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

olcott <polcott2@gmail.com> writes:

On 3/29/2024 8:21 PM, Keith Thompson wrote:

olcott <polcott2@gmail.com> writes:

On 3/29/2024 7:25 PM, Keith Thompson wrote:

[...]

What he either doesn't understand, or pretends not to understand, is
that the notation "0.999..." does not refer either to any element of
that sequence or to the entire sequence.  It refers to the *limit* of
the sequence.  The limit of the sequence happens not to be an element of
the sequence, and it's exactly equal to 1.0.
>

In other words when one gets to the end of a never ending sequence
(a contradiction) thenn (then and only then) they reach 1.0.

No.
>
You either don't understand, or are pretending not to understand,
what the limit of sequence is.  I'm not offering to explain it to
you.

>
I know (or at least knew) what limits are from my college calculus 40
years ago. If anyone or anything in any way says that 0.999... equals
1.0 then they <are> saying what happens at the end of a never ending
sequence and this is a contradiction.

 Apparently you've forgotten what limits are.  I'm still not offering to
explain them.
 

This is all stated in terms of the real numbers, which are a well
defined set.  There are other systems with different properties.  If we
were talking about the hyperreals, for example, olcott's statement might
be correct (though I'm not sure of that).  But olcott seems to be
insisting, quite incorrectly, that his statements apply to the reals.

>
Pi exists at a single geometric point on the number line.

Irrelevant.

>
One geometric point to the left or to the right is incorrect.

 You apparently think there's a geometric point immediately to the left
of pi.  Real numbers don't work that way.
 Say the numeric value corresponding to the geometric point immediately
to the left of pi on the real number line is x.  What is the real value
of (pi+x)/2?  Is it greater than x?  Is it less than pi?
 I'm going to drop out of this discussion unless someone says something
sufficiently interesting.