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

On 10/26/2024 12:04 PM, WM wrote:

On 26.10.2024 05:21, Jim Burns wrote:

On 10/25/2024 3:15 PM, WM wrote:

Mainly, among other points, the claim that
all unit fractions can be defined and the claim that
a Bob can disappear in lossless exchanges.

>
The proof that all unit fractions can be defined
is to define them
as reciprocals of positive countable.to.from.0 numbers.
>
That describes all of them and only them.

>
No, you falsely assume that
all natnumbers can be defined.

I assume that
natural numbers are finite ordinals, and
that a finite ordinal is predecessored or is 0  and
  each of its priors is predecessored or is 0, and
that an ordinal is successored and
   a set of ordinals is minimummed or is {}

Almost all unit fractions
cannot be discerned by definable real numbers.

>
All unit fractions are reciprocals of
positive countable.to.from.0 numbers.

>
That does not change the facts.

>
Clearly, the disagreement of proofs
does not change which claims you call facts.

>
Try to define more unit fractions
than remain undefined.

𝗖𝗹𝗮𝗶𝗺𝘀 exist which follow
  a 𝗰𝗹𝗮𝗶𝗺 of being countable.to.from.0
which are unaddressed by
  a 𝗰𝗹𝗮𝗶𝗺 of being discernable.or.visible.or...
and,
because they _literally_ follow
  true.or.not.first.false.ly,
its followers must be true 𝗰𝗹𝗮𝗶𝗺𝘀 about
  the truly countable.to.from.0

Try to understand the function NUF(x)
which starts with 0 at 0 and
after 1 cannot change to 2 without
pausing for an interval consisting of
uncountably many real points.

⎛ ℕ consists of points countable.to.from.0
⎜ ℤ consists of differences of ℕ
⎜ Q consists of quotients (w/o 0.denom) of ℤ
⎝ ℝ consists of points between splits of ℚ
⅟ℕ consists of reciprocals of non.0 ℕ
NUF(x) = |⅟ℕ∩(∞,x]|
∀ᴿx > 0: ∀n ∈ ℕ: ⅟⌈n+⅟x⌉ ∈ ⅟ℕ∩(∞,x]
NUF(0) = 0  ∧  ∀ᴿx > 0: NUF(x) = |ℕ|

Don't you claim that
every unit fraction can be discerned,
i.e. separated from the smaller ones
by a real number?

>
½⋅(⅟n+⅟(n+1)) existing is discerning?
I did not know that.

>
Giving a real number
_by its digits or as a fraction_
which is between two unit fractions discerns
the larger one and its larger predecessors.

⎛ ℕ consists of points countable.to.from.0
⎜ ℤ consists of differences of ℕ
⎜ Q consists of quotients (w/o 0.denom) of ℤ
⎝ ℝ consists of points between splits of ℚ
Consider two points x,x′ in ℝ
Either  x = x′
or there is a rational between x and x′
Each rational can be given as a fraction.
Each two points x and x′ are discernable.