Re: Simple enough for every reader?

Am Fri, 04 Jul 2025 14:23:57 +0200 schrieb WM:

On 04.07.2025 13:18, joes wrote:

Am Fri, 04 Jul 2025 12:15:16 +0200 schrieb WM:

On 04.07.2025 10:38, joes wrote:

Am Thu, 03 Jul 2025 21:10:39 +0200 schrieb WM:

On 03.07.2025 16:12, joes wrote:

Am Thu, 03 Jul 2025 15:08:25 +0200 schrieb WM:

>

The rule of subset proves that every proper subset has fewer
elements

No such rule for infinite sets.

For all sets.

No, this is not an accepted theorem of mainstream mathematics.

But it is correct. See the last paragraph. Can you contradict it?

It is incorrect: infinite sets have subsets of the same cardinality.

But cardinality is nonsense because mathematics contradicts it.

Cardinality IS mathematics.

No, you can number the primes.

Yes, there are only few known primes.

...you can enumerate them in the sense of a bijection to N: there is
a first prime, a second and so on for every natural, which is
infinitely many.

I mentioned the *known* primes.

I didn't, and you didn't the first time.

I said: Yes. If all prime numbers could be known, the same contradiction
would arise.

All primes can be known and there is no contradiction.

You: Primes are not infinite themselves.
Me: But the known primes are a potentially infinite collection.

Yeah, that we know only finitely many primes has nothing to do with that.

I am aware that we don't know all primes, but you can't mean that.
There aren't more naturals than primes,
one doesn't run out of primes when counting them.

That shows that this "bijection" does not capture all natural numbers.

Huh? There is a prime to every natural.

And half of infinity is still infinite.

That does not change the correct mathematics: For every interval
[0, 2n] there are half as many even numbers as natural numbers.
That is true also in the limit.

That is what I said.

No. You said above "There aren't more naturals than primes".

Yes, Aleph_0 of both.

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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.