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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Tue, 8 Oct 2024 17:26:43 +0200
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On 08.10.2024 15:24, joes wrote:
> Am Tue, 08 Oct 2024 12:51:03 +0200 schrieb WM:
>> On 07.10.2024 17:18, joes wrote:
>>> Am Mon, 07 Oct 2024 11:08:33 +0200 schrieb WM:
>>>> On 07.10.2024 10:05, joes wrote:
>>>>> Am Mon, 07 Oct 2024 09:41:23 +0200 schrieb WM:
>>>>>> On 06.10.2024 17:48, Alan Mackenzie wrote:
>>>>>> even an unbounded sequence does not get longer when shifted by one
>>>>>> step.
>>>>> Nor does it get shorter, it stays infinite.
>>>> It keeps all its elements but not more.
>>> „More” being a different kind of infinity, namely at least uncountable.
>> Nonsense.
> Only according to your broken concept of cardinality, by which N u {a}
> is „bigger” than N, and N\{1} u {a} can’t even be compared.
> 
>>> All ω+k are equally infinite.
>> Nonsense.
> I believe it is called the order type.
> 
>>>>> Bijection is not about completeness, countability is.
> 
>>>>> Of course stopping after a finite number, which potential infinity
>>>>> seems to mean, is not „complete” in that sense. Hilbert’s Hotel is
>>>>> actually infinite, it already holds infinite guests.
>>>> Name them by all the natural numbers. Then no further guest can
>>>> appear.
>>> It can, if I begin numbering with 2. The cardinality of N\{1} can’t be
>>> finite.
>> Cardinality is nonsense.
> Isn’t N\{1} finite? It has ω-1 elements.

But after the visble natural numbers the dark domain comes, and that is 
what prevents to see the end (which is dark too).
>> That shows my point. Infinite sets can be moved. 0.999...999 moved gives
>> 9.99...9990.
> You have not indicated what this notation means. Where does the zero come
> from?

The last natural index has lost its 9 by shifting to the left-hand side. 
Hence there is nothing remaining.
> 
>> Another point is this: [0, 1) moved gives (0, 1].
> Can you generalise this?

What example do you have in mind?
> 
Regards, WM