Re: The non-existence of "dark numbers"

Am Thu, 27 Mar 2025 20:50:00 +0100 schrieb WM:

Am 27.03.2025 um 15:58 schrieb joes:

Am Wed, 26 Mar 2025 21:09:03 +0100 schrieb WM:

 

At finite steps, but not in the limit.

Countability is proved without limits. Bijections are one-to-one
without limit.

I have no idea what you are talking about. What can be proved to be
countable without resorting to limits?

Limits deviate from countability.

You are writing way too short. There are countable infinities,
whether you like that word or not.

I didn't say anything about bijections not being one-to-one, nor about
them having limits??? The transformation from the initial to the final
matrix can not be achieved in finitely many steps.

That is true but contradicts Cantor's theory according to which every
pair has a finite index. Every finite index n has only n-1 predecessors.

There's no contradiction. The limit is not a term of the sequence.

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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.