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

"Fred. Zwarts" <F.Zwarts@HetNet.nl> writes:

Op 29.mrt.2024 om 16:46 schreef olcott:

On 3/29/2024 8:13 AM, Richard Damon wrote:

On 3/28/24 11:50 PM, olcott wrote:

On 3/28/2024 10:36 PM, Keith Thompson wrote:

olcott <polcott2@gmail.com> writes:
[...]

It seems dead obvious that 0.999... is infinitesimally less than 1.0.

>
Yes, it *seems* dead obvious.  That doesn't make it true, and in
fact it
isn't.
>

>
0.999... means that is never reaches 1.0.
and math simply stipulates that it does even though it does not.

>
>
0.999... isn't a "number" in the Real Number system, just an
alternate representation for the number 1.
>

That is not true. 0.999... means never reaches 1.0

>
Maybe for olcott's unspecified olcott numbers. For real numbers
0.999... equals 1.0. There are many proofs. See e.g.

https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

olcott almost has a point, in that the sequence of values 0.9, 0.99,
0.999, 0.9999, ... (continuing in the obvious manner) never reaches
1.0.  No element of that unending sequence of real numbers is exactly
equal to 1.0.

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.

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.

If he's talking about the reals, he's wrong.  If he's talking about
something other than the reals, he's boring.  Either way, he will not
change his mind.  Attempts to explain limits and real numbers to him
will fail.

--
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 */