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

On 09.01.2025 20:46, Jim Burns wrote:

On 1/9/2025 1:25 PM, WM wrote:

On 09.01.2025 18:52, Jim Burns wrote:

On 1/8/2025 9:31 AM, WM wrote:

 

(Losing all numbers but
keeping infinitely many
can only be possible if
new numbers are acquired.)

>
No.

>
Losing all numbers but keeping infinitely many
is impossible in inclusion-monotonic sequences.
>

Sets do not change.

>
But the terms (E(n))
differ from their successors by one number.

 Each end.segment is larger than
any ordinal smaller.than fuller.by.one sets.

What makes it so?

ℕ is the set of such ordinals.

Do all elements of ℕ leave the endsegments?

Finite means like a natural number.

 ...which doesn't define 'natural number'.

In potential infinity, natural numbers are the basis of mathematics, mimiced by Peano. No definition necessary. In actual infinity natural numbers are positive integers smaller than ω.

Do you (WM) disagree with
'finite' meaning
'smaller.than fuller.by.one sets'?

That is also true for infinite sets.
Regards, WM