember 2022 um 21:01:47 UTC+1:

Useless stuff.

I have reported your last useful statement in de.sci.mathematik as the seventh counter-argument:

7) Bob does not step out of the matrix. Insisting that Bob either stays in the matrix or steps out of the matrix is insisting on Bob-conservation. The set of all exchanges does not have
Bob-conservation. [J. Burns, https://groups.google.com/g/sci.math/c/LHEV4iqs5bM]
Bob {{Abkürzung für das zuerst auf dem Bruch 1/2 liegende O}} tritt nicht aus der Matrix heraus. Darauf zu bestehen, dass Bob entweder in der Matrix bleibt oder aus der Matrix heraustritt, bedeutet auf Bob-Erhaltung zu bestehen. Die Menge aller Vertauschungen hat keine Bob-Erhaltung.

WM: Darauf zu verzichten, dass Bob entweder in der Matrix bleibt oder aus der Matrix heraustritt, bedeutet einen Verzicht auf Logik.

Regards, WM

On 12/2/2022 6:13 AM, WM wrote,
responding to Jim Burns schrieb am Donnerstag,
1. Dezember 2022 um 21:01:47 UTC+1:

> Useless stuff.
>
> I have reported your last useful statement in
> de.sci.mathematik as the seventh counter-argument:
>
> 7) Bob does not step out of the matrix.
> Insisting that Bob either stays in the matrix
> or steps out of the matrix is insisting on
> Bob-conservation. The set of all exchanges
> does not have Bob-conservation.
> [J. Burns,
> https://groups.google.com/g/sci.math/c/LHEV4iqs5bM]
>
> Bob
> {{Abkürzung für das zuerst auf dem Bruch 1/2
> liegende O}}
> tritt nicht aus der Matrix heraus.
> Darauf zu bestehen, dass Bob entweder
> in der Matrix bleibt oder
> aus der Matrix heraustritt,
> bedeutet auf Bob-Erhaltung zu bestehen.
> Die Menge aller Vertauschungen hat
> keine Bob-Erhaltung.

Okay.

> WM:
> Darauf zu verzichten,
> dass Bob entweder in der Matrix bleibt
> oder aus der Matrix heraustritt,
> bedeutet einen Verzicht auf Logik.

<WM>
| To forgo Bob either staying in the Matrix
| or stepping out of the Matrix is
| to forego logic.
| | https://translate.google.com

There is a more-common, less-amusing term for
"Bob-conservation". I have reasons for not using it.

A set (collection) has _Bob-conservation_
if and only if
there is no match of its elements to
a proper (unequal) subset or to a proper
superset.

A FISON
-- Finite Initial Segment Of Naturals --
is a totally-ordered set (collection) which
begins at 0
ends somewhere
and for which, for each split,
there is some i last-before that split
with its successor i⁺⁺ first-after that split.

| ∀i : i⁺⁺ ≠ 0
| ∀i, ∀j ≠ i : i⁺⁺ ≠ j⁺⁺

For each FISON, there is a unique FISON-end.
For each FISON-end, there is a unique FISON.

Because of that match of FISON to FISON-ends,
FISON-ends can be used to represent FISONs, and
FISONs can be used to represent FISON-ends.

Assuming that, by "dark number", you mean
something not in any FISON,
each element of the set of all FISON-ends
is not a dark number.

FISON-ends are what are usually meant by
"natural number" but (we see) FISONs could serve
as representations of natural numbers as well.

<JB>
> Insisting that Bob either stays in the matrix
> or steps out of the matrix is insisting on
> Bob-conservation. The set of all exchanges
> does not have Bob-conservation.

Each FISON has Bob-conservation.
It has no match to any other FISON.

Each set which matches some FISON has
Bob-conservation.
It has no match to any proper subset or
proper superset.
It has no match to any other FISON.

However,
the set of all FISONs and
the set of all FISON-ends
have matches to proper subsets and
to proper supersets.

There can be matches between a set with Bob
and a proper subset, a set without Bob.
These sets do not have Bob-conservation.

Also, recalll that
these sets have no dark numbers.
Therefore,
"their dark numbers" is not the reason
that they do not have Bob-conservation.

Jim Burns schrieb am Freitag, 2. Dezember 2022 um 20:41:35 UTC+1:

> There can be matches between a set with Bob
> and a proper subset, a set without Bob.

This is wrong. If an identity mapping between a set and itself exists, then every transposition preserves every element. Where should be the first deletion?

Regards, WM

On 12/3/2022 4:26 AM, WM wrote:
> Jim Burns schrieb am Freitag, 2. Dezember 2022 um 20:41:35 UTC+1:
>
>> There can be matches between a set with Bob
>> and a proper subset, a set without Bob.
>
> This is wrong. If an identity mapping between a set and itself exists, then every transposition preserves every element. Where should be the first deletion?

Bob deletes WM. Done.

>
> Regards, WM
>
>
>

On 11/16/2022 3:26 AM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 16. November 2022 um 00:33:11 UTC+1:
>> On Tuesday, November 15, 2022 at 3:40:36 PM UTC+1, Sergi o wrote:
>>
>>> you use the word "define" or "definable" however you mean "finite" which is wrong.
>>
>> Indeed! Whenever he talks about "definable XXX" he means _finitely many_ XXX.
>
> Of course. Nobody can define infinitely many elements individually.
>>
>> And his darkies are "all the others" (which "makes" a set infinite).
>
> Of course. Defined numbers will never make a set infinite.
>
> Regards, WM

Of course. Anybody can define infinitely many elements individually.

Defined numbers will always make a set infinite.