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

WM formulated the question :

On 19.01.2025 11:42, FromTheRafters wrote:

WM presented the following explanation :

On 18.01.2025 12:03, joes wrote:

Am Fri, 17 Jan 2025 22:56:13 +0100 schrieb WM:

>

Correct. If infinity is potential. set theory is wrong.

And that is why set theory doesn't talk about "potential infinity".

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Nevertheless it uses potential infinity.

 No, it doesn't.

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Use all natnumbers individually such that none remains. Fail.

This makes no sense.

All "bijections" yield the same cardinality because only the potentially infinite parts of the sets are  applied.

 No, it is because these bijections show that some infinite sets' sizes can be shown to be equal even if no completed count exists.

>
They appear equal because no completed count exists.

No, they are the same size when it is shown there is at least one bijection. Still, no counting necessary.

All natnumbers in bijections have ℵ₀ not applied successors.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
Only potential infinity is applied.

You mean that only finite sets are involved.

In actual infinity all natnumbers would be applied:
ℕ \ {1, 2, 3, ...} = { }

That is simply 'infinity' which means not finite.

But that is not possible in bijections.

Sure it is, when they are infinite.