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

On 20.01.2025 15:34, Jim Burns wrote:

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

⎜ 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.

Sets like ℕ and ℚ are infinite and countable!

"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"?

The reason is that the infinite set follows beyond:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

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.

You cannot see it. But there are more things in heaven and earth than you can see.

 

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

But there are infinite sets of different size. |ℚ| = 2|ℕ|^2 + 1.
Regards, WM