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

On 1/10/25 12:33 PM, WM wrote:

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.

So? We can know any of the eleements of the set, and none got added, as we can see that the result of multiplying any element of the set will result in another value that was always a member of the set.

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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.

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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.

It can't "distinguish" there size, as there isn't an actual difference is size, since there *IS* a one-to-one mapping between the two, which is the DEFINITION of "same size".
It can distinguish the sets, The sets are different but the same size. You are just tripping up by the INCORRECT assumption of your naive mathematics that a proper subset needs to have less members then its super set. That property only applies to FINITE sets, not infinitie sets. When you start from error, you get error.
Note, in your examples, you use finite sets, but then try to generalize to the infinite set, but fail, because you logic doesn't work for that set.

 Any questions about mathematics?

I wouldn't ask you, because you clearly don't know how mathematics works.
Msybe if I had a question about your naive mathematics, but who cares about broken systems.

 Regards, WM