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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 17:21:24 -0400
Organization: i2pn2 (i2pn.org)
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On 10/8/24 3:30 PM, Chris M. Thomasson wrote:
> On 10/7/2024 5:57 PM, Richard Damon wrote:
>> On 10/7/24 8:38 PM, Chris M. Thomasson wrote:
>>> On 10/7/2024 4:01 PM, Richard Damon wrote:
>>>> On 10/7/24 4:27 PM, Chris M. Thomasson wrote:
>>>>> On 10/7/2024 4:17 AM, Richard Damon wrote:
>>>>>> On 10/7/24 7:13 AM, joes wrote:
>>>>>>> Am Mon, 07 Oct 2024 10:13:21 +0200 schrieb WM:
>>>>>>>> On 06.10.2024 17:55, joes wrote:
>>>>>>>>> Am Sun, 06 Oct 2024 17:26:07 +0200 schrieb WM:
>>>>>>>>>> On 06.10.2024 16:52, Alan Mackenzie wrote:
>>>>>>>>>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>>>>>>
>>>>>>>>>>> This idea of time may be what misleads the mathematically 
>>>>>>>>>>> less adept
>>>>>>>>>>> into believing that 0.999... < 1.
>>>>>>>>>> That is true even in actual infinity.
>>>>>>>>>> We can add 9 to 0.999...999 to obtain 9.999...999. But 
>>>>>>>>>> multiplying
>>>>>>>>>> 0.999...999 by 10 or, what is the same, shifting the digits 9 
>>>>>>>>>> by one
>>>>>>>>>> step to the left-hand side, does not increase their number but 
>>>>>>>>>> leaves
>>>>>>>>>> it constant: 9.99...9990.
>>>>>>>>>> 10*0.999...999 = 9.99...9990 = 9 + 0.99...9990 < 9 + 
>>>>>>>>>> 0.999...999 ==>
>>>>>>>>>> 9*0.999...999 < 9 as it should be.
>>>>>>>>> In actual infinity, there is no last 9 (that would not be 
>>>>>>>>> infinite).
>>>>>>>> Actual means all, but not more. This implies a last before ω. The
>>>>>>>> infinity means an end cannot be determined. It is produced by 
>>>>>>>> the dark
>>>>>>>> numbers.
>>>>>>> Actually infinite means infinite, which doesn’t change when you 
>>>>>>> add or
>>>>>>> subtract a finite number.
>>>>>>>
>>>>>>
>>>>>> Actual infinity doesn't exist for us finite beings.
>>>>>
>>>>> What do you mean? Are you trying to go the WM route where there is 
>>>>> a largest natural number... We just can't see it yet because it's 
>>>>> dark? Oh shit.
>>>>
>>>> I am allowing that an INFINITE being MIGHT be able to comprehend 
>>>> something like an actual infinity. But this can not possibly be done 
>>>> by a finite being.
>>>
>>> Well, us as finite beings know that there is not a largest natural 
>>> number... That right there is a basic understanding of the infinite: 
>>> Fair enough?
>>
>> Yes, us "smart" finite beings understand that the logic of there 
>> needing to be a "lowest value" unit fraction is incorrect, even if the 
>> "logic" of approaching the "existing" list of points from the low side 
>> would seem to imply that there should be a first.
>>
>> We can't really "see" what is happening there, in a fixed creation 
>> point of view, which is what makes that concept not "real" for us.
> 
> Humm... Well, I know that there is no smallest unit fraction, yet there 
> are infinitely many of them that get closer and closer to zero, yet 
> never reach it because no unit fraction will ever actually equal zero. 
> That right there helps me as a finite being to understand some basic 
> facts about infinity. And yes, WM, actual infinity... ;^)

Yes, and we understand that by understanding the process of potential 
infinity, that for each number there is a smaller number.

In the "Actual Infinity" the numbers are all there so it would seem by 
logic that there needs to be a smallest one, but there isn't.

Because we just can't imagine how that would work, because the rules of 
the infinite are different than what we are used to perceiving.