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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Tue, 8 Oct 2024 14:01:18 -0400
Organization: i2pn2 (i2pn.org)
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On 10/8/24 11:26 AM, WM wrote:
> 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).

Where?

The "visible" numbers, per you definition are ALL the numbers, as all of 
them can be used individually and are selectable.

Thus, there aren't any left to be dark, except the ones that don't 
actually exist.

>>> 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.

How did it "lose" it, I thought you claim was that it was unchangable?

Remember, infinite means without end, so you can't have and end to it.

>>
>>> Another point is this: [0, 1) moved gives (0, 1].
>> Can you generalise this?
> 
> What example do you have in mind?
>>
> Regards, WM