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

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... 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
"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)".

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.

<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>.

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

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.

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Medtronic
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