Re: How many different unit fractions are lessorequal than all unit fractions? (repleteness)

On 9/25/2024 9:50 PM, Ross Finlayson wrote:

On 09/25/2024 02:00 PM, Jim Burns wrote:

On 9/25/2024 2:44 PM, Ross Finlayson wrote:

How would you define "atom"
the otherwise "infinitely-divisible"?

It's seems quite Aristotlean to be against atomism,

I am anti.atomized.ℝ (complete ordered field)
You seem to read that as anti.atomized.anything.
How do I make claims to you (RF) which
are only about the things I intend?
For example, how do I claim to you (RF),
for a right triangles but not for any triangle, that
⎛ the square of its longest side equals
⎝ the sum of the squares of its two other sides
?

yet, at the same time
it's a very useful theory,
for example, with Democritan chemistry, atomic chemistry,
and stoichiometry.
>
This is foundations under consideration here,
not merely "pre-calc".

>
I think we don't choose foundations which
choose for us what is to be built on them.
(I think they shouldn't, so, Yay!)
>
ℝ is anti-atomic, that is, without infinitesimals.
>
And yet, ℝ is very useful for describing solutions to
the hydrogen.atom Hamiltonian ̂H = ̂p²/2m - e²/̂r
>
The periodic table and the complete ordered field
seem to connect differently from
the way i which you (RF) think they connect.

"you know I wrote a paper about the
infinitesimals with measurable character
in Nelson's Internal Set Theory, IST,
that IST, was co-consistent, with ZFC,
Zermelo-Fraenkel set theory".

I'd bet a large amount of money that those
infinitesimals with measurable character
aren't in the complete ordered field,
but instead are in something else.
Am I wrong?

About foundations, then it seems
that reflects on what Leibniz and others call,
"the fundamental question of meta-physics",
then why there's any one theory at all.
(Truth, ....)

I think we don't choose foundations which
choose for us what is to be built on them.
There is the question of what the continuum is.
Ideally,
that would not be settled by
which foundation we have chosen,
but by exo.logical considerations,
and, ideally,
it is _those considerations_ which are reflected in
properties like least.upper.bound.
If
we want a continuum containing the rationals and
in which crossing curves intersect,
then
we do not have points between 0 and unit.fractions.
If
we want something else,
then
something else might well be true.