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

On 12/16/2024 03:56 AM, joes wrote:

Am Mon, 16 Dec 2024 09:27:19 +0100 schrieb WM:

On 15.12.2024 21:20, joes wrote:
>

Duh. All naturals are finite. You need to actually remove all inf.many
of them.

That is not possible with definable naturals:

Duh again. No natural is infinite.
>

And numbers which succeed ∀k ∈ ℕ

There is no natural larger than all others.
>

>
Oh, one of my podcasts next week will have
"natural infinities", because there's no
standard model of integers, only fragments
or extensions, making that there _are_ natural
models with infinite members.
>
It's a simple consequence of comprehension
and quantification, in theories that don't
define it away.
>
How could there be a natural integer smaller
than all others? Of course you may know
that "1" is a very large cardinal, then
as with regards to whether "0" exists at all,
whether it's finite, and otherwise via
comprehension and quantification,
"natural zeros".