Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 1/19/2025 5:52 AM, WM wrote:

On 18.01.2025 19:38, Jim Burns wrote:

On 1/18/2025 3:41 AM, WM wrote:

On 18.01.2025 00:08, Jim Burns wrote:

The finite extends
much further than you (WM) think it does.
Infinitely further than you think it does.

>
No.
As long as
you deny Bob's existence and violate logic
you are not a reliable source.

>
https://en.wikipedia.org/wiki/Finite_set
⎛
⎜ Informally, a finite set is a set which
⎜ one could in principle count and finish counting.

Cantor claims this also for infinite sets:

The finite extends
much further than you (WM) think it does.
Infinitely further than you think it does.
⎜ 1.
⎜ S is a finite set.
⎜ That is,
⎜ S can be placed into
⎜ a one-to-one correspondence with
⎜ the set of those natural numbers less than
⎜ some specific natural number.

"The infinite sequence thus defined has
the peculiar property to contain
the positive rational numbers completely, and
each of them only once
at a determined place."
[G. Cantor, letter to R. Lipschcitz c(19 Nov 1883)]
>

There is no step from finite to infinite.

>
Not in the visible domain.

How do we know "not in visible domain"?
Because there is an argument.
That same argument does not
distinguish visibleᵂᴹ and darkᵂᴹ.
Yes,
there is no step from finite to infinite
in the visibleᵂᴹ domain.
As well,
there is no step from finite to infinite
in the visibleᵂᴹ.or.darkᵂᴹ domain.
There is no step from finite to infinite
anywhere.

But there is no loss in lossless exchange - even in the dark domain.

In the visibleᵂᴹ.or.darkᵂᴹ domain:
⎛
⎜ There are no sets A ≠ Aᵃ and B ≠ Bᵇ such that
⎜⎛ A is smaller than B but
⎜⎝ Aᵃ is not.smaller than Bᵇ
⎜
⎜ For readability, I have written
⎜ Aᵃ for A∪{a}, Bᵇ for B∪{b}
⎜
⎜ Let B = Aᵃ
⎜
⎜ There are no sets A ≠ Aᵃ ≠ (Aᵃ)ᵇ such that
⎜⎛ A is smaller than Aᵃ but
⎜⎝ Aᵃ is not smaller than (Aᵃ)ᵇ
⎜
⎜ The finite extends
⎜ much further than you (WM) think it does.
⎜ Infinitely further than you think it does.
⎜
⎜ For sets A ≠ Aᵃ
⎜ if A is smaller than Aᵃ
⎜ then A is finite.
⎜
⎜ There are no sets A ≠ Aᵃ such that
⎜⎛ A is finite but
⎜⎝ Aᵃ is not finite.
⎜
⎜ There is no step from finite to infinite.
⎜
⎝ In the visibleᵂᴹ.or.darkᵂᴹ domain.

There lies your fault.