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NNTP-Posting-Date: Wed, 03 Apr 2024 02:58:35 +0000
Subject: =?UTF-8?Q?Re:_Definition_of_real_number_=e2=84=9d_--infinitesimal--?=
Newsgroups: comp.theory
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Tue, 2 Apr 2024 19:58:37 -0700
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On 04/02/2024 04:33 PM, Keith Thompson wrote:
> Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
>> On 02/04/2024 19:29, Keith Thompson wrote:
>>> Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
>>>> On 02/04/2024 02:27, Keith Thompson wrote:
>>>>> olcott <polcott333@gmail.com> writes:
>>>>>> On 4/1/2024 6:11 PM, Keith Thompson wrote:
>>>>>>> olcott <polcott333@gmail.com> writes:
>>>>>>> [...]
>>>>>>>> Since PI is represented by a single geometric point on the number line
>>>>>>>> then 0.999... would be correctly represented by the geometric point
>>>>>>>> immediately to the left of 1.0 on the number line or the RHS of this
>>>>>>>> interval [0,0, 1.0). If there is no Real number at that point then
>>>>>>>> there is no Real number that exactly represents 0.999...
>>>>>>> [...]
>>>>>>> In the following I'm talking about real numbers, and only real
>>>>>>> numbers -- not hyperreals, or surreals, or any other extension to the
>>>>>>> real numbers.
>>>>>>> You assert that there is a geometric point immediately to the left
>>>>>>> of
>>>>>>> 1.0 on the number line.  (I disagree, but let's go with it for now.)
>>>>>>> Am I correct in assuming that this means that that point corresponds
>>>>>>> to
>>>>>>> a real number that is distinct from, and less than, 1.0?
>>>>>>
>>>>>> IDK, probably not. I am saying that 0.999... exactly equals this number.
>>>>> "IDK, probably not."
>>>>> Did you even consider taking some time to *think* about this?
>>>>
>>>> PO just says things he thinks are true based on his first intuitions
>>>> when he encountered a topic. He does not "reason" his way to a new
>>>> carefully thought out theory or even to a single coherent idea. Don't
>>>> imagine he is thinking of hyperreals or anything - he just "knows"
>>>> that obviously any number which starts 0.??? is less than one starting
>>>> 1.??? - because 0 is less than 1 !! Or whatever, it really doesn't
>>>> matter.
>>> I don't think he's explicitly said that any real number whose
>>> decimal
>>> representation starts with "0." is less than one starting with "1." --
>>> but if said that, he'd be right.
>>
>>    0.999...  = 1.000...  (so he'd be wrong)
>>
>>> What he refuses to understand is that the notation "0.999..." is not
>>> a
>>> decimal representation.  The "..."  notation refers to the limit of a
>>> sequence, and of course the limit of a sequence does not have to be a
>>> member of the sequence.  Every member of the sequence (0.9, 0.99, 0.999,
>>> 0.9999, continuing in the obvious manner) is a real (and rational)
>>> number that is strictly less than 1.0.  But the limit of the sequence is
>>> 1.0.  Sequences and their limits can be and are defined rigorously
>>> without reference to infinitesimals or infinities,
>>
>> Ah, I see - you're trying to say that 1.000... is a decimal
>> representation, but not 0.999...?, which would make sense of why you
>> think PO would be right above.  That's a new one on me, but I don't go
>> for that argument at all.
>
> No, I was trying to say that "0.999" is a decimal representation
> (representing exactly 999/1000, or 9/10 + 9/100 + 9/1000), but
> "0.999..." is something fundamentally different, where the "..." denotes
> a limit.
>
> But now that I think about it, "most" real numbers don't have a finite
> decimal representation.  Rational numbers that are an integer multiple
> of an integer power of 10 have finite representations, other rational
> numbers have an indefinitely repeating representation that denotes a
> limit (e.g., "0.333..."), and irrational numbers can only be
> approximated arbitrarily closely in decimal.
>
>> 0.999... is a decimal representation for the number 1, shortened by
>> ... which means "continuing in the obvious fashion" or equivalent
>> wording.  I.e. 0.999... is the decimal where every digit after the
>> decimal point is a 9.  It represents the number 1, as does 1.000....
>> Yes, there are two ways to represent the number 1 as an infinite
>> decimal.  Not a problem.
>
> Good point.  And I've probably muddied the waters enough.  (If olcott
> claims that this discussion means he's right, I'll just ignore him.)
>
>> Anyhow, I have a BA in mathematics, so I understand limits etc..  :)
>> I was posting to explain why you're wasting your time trying to
>> explain abstract ideas to PO, but it's fine with me if people want to
>> do that for whatever reason.
>>
>> Mike.
>> ps. of course, someone could make a rule that infinitely repeating 9s
>> in a decimal expansion is outlawed, but that's not normal practice
>> AFAIK.  People just accept there are two representations of certain
>> numbers.
>>
>>> It can be genuinely difficult to wrap your head around this.  It
>>> *is*
>>> counterintuitive.  And thoughtful challenges to the mathematical
>>> orthodoxy, like the paper recently discussed in this thread, can be
>>> useful.  But olcott doesn't offer a coherent alternative.
>>> [...]
>

My coherent alternative is that there are at least three models
of continuous domains, so, different sets in the set theory,
though that make for extent, density, completeness, and measure,
so that real analytical character characterizes continuous domains.


Notions like "Dirac delta" really help to visualize what are
non-real functions, "not-a-real-function", that however have
real analytical character. It's built as a limit of real
functions, while of course, it results that it only has
exactly what results its real analytical character as
not-a-real-function.

Then, the measure problem, or Vitali's construct, then
about Vitali and Hausdorff's approach to Banach-Tarski,
really do get into conditions in continuity that result
the quasi-invariant measure theory, up after Ramsey theory.



The other day, for example, I prompted one of the new online
mechanical reasoners to show that the continuum limit of
n/d for naturals as d goes to infinity, looks like [0,1].

So, it's not counterintuitive, it just sort of develops
that there are not-a-real-function's that have real
analytical character, and so they're pretty special.


Having a mathematics degree helps a lot to demonstrate
at least a usual modicum of acquired knowledge, but it
was in physics class one day when the professor introduced
"continuum limit". Later it came up a lot in real analysis,
mostly about the concept of Dirac delta, the unit impulse function.