Re: There is a first/smallest integer (in Mückenland)

On 7/18/2024 4:55 PM, WM wrote:

Le 18/07/2024 à 21:16, Jim Burns a écrit :

On 7/18/2024 2:28 PM, WM wrote:

Le 18/07/2024 à 19:00, Jim Burns a écrit :

Jumps "at" a point are between
 nearby points.
WM admitted that much in a recent post,
but changed what "change" means to him.

>
A claim for all x > 0 is
a claim for all points of the interval (0, oo).
The claim
"between 0 and every point of (0, oo) NUF(x) = ℵo"
implies the existence of
ℵo unit fractions between 0 and (0, oo)
and is false.

>
The unit.fractions between 0 and different points x
are not the same unit.fractions.

>
Not claimed that they are the same.

For different points x₁ x₂
the unit.fractions ⅟ℕ∩(0,x₁] and ⅟ℕ∩(0,x₂] are
step.down non.⅟⌈⅟x₁⌉.step.up well.ordered.by.>
and
step.down non.⅟⌈⅟x₂⌉.step.up well.ordered.by.>
⅟ℕ∩(0,x₁] and ⅟ℕ∩(0,x₂] are order.isomorphic to ℕ
step.up non.0.step.down well.ordered.by.<
|⅟ℕ∩(0,x₁]| = |⅟ℕ∩(0,x₂]| = |ℕ| = ℵ₀

But if
for all points x > 0
there are ℵo smaller unit fractions,
then
for the interval (0, oo)
there are ℵo smaller unit fractions.

No.
For each point x > 0
there are
step.down non.⅟⌈⅟x⌉.step.up well.ordered.by.>
unit.fractions in (0,x] which,
because
step.down non.⅟⌈⅟x⌉.step.up well.ordered.by.>
are ℵ₀.many.
No unit fractions are smaller than (0,∞)

Or is there a point in (0, oo) which
is not an x > 0?

No, but
there is no point x in (0,∞) such that
⅟⌈⅟x⌉ isn't = max.(⅟ℕ∩(0,x])
and
unit.fractions in ⅟ℕ∩(0,x] aren't
step.down non.⅟⌈⅟x⌉.step.up well.ordered.by.>
and
unit.fractions in ⅟ℕ∩(0,x] aren't ℵ₀.many.