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

On 10.01.2025 13:41, Richard Damon wrote:

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

Fine. Then the set of natural numbers is completed. Multiply all natural numbers by 2. The set of even numbers then doubles. The domain below ω is unable to absorb new numbers. What happens to the newly created even numbers?

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.

 

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they are the smallest infinite set.

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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. But mathematics. Every set {1, 2, 3, 4, 5, ..., n} contains roughly twice the even numbers. This holds for all n. More are not available. Hence it holds for the infinite set.
Any questions about mathematics?
Regards, WM