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

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".

Afterwards it contains further elements. Same can be accomplished by
multiplying all natural numbers by 2.

There is no natural n such that 2*n is not natural.

That it true for every definable number and proves potential infinity.
It contradicts actual infinity. If ***all*** natural numbers are
doubled, then you get more than you started with or you did not start
with ***all*** natural numbers.

Bzzt wrong. It proves it's true for infinitely many numbers,
contradicting a finite size. If you double every single one, you are
using inf.many and not more (not even with finitely many. What would
the preimage be?).

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.