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

On 30/03/2024 01:11, olcott wrote:

In other words when one gets to the end of a never ending sequence
(a contradiction) thenn (then and only then) they reach 1.0.

Zeno knew better than that some 2400 years ago.  He knew that
Achilles caught the tortoise, even though it was an infinite number of
steps;  it depends on how fast you complete/encounter the elements of
the sequence.  If you travel along the real number line at 1 unit/hour,
starting at 0 at noon, then you reach 0.9 at 12:54, 0.99 at 12:59:24,
0.999 at 12:59:56.4, 0.9999 at 12:59:59.64, 0.99999 at 12:59:59.964,
0.999999 at ..., and this "never ending sequence" is completed by 13:00.
Luckily, none of this matters, as "0.999..." viewed as a real number is
not defined as a "never ending sequence", but as the limit of a sequence
whose terms are defined.
If you switch from the reals to the surreals or hyperreals, then
you may prefer the way "0.999..." is treated, but you have simply moved
the "never-ending" problem to somewhere else, and the solution is the
same -- you need to sub-divide time appropriately to complete many steps
in a short time.  Time is inexorable in the Real World, but messwithable
in theoretical physics and in mathematics.
--
Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Praetorius