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

On 28/03/2024 16:07, 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, but that will depend on what is meant by
"0.999..", 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.

      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
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.
To repeat [to both you and Wij]:  *Show us your axioms, and this
may perhaps be worth discussing.*  In particular, we need to know where
and why you are departing from standard axiomatisations of the reals.
[For the latter, simplest is to google for "axioms of real numbers",
which throws up dozens of articles ranging from elementary to extremely
advanced.]
--
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