Re: The non-existence of "dark numbers"

On 3/17/2025 12:11 PM, WM wrote:

On 17.03.2025 05:23, Jim Burns wrote:

Do you (WM) know what 'finite' means?

In mathematics, particularly set theory,
a finite set is a set that has a finite number of elements.
Informally, a finite set is a set which
one could in principle count
and finish counting.
(Wikipedia)

 From the same page:
⎛ Any proper subset of a finite set S is finite
⎜ and has fewer elements than S itself.
⎜ ...
⎝ The union of two finite sets is finite,
A set which is finite in Wikipedia's sense
has fuller.by.one sets which are larger and finite
and emptier.by.one sets which are smaller and finite.
The set of all finite.sizes does not have
a finite.size.
⎛ Assume otherwise.
⎜ Assume 𝔊 is the finite.size of
⎜ the set of finite.sizes.
⎜
⎜ 𝔊+1 is also a finite.size.
⎜ The subset of finite.sizes ≤ 𝔊+1
⎜ is larger than 𝔊 and
⎜ is larger than its superset:
⎜ the set of all finite.sizes.
⎜
⎜ However,
⎜ a subset isn't larger than its superset.
⎝ Contradiction.
Therefore,
the set of all finite.sizes does not have
a finite.size.
If
the set of all finite.sizes
has fuller.by.one sets which are larger and finite
and emptier.by.one sets which are smaller and finite,
then
the set of all finite.sizes
has a finite.size in
the set of all finite.sizes.
It doesn't have a finite.size.
Therefore, the set of all finite.sizes
has fuller.by.one sets which are NOT larger
and emptier.by.one sets which are NOT smaller.

No swap exists immediately before all the swaps.
No visibleᵂᴹ and no darkᵂᴹ.

>
Therefore:
If Bob disappears, it happens at a finite index.

The set of all finite.sizes is same.sized as
the set of all finite sizes and Bob.
No single swap changes the size of the set.
No finite set of swaps changes the size of the set.
Also,
no infinite set of swaps changes the size of the set,
although,
(only) in the case of infinite sets,
sufficient swaps can change a set to an emptier set,
but only to an emptier set which is not smaller.

Name it or confess that you have used a foolish idea.

Do you agree with Wikipedia?
Do you agree that
the set of all finite.sizes has
has fuller.by.one sets which are NOT larger
and emptier.by.one sets which are NOT smaller ?