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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The reality of sets, on a scale of 1 to 10 [Was: The
 non-existence of "dark numbers"]
Date: Wed, 2 Apr 2025 16:14:09 +0200
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On 27.03.2025 22:45, joes wrote:
> Am Thu, 27 Mar 2025 21:11:00 +0100 schrieb WM:
>> Am 26.03.2025 um 23:18 schrieb Moebius:
>>> Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:
>>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>
>>>>> The number of [...] rooms in Hilbert's hotel is infinite but [...]
>>>>> grow[s].
>>> No, it doesn't.
>> And the number of guests?
> Depends on how you look at it. The hotel is always "full" in that all
> rooms are occupied; yet it can always accomodate more guests (but not
> uncountably many). There are "exactly" as many rooms as guests at any
> point, so far as that makes sense for infinities.

That means not fixed infinity. The alternative is potential infinity.
> 
>> Real fools like really counterintuitive "results".
> The whole point of Hilbert's Hotel is illustrating the counterintuitivity
> of infinite cardinalities.

It illustrates the confusion between potential and actual infinity.

"so daß jedes Element der Menge an einer bestimmten Stelle dieser Reihe 
steht" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen 
mathematischen und philosophischen Inhalts", Springer, Berlin (1932) S. 152]

If a new element enters, then the old order must be given up. A new 
number is required.

Regards, WM