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

On 10/01/2024 10:00 AM, Jim Burns wrote:

On 9/29/2024 5:04 PM, Ross Finlayson wrote:

On 09/29/2024 12:02 PM, Jim Burns wrote:

On 9/26/2024 4:50 PM, Ross Finlayson wrote:

On 09/26/2024 06:48 AM, Jim Burns wrote:

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
?

>

I enjoy reading that, Jim, if I may be so familiar,
I enjoy reading that because it sounds _sincere_,
and, it reflects a "conscientiousness", given what
there is, given the milieu, then, for given the surrounds.

>

Thank you.

>

A right triangle: either has one right angle,
or two angles that sum to a right angle.
>
The sum of the angles is the angle of the sum.
>
Then, as with regards to atoms,

>
Perhaps I am taking more out of your posts
than you are putting in.
I say:
the complete ordered field doesn't have
infinitesimals.
You say:
There are these other systems, they have
infinitesimals.
It is very unclear to me what you intend for me
to take from that information.
>
I hope this will help me understand you better.
Please accept or reject each claim and
-- this is important --
replace rejected claims with
what you _would_ accept.
>
⎛ ℝ, the complete ordered field, is
⎝ the consensus theory in 2024 of the continuum.
>
⎛ ℝ contains ℚ the rationals and
⎜ the least upper bound of
⎝ each bounded nonempty subset of ℚ and of ℝ
>
( The greatest lower bound of ⅟ℕ unit fractions is 0
>
⎛ A unit fraction is reciprocal to a natural>0
⎜
⎜ A set≠{} ⊆ ℕ naturals holds a minimum
⎜ A natural≠0 has a predecessor.natural.
⎜ A natural has a successor.natural.
⎜
⎜ The sum of two naturals is a natural
⎝ the product of two naturals is a natural.
>
⎛ There are no points in ℝ
⎜ between 0 and all the unit fractions
⎝ (which is what I mean here by 'infinitesimal').
>
Thank you in advance.
>
>

Here it's that "Eudoxus/Dedekind/Cauchy is
_insufficient_ to represent the character
of the real numbers".
Then, that there are line-reals and signal-reals
besides field-reals, has that of course there are
also models of line-reals and signal-reals in the
mathematics today, like Jordan measure and the ultrafilter,
and many extant examples where a simple deliberation
of continuity according to the definitions of
line-reals or signal-reals, results any contradictions
you might otherwise see as arriving their existence.
Then, besides noting how it's broken, then also
there's given a reasoning how it's repaired,
resulting "less insufficient", or at least making
it so that often found approaches in the applied,
and their success, make the standard linear curriculum,
unsuited.
Then, I think it's quite standard how I put it,
really very quite standard.