Re: Does the number of nines increase?

On 7/15/2024 8:21 AM, WM wrote:

Le 14/07/2024 à 17:04, Moebius a écrit :

Am 14.07.2024 um 14:58 schrieb WM:

Le 13/07/2024 à 21:37, Moebius a écrit :

Am 13.07.2024 um 17:56 schrieb WM:

Le 13/07/2024 à 04:38, Moebius a écrit :

Am 12.07.2024 um 19:29 schrieb WM:

Le 10/07/2024 à 19:58, Jim Burns a écrit :

When you say that
NUF(x) doesn't changeᵂᴹ at 0
you mean that

>
it does not change its value from x < 0,
namely NUF(0) = 0.

>
But it changes its value from aleph_0
for each and every x > 0
to 0 at x = 0,

>
That claim is wrong,

>
Nope.

>
That claim is wrong,

>
Nope.

>
That claim has been disproved.

You (WM) are confused about what that claim is.

Something that is before each and every x > 0
sits at zero or at the negative real line.

⎛ That claim is not that
⎜ one unit fraction or ℵ₀.many unit.fractions
⎝ sit at zero or at the negative real line.
Each ℵ₀.sequenceᴶᴮ matches each ℵ₀.sequenceᴶᴮ
element.to.element.
Each ℵ₀.sequenceᴶᴮ holds ℵ₀.many.
For ℵ₀.sequenceᴶᴮ A ordered by '<'
⎛ for each nonempty B ⊆ A: firstᑉ.B ∈ B
⎜ for each j ∈ A: first.afterᑉ.j ∈ A\{firstᑉ.A}
⎝ for each k ∈ A\{firstᑉ.A}: last.beforeᑉ.k ∈ A
For each real x > 0
the unit fractions ⅟ℕ∩(0,x] ordered by '>'
is an ℵ₀.sequenceᴶᴮ and holds ℵ₀.many
...because...
For each nonempty B ⊆ ⅟ℕ∩(0,x]
⅟B = {⅟u: u ∈ B} is a nonempty subset of ℕ
exists min.⅟B ∈ ⅟B ⊆ ℕ
⅟(min.⅟B) = max.B ∈ B
max.(⅟ℕ∩(0,x]) = ⅟⌈⅟x⌉
For each u ∈ ⅟ℕ∩(0,x]: ⅟(1+⅟u) ∈ ⅟ℕ∩(0,x]\{⅟⌈⅟x⌉}
For each v ∈ ⅟ℕ∩(0,x]\{⅟⌈⅟x⌉}: ⅟(-1+⅟v) ∈ ⅟ℕ∩(0,x]
Therefore,
for each real x > 0
the unit fractions ⅟ℕ∩(0,x] ordered by '>'
is an ℵ₀.sequenceᴶᴮ and holds ℵ₀.many
and NUF(x) = |⅟ℕ∩(0,x]| = ℵ₀
∀ᴿx>0: NUF(x) = ℵ₀