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

Op 02.apr.2024 om 03:52 schreef olcott:

On 4/1/2024 8:27 PM, 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?
>

 Whether it is a real number or not is moot to me.
My key point is that 0.999... = 1.0 is categorically impossible.

It is only impossible for olcott because olcott seems to be unable to learn what 0.999... = 1.0 means for real numbers. He sticks to another interpretation and is not able to reason in the context of real numbers, even when olcott's interpretation has contradictory consequences.