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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
Date: Sun, 19 Jan 2025 08:29:30 -0500
Organization: Peripheral Visions
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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".
>>>
>>> Nevertheless it uses potential infinity.
>> 
>> No, it doesn't.
>
> 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.