Re: Definition of real number ℝ --infinitesimal--

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.
[...]

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Medtronic
void Void(void) { Void(); } /* The recursive call of the void */