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

Op 03.apr.2024 om 17:16 schreef olcott:

On 4/3/2024 3:42 AM, Fred. Zwarts wrote:

Op 02.apr.2024 om 23:52 schreef olcott:

On 4/2/2024 4:20 PM, Mike Terry wrote:

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)
>

>
In other words you simply choose to "not believe in"
the notion of infinitesimal difference. That doesn't
actually make it go away.
>

>
It is not a matter of 'believe-in'. In the real number system there are no infinitesimal differences.

 So when they do occur

In the real number system they do not occur. Olcott is fighting windmills again.

they cannot be expressed so the convention is to ignore them.

Things that do not occur, don't need to be ignored.  Olcott is fighting windmills again.

Infinitesimal differences cannot simply be ignored on the
basis the Real number cannot express them.
 The Sapir–Whorf hypothesis, also known as the linguistic relativity
hypothesis, refers to the proposal that the particular language one
speaks influences the way one thinks about reality.
https://www.sciencedirect.com/topics/psychology/sapir-whorf-hypothesis
 

There always a finite ε is used. Apparently olcott is talking about his undisclosed olcott-numbers, but he keeps it as a secret what it means.
>