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

On 11/2/2024 5:28 PM, WM wrote:

On 02.11.2024 20:52, Jim Burns wrote:

On 11/2/2024 1:50 PM, WM wrote:

On 02.11.2024 15:10, Jim Burns wrote:

On 11/1/2024 1:24 PM, WM wrote:

Hence
at least ℵ₀ points with
ℵ₀ intervals of uncountably many points
must be between 0 and x₀.

>

That cannot happen at x₀ = 0.

>

Each x₀ > 0 is an x₀ where it happens.

>
No.

Yes.
x₀ ≥ ⅟⌈n+⅟x₀⌉ > 0
E pur si muove.

At least ℵ₀ points
with ℵ₀ intervals of uncountably many points
must be between 0 and x₀.
This criterion is not satisfied by every point x > 0,

No.
x₀ ≥ ⅟⌈n+⅟x₀⌉ > ⅟⌈n+1+⅟x₀⌉ > 0
E pur si muove.

but it is satisfied by
every definable or visible point x > 0.

every point.between.splits > 0 of
differences of ratios of countable.to numbers.

Why do you not see this requirement?

I see that you require us to be talking about
what we are not talking about:
points which _aren't_ between.splits > 0 of
differences of ratios of countable.to numbers.
i.
That's not a thing.
You are not in command of what we do and don't talk about.
ii.
_Even if you were in command_
The reason that, for each real x > 0
there are ℕ.many unit fractions between 0 and x
is not that _we say so_
Our (hypothetical) not.saying.so can't change it being so.