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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 09:31:18 -0400
Organization: i2pn2 (i2pn.org)
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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.

> 
>> 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.

The problem is then that the objects are "fixed", The problem here is 
that our concepts of such don't really handle "infinite" objects, we can 
think about them "going on to infinity" and it sort of fades out of view 
in the distance, but since we have never really sensed "infinity" we 
have no way to actually fully understand it.

When we understand the "potential infinity" we see that it goes on 
forever, and don't feel a need to get there to see it, since we know we 
can't.

In presuming it is fixed, he presumes he can get to that infinite point, 
because everything he knows is finite (just like all we have actually 
seen is finite) and his logic is based on that intuition.

WM's logic seems to be based on the presumption that:

If actual infinity exists ... (with the assumption that if it exists, if 
follows the logic we know)

Since such a thing does NOT exist, at least as far as our logic can 
handle, NONE of his conclusions apply, since the logic was based on a 
untrue premise.

In WM's mind, anything that is "fixed" has ends, but an infinite thing 
doesn't have all those ends