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

On 9/25/2024 11:51 AM, WM wrote:

On 25.09.2024 06:59, Jim Burns wrote:

On 9/24/2024 4:37 PM, WM wrote:

On 24.09.2024 22:19, Jim Burns wrote:

Thus
increasing NUF(x) from 0 to infinity
WITH intermediate steps
is gibberish,

>
The only alternative would by
infinitely many unit fractions at one point.

>
The correct (different) alternative is:
infinitely.many unit.fractions
at one point per unit.fraction,

>
That means NUF increases by 1
at every point occupied by a unit fraction.

There are numbers (cardinalities) which increase by 1
and other numbers (cardinalities), which
don't increase by 1.
We call them, respectively, 'finite' and 'infinite'.
You (WM) apparently think that means something else,
something like "reallyreallyreally big".
For each positive point x
for each number (cardinality) k which can increase by 1
there are more.than.k unit.fractions between 0 and x
0 > ⅟⌈k+1+⅟x⌉ < ... < ⅟⌈1+⅟x⌉ < x
For each positive point x
the number (cardinality) of unit.fractions between 0 and x
is not
any number (cardinality) which increases by 1
Instead, it is
a number (cardinality) which doesn't increase by 1.

That means NUF increases by 1
at every point occupied by a unit fraction.

At each point x occupied by a unit.fraction,
NUF(x)  =  |⅟ℕ∩(0,x]|  >  |⟨⅟⌈k+1+⅟x⌉,...,⅟⌈1+⅟x⌉⟩|  is
a number (cardinality) which doesn't increase by 1.

with infinitely.many points in all.

>
NUF(x) distinguishes all points.

∀ᴿx₁>0: ∀ᴿx₂>0:  NUF(x₁) = ℵ₀ = NUF(x₂)

Although
no more than finitely.many points
can be stepped.through end.to.end
we don't require these points to do more than _exist_
>
More.than.finitely.many can _exist_

>
Yes, but they are dark.

Each positive point is undercut by
some finite.unit.fraction.
[Archimedean property (Otto Stolz)]

[...] I will repeat it on and on, [...]

Repetition is apparently
what you (WM) think mathematics is:
🛇 "I refuse to concede" ⇒
🛇 "I have a theorem"