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?

Why couldn't the domain below omega absorb it? That domain is INFINITE, and thus has room for them.
No even numbers were "newly created" because they all existed before when the actual infinity was made.
Of course, you can't make that actual infinity one at a time, as that would take forever, and that is what you logic needs to do, so it first needs to speed that eternity building the set of Natural Numbers, so it can never get to the second step of trying to double them all.

 

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.

Right, and *EVERY* Natural Number has another Natural Number that is twice it, so no new Natural Numbers need to be created. You are just stuck having failed to finish the infinite task of creating the Natural Numbers, so of course you fail at the second.

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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.

And for an infinite cardinality, twice it is still the same size so there is room for them.
Your problem is you keep on thinking with rules that only work for finite numbers, because you are too dumb to understand the infinite, and too dumb to understand that you don't understand.

 Any questions about mathematics?
 Regards, WM