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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: Mon, 28 Oct 2024 20:52:22 +0100
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On 28.10.2024 12:21, Richard Damon wrote:
> On 10/28/24 6:36 AM, WM wrote:

>> NUF increases by 1 or more, but more would violate mathematics.
> 
> No, NUF(x) jumps from 0 to Aleph_0 in the domain of finite numbers, 
> because there is no finite x where it has the value of 0.

It has the value 0 for all x =< 0. And it cannot jump by more than 1 at 
any point.
> 
> It is just a "undefined" function.

No, but the first steps happen at undefinable x.
> 
>>
>>>>> Hint: For each and every x e IR, x <= 0: NUF(x) = 0
>>>>> and for each and every x e IR, x > 0: NUF(x) = aleph_0.
>>>>
>>>> That is blatantly wrong because it would require that ℵo unit 
>>>> fractions exist between 0 and each and every x > 0, i.e., the open 
>>>> interval (0, 1].
>>
>>> Right, like is what happens.
>>>
>>> It may seem strange to a person stuck in finite logic, but is true 
>>> when you understand how infinity works.
>>
>> This infinity between 0 and (0, 1] is not what I can accept.
> 
> Note, it isn't an "infinity between" it is that the "bottom" of (0, 1] 
> doesn't exist as a definable point.

That is true. The bottom is dark.

Regards, WM