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

On 10/28/2024 11:31 AM, Jim Burns wrote:

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.

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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.
>
>

Of course you would mean "finite ordinals as
a brief summary model of discernibles the members"
of "the closure of all relations that make things
numbers, all those sets, too, all the related things",
"the" "numbers".
One might assume "the finite, whole, natural, non-negative,
counting integers, including zero, have a model of a
representation of each their indidividuals, as a
set of finite ordinals, suitable as a value to indicate
a member".
Then, where it's so that extensionally they "are", ...,
"the natural integers", is when for all purposes they
aren't not. It's not always so, yet, for a particular
broad class of theorems call "inductive counting arguments"
it's quite so.
It's sort of like that "TND" is sort of a class,
not necessarily a property of all propositions,
that it's particularly tractable when they are,
they aren't always.
It's like, if Chrysippus had been around so
that Plotinus and the Neo-Platonist Stoics
hadn't made it so Plotinus' weasel-words about
all the Aristotlean didn't result that Chrysippus'
moods would better reflect the modality (moo-dality)
thus that Russell et alia wouldn't be able to say
that "well Plotinus is classical so material implication
is classical" when it's so instead that the classical
already at least _solved_ that away in Chrysippus modal logic,
then maybe it wouldn't all along been "classical QUASI-modal logic",
as ruined the intuition of a century of scholastics.