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

On 11/5/2024 11:49 AM, WM wrote:

On 04.11.2024 22:46, Jim Burns wrote:

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

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

>
That requires that x₀ is definable.

x₀ is one of
  (ℝ) points.between.splits of
  (ℚ) differences of
  (ℚ⁺) ratios of
  (ℕ) countable.to from.1,
n is one of
  (ℕ) countable.to from.0.
x₀ ≥ ⅟⌈n+⅟x₀⌉ > ⅟⌈n+1+⅟x₀⌉ > 0

But it is dark.

Whatever 'dark' means,
x₀ is one of
  (ℝ) points.between.splits of
  (ℚ) differences of
  (ℚ⁺) ratios of
  (ℕ) countable.to from.1,
n is one of
  (ℕ) countable.to from.0.
i.
You could be talking about not.that,
in which case,
you aren't denying my claim, which is about that,
  no matter what it seems like you're doing.
ii.
You could be talking about that,
in which case,
x₀ ≥ ⅟⌈n+⅟x₀⌉ > ⅟⌈n+1+⅟x₀⌉ > 0
We know
x₀ ≥ ⅟⌈n+⅟x₀⌉ > ⅟⌈n+1+⅟x₀⌉ > 0
is true because
it is a claim in a finite sequence of only
true.or.not.first.false claims,
and
each claim in such a sequence must be true.

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.

>
Of definable splits.

Of splits of
  (ℝ) points.between.splits of
  (ℚ) differences of
  (ℚ⁺) ratios of
  (ℕ) countable.to from.1.