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

Am Sun, 19 Jan 2025 11:59:47 +0100 schrieb WM:

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".

Nevertheless it uses potential infinity.

No, it doesn't.

Use all natnumbers individually such that none remains. Fail.

Set theory doesn't use "potential infinity".

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

Quite the opposite.

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.

The "complete count" is infinite.

All natnumbers in bijections have ℵ₀ not applied successors.

A bijection is not meant to be thought about sequentially?

∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo Only potential infinity is
applied.

This is not infinity.

In actual infinity all natnumbers would be applied:
ℕ \ {1, 2, 3, ...} = { }
But that is not possible in bijections.

It absolutely is. Just give a rule for every natural.

--
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.