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

On 28/03/2024 16:53, olcott wrote:

Yet it seems that wij is correct that 0.999... would seem to
be infinitesimally < 1.0.

     That /cannot/ be correct in the "real" numbers, in which there
are no infinitesimals [basic axiom of the reals].  In other systems of
numbers, it could be correct,

Yes.

but that will depend on what is meant by
"0.999..",

Approaching yet never reaching 1.0.

That is a property of the numbers 0.9, 0.99, 0.999 and so
on arranged as a sequence [and of many other sequences], but is not
/yet/ a value.  Not until you explain what you mean.  In conventional
mathematics, it is usually taken to mean the limit of that sequence
expressed as a real number, where "limit" has a precise meaning as
discussed and formalised in the 19thC.  That limit is 1.  Not a tiny
bit less than one, not some new sort of object, but 1, exactly.  You
and Wij may find that surprising, or even nonsensical, but it is what
the mathematics tells us from the axioms of the real numbers and from
the definition of "limit".  If you want the answer to be different,
then that must follow from different axioms and definitions.  Until
you and/or Wij tell us what those are, there is nothing further useful
to be said.

and note that if you appeal to something that mentions limits
to define this, then you have to explain how infinite and infinitesimal
numbers are handled in the definition.

Again, there are no infinite or infinitesimal real numbers, so
if you want an infinitesimal in your answer, it is incumbent on you to
explain what you are using /other than/ conventional maths.

                  One geometric point on the number line.
[0.0, 1.0) < [0.0, 1.0] by one geometric point.

     Until you describe the axioms of what you mean by "geometric
point" and "number line", this is meaningless verbiage.  Give your

Of course by geometric point I must mean a box of chocolates and by
number line I mean a pretty pink bow. No one would ever suspect that
these terms have their conventional meanings.

I didn't ask what "geometric point" and "number line" are, but
what axioms you think they have.  In conventional mathematics, those two
intervals have /exactly/ the same measure even though they are not
exactly the same sets of points.  If you get a different answer [and
have not simply made a mistake], it /must/ be because you are using
different axioms.  What are they?

axioms, and it becomes possible to discuss this.  Until then, we are
entitled to assume that you and Wij are talking about the "traditional"
"real" numbers [as used in engineering, etc.] in which there are no
infinitesimals, and so the only interpretation we can make of the size
of "one geometric point" is the usual "measure", which is zero.

Yet it is never actually zero because it is possible to specify a
line segment that is exactly one geometric point longer than another.
[0.0, 1.0] - [0.0, 1.0) = one geometric point.

But "one geometric point" has measure zero.  Not "never actually
zero", but actually and really zero.  Unless, that is, you are using some
different and as yet unexplained axioms/definitions.  Which are ...?
--
Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Couperin