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

On 10.01.2025 22:58, joes wrote:

Am Fri, 10 Jan 2025 22:30:52 +0100 schrieb WM:

On 10.01.2025 21:03, joes wrote:

Am Fri, 10 Jan 2025 20:22:44 +0100 schrieb WM:

>

If actual infinity is invariable, then we can take the set ℕ and
consider it as invariable. When we add further elements, it grows.

The set N u {w}, for example, is not equal to N. You can't call that
invariable.

Every set in ZF is invariable.

Right. N stays N even when you construct a new set N u {k} (which is
a different set only if k !e N}. So N doesn't "grow".

That contradicts the possibility of bijections, because if it has been finished, you can add further elements and violate completeness. But since you cannot comprehend this, look at the simpler theorem:
Every union of FISONs which stay below a certain threshold stays below that threshold.
I never met a student who doubted this. And I am sure I will never meet a contrary opinion in a sober brain.
Regards, WM