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

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... denotes a *limit*.  In particular, it's the limit of the value
as the number of 9s increases without bound.  That's what the notation

That is how it has been misinterpreted yet it has always meant
infinitesimally less than 1.0.

"0.999..." *means*.  (There are more precise notations for the same
thing, such as "0.9̅" (that's a 9 with an overbar, or "vinculum") or
"0.(9)".
 

I already know all that.

You have a sequence of numbers:
      0.9
     0.99
     0.999
     0.9999
     0.99999
     ...
 Each member of that sequence is strictly less than 1.0, but the *limit*
is exactly 1.0.  The limit of a sequence doesn't have to be a member of
the sequence.  The limit is, informally, the value that members of the
sequence approach arbitrarily closely.
 

Yet never reaching.

<https://en.wikipedia.org/wiki/Limit_of_a_sequence>
 

That we can say this in English yet not say this in conventional
number systems proves the need for another number system that can
say this.

 Then I have good news for you.  There are several such systems, for
example <https://en.wikipedia.org/wiki/Hyperreal_number>.
 

Infinitesimally less than 1.0 means one single geometric point
on the number line less than 1.0.

If your point is that you personally like hyperreals better than you
like reals, that's fine, as long as you're clear which number system
you're using. 

The Infinitesimal number system that I created.

If you talk about things like "0.999..." without
qualification, everyone will assume you're talking about real numbers.
 

It is already the case that 0.999...
specifies Infinitesimally less than 1.0.

And if you're going to play with hyperreal numbers, or surreal numbers,
or any of a number of other extensions to the real numbers, I suggest
that understanding the real numbers is a necessary prerequisite.  That
includes understanding that no real number is either infinitesimal or
infinite.
 Disclaimer: I'm not a mathematician.  I welcome corrections.
 

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer