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

Am Fri, 10 Jan 2025 18:33:31 +0100 schrieb WM:

On 10.01.2025 13:41, Richard Damon wrote:
 

"Nubmers" or "Sets" don't evolve.

They either are members, or not.

Fine. Then the set of natural numbers is completed. Multiply all natural
numbers by 2. The set of even numbers then doubles.

Very wrong. The set {2*k for k e N} = G = 2N = {2*1, 2*2, 2*3, ...} =
{2, 4, 6, ...} is a proper subset of N. I have no idea what you think.

The domain below ω
is unable to absorb new numbers. What happens to the newly created even
numbers?

None are "created". The multiples of 4 are ALSO in the original set.

You seem to THINK that sets,
particularly "potentially infinite" set "evolve" in that numbers get
added to them as you move along the generator, but the set doesn't
change, only our knowledge of the set.

The multiplication above concerns the set, not only the numbers we know.

Then don't talk about "creation".

they are the smallest infinite set.

The set of prime numbers is infinite but smaller because it is a
proper subset. It has less than 1 % content.

It may SEEM smaller, but it turns out it is of the same countable
infinite cardinality.

It turns out that countable cardinality is not able to distinguish the
sets* of natural numbers and of even numbers.

*sizes of the sets

But mathematics. Every set
{1, 2, 3, 4, 5, ..., n} contains roughly twice the even numbers.

Could you formalise what you mean here?

This holds for all n.  Hence it holds for the infinite set.

No. The limit of the ratio is different from the ratio of the limit.

--
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.