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

On 9/11/2024 3:53 PM, WM wrote:

On 11.09.2024 20:52, Jim Burns wrote:

⅟ℕᵈᵉᶠ does not change. (Sets do not change.)

>
Because their elements,
here: the points
do not change.

Mathematical sets do not change
because
mathematical sets are immaterial;
thus, a different set merely to hold
slightly different elements
weighs nothing and costs nothing.
Mathematical sets do not change
because,
in an all.not.first.false finite claim.sequence,
claims about
mathematical sets both with and without some element
are NOT reliably.true.
Mathematical sets do not change
because,
in an all.not.first.false finite claim.sequence,
claims about
mathematical sets NOT both with and without some element
ARE reliably.true,
even if they're about infinitely.many never.seen
-- and, in that way,
unchanging sets can serve as a finite tool with which
to explore infinity.

β = glb.⅟ℕᵈᵉᶠ
>
Positive β requires impossibilities.
It requires that ½.β
both IS and IS NOT
undercut by a visibleᵂᴹ unit.fraction.

>
In some dark cases ½.β does not exist.

For each β, ½⋅β exists
among points (in ℝ) situating
splits (of ℚ) of differences of ratios of
of points (in ℕ) countable.to from 1,
which is to say well.ordered,
with each point successored and predecessored
except the first point, 1, only successored.
If darkᵂᴹ β allows ½⋅β to not.exist,
then darkᵂᴹ β not.exists among the points of ℝ
as described here.
For each of those β, ½⋅β exists.
However,
no ½⋅β exists for positive β = glb.⅟ℕᵈᵉᶠ
Therefore,
none of those β are positive β = glb.⅟ℕᵈᵉᶠ
0 = glb.⅟ℕᵈᵉᶠ
No gap exists between 0 and ⅟ℕᵈᵉᶠ
No points not undercut by points of ⅟ℕᵈᵉᶠ

But you claim that if β exists, then ½.β exists.
And if ½.β exists, then β/4 exists, and so on.

There's a proof.

That is potential infinity.

You (WM) apparently only say "potential infinity"
when you've recognized that you've lost an argument.
I accept your concession.

If all exists, then a smallest exists.

In ℝ.points situating ℚ.splits of
differences.of.ratios of countable.to ℕ.points,
no smallest exists.

We use "approach" to say "no gap".

For example,
⎛ ⅟ℕᵈᵉᶠ approaches 0
⎜ 0 = glb.⅟ℕᵈᵉᶠ
⎜ 0 ≤ᵉᵃᶜʰ ⅟ℕᵈᵉᶠ
⎝ ¬∃ᴿx: 0 < x ≤ᵉᵃᶜʰ ⅟ℕᵈᵉᶠ

And infinite set that approaches it
arbitrarily close.
It is an unbounded end.

>
Real points on the real line
either are there or are not there.

>
Sets of real.line.points and
real.line.points
are not the same things.

>
But we need not use sets at all
if we use the points.

If you use points as sets, expect gibberish.
For each point x in (0,1]: 0 < x/2 < x
No point exists between 0 and (0,1]