Re: Does the number of nines increase?

On 7/10/2024 1:58 PM, Jim Burns wrote:

On 7/10/2024 12:02 PM, WM wrote:

Le 09/07/2024 à 20:51, Jim Burns a écrit :

On 7/9/2024 8:08 AM, WM wrote:

 

A change of NUF(x) happens at point x if
for all y < x NUF(x) > NUF(y).

>
Then ceiling(x) _doesn't_ changeᵂᴹ "at" 0

>
Then what ever. It is my definition.

 Very much like Humpty Dunpty and "glory".
[1]

| “I don't know what you mean by 'glory,'"
| Alice said.
|
| Humpty Dumpty smiled contemptuously.
| "Of course you don't
| ---till I tell you. I meant
| 'there's a nice knock-down argument for you!'"
|
| "But glory' doesn't mean
| 'a nice knock-down argument,'"
| Alice objected.
|
| "When I use a word,"
| Humpty Dumpty said, in rather a scornful tone,
| "it means just what I choose it to mean
| ---neither more nor less."
|
| "The question is,"
| said Alice,
| "whether you can make words mean so many different things."
|
| "The question is,"
| said Humpty Dumpty,
| "Which is to be master---that's all."
|
-- Lewis Carroll

When you say that
NUF(x) doesn't changeᵂᴹ at 0
you mean that
NUF(x) doesn't changeᵂᴹ like ceiling(x) doesn't.
 However,
your argument depends upon
it being unclear what you (WM) mean.
Not the best argument.
 

Relevant is this and only this:
NUF(0) = 0,
and the first step happens  at x > 0.
Like every step it is a step by 1.

>
If NUF(x) = 0  at  x > 0
then
Contradiction.

>
Not in dark numbers.

 For the unit.fractionsⁿᵒᵗᐧᵂᴹ.set ⅟ℕⁿᵒᵗᐧᵂᴹ
glb.⅟ℕⁿᵒᵗᐧᵂᴹ > 0 is contradictory.
 Each nonempty ⅟ℕⁿᵒᵗᐧᵂᴹ.subset S
holds a largest S.element.
 Each ⅟ℕⁿᵒᵗᐧᵂᴹ.element u (including 1) has
a next.smaller ⅟ℕⁿᵒᵗᐧᵂᴹ.element ⅟(1+⅟u)
 Each ⅟ℕⁿᵒᵗᐧᵂᴹ.element v (excluding 1) has
a next.larger ⅟ℕⁿᵒᵗᐧᵂᴹ.element ⅟(-1+⅟v)
 However,
your argument depends upon
it being unclear what you (WM) mean.
Not the best argument.