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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Tue, 8 Oct 2024 23:01:41 -0700
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On 10/8/2024 3:46 PM, FromTheRafters wrote:
> It happens that Chris M. Thomasson formulated :
>> On 10/8/2024 2:21 PM, Richard Damon wrote:
>>> On 10/8/24 4:17 PM, Chris M. Thomasson wrote:
>>>> On 10/8/2024 7:10 AM, Richard Damon wrote:
>>>>> On 10/8/24 10:00 AM, Alan Mackenzie wrote:
>>>>>> Richard Damon <richard@damon-family.org> wrote:
>>>>>>> On 10/8/24 9:11 AM, Alan Mackenzie wrote:
>>>>>>>> Richard Damon <richard@damon-family.org> wrote:
>>>>>>>>> On 10/8/24 5:42 AM, Alan Mackenzie wrote:
>>>>>>>>>> Richard Damon <richard@damon-family.org> wrote:
>>>>>>>>>>> On 10/7/24 7:44 AM, Alan Mackenzie wrote:
>>>>>>>>>>>> Richard Damon <richard@damon-family.org> 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.
>>>>>>
>>>>>>>>>>>> English language tip: The "Actually" in that sentence was not
>>>>>>>>>>>> attached to the word "infinite", it meant something like 
>>>>>>>>>>>> "This is
>>>>>>>>>>>> really true:".
>>>>>>
>>>>>>>>>>> But all his reference to the word "Actually" are part of his 
>>>>>>>>>>> trying to
>>>>>>>>>>> define the term "Actual Infinity".
>>>>>>
>>>>>>>>>> OK, maybe you're right, there.  The semantics are a bit 
>>>>>>>>>> ambiguous.
>>>>>>>>>> Joes is not a native English speaker.  Apologies to Joes.
>>>>>>
>>>>>>>>>>>> Anyhow, what do you mean when you say that "actual infinity
>>>>>>>>>>>> doesn't exist"?  I think we established over the weekend 
>>>>>>>>>>>> that for
>>>>>>>>>>>> a mathematical entity not to exist, it must cause a 
>>>>>>>>>>>> contradiction.
>>>>>>>>>>>> Or something like that.
>>>>>>
>>>>>>>>>>>> So what contradiction would the existence of actual infinity 
>>>>>>>>>>>> cause?
>>>>>>
>>>>>>>>>>> It implies that there exists a first positive real, rational 
>>>>>>>>>>> number or
>>>>>>>>>>> unit fraction for one (at least the way WM uses it).
>>>>>>
>>>>>>>>>> Whoa!  There're rather a lot of argument steps missing there.  
>>>>>>>>>> Just
>>>>>>>>>> because WM asserts the existence of both actual infinity and a 
>>>>>>>>>> first
>>>>>>>>>> strictly positive unit fraction doesn't mean the one implies 
>>>>>>>>>> the other.
>>>>>>
>>>>>>
>>>>>>>>> It does in his logic, which is all that matters to him.
>>>>>>
>>>>>>>> Were we talking about WM's "logic", just there?  I don't think I 
>>>>>>>> was.
>>>>>>
>>>>>>> But my quote came from his logic.
>>>>>>
>>>>>> Ah, OK.
>>>>>>
>>>>>>>>> Yes, it is a wrong conclusion, but that error is based on his 
>>>>>>>>> initial
>>>>>>>>> assumption that something could be used that isn't available as a
>>>>>>>>> understandable entity to us finite beings.
>>>>>>
>>>>>>>> I think infinity is understandable.  I think I understand it.  My
>>>>>>>> position is that the distinction between "potential infinity" 
>>>>>>>> and "actual
>>>>>>>> infinity" is bogus.  It makes no difference in mathematics, 
>>>>>>>> which is
>>>>>>>> probably why the terms have vanished from mathematical discourse.
>>>>>>
>>>>>>> The difference is that in his "actual infinity" the generation 
>>>>>>> process
>>>>>>> is complete and nothing can change.
>>>>>>
>>>>>> In a sense, the "actual infinity" _is_ the generation process, 
>>>>>> which is
>>>>>> fixed and complete.
>>>>>
>>>>> But only after that infinite process has completed. Finite beings 
>>>>> can't actually SEE that result.[...]
>>>>
>>>> When does an infinite process finally complete? If it does, well, it 
>>>> was NOT infinite in any way shape, or form... Right?
>>>
>>> The infinite process completes in the infinite, thus beyond what we 
>>> can perceive, but only dimly imagine.
>>>
>>> That is why we can't have actual infinity, we can't get to the 
>>> infinite to see it done.
>>
>> since a step-by-step process for the naturals aka:
>>
>> 1, 1+1, 1+1+1, ...
>>
>> Will never end, well... That does not mean there is a largest natural. 
>> WM is strange on this aspect. See, this right here is pondering on the 
>> infinite from a finite being... ;^)
> 
> I suggest forgetting the whole 'finite being' idea. Mathematics is 
> filled with abstractions of things that are not possible in real life 
> and time. It is not necessary to build the natural numbers step by step 
> in sequence in an impossible supertask. It is an object in mathematics, 
> boom there it is. :)

Touche! Thanks. :^)