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

On 10/26/2024 12:35 PM, Chris M. Thomasson wrote:

On 10/26/2024 9:04 AM, 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.

>
Huh? Confusing to me. Humm... Are you trying to suggest that a natural
number can _not_ be a natural number?
>
[...]
>

Well, you can know there are usually considered standard
and also non-standard models of integers with regards to
whether they also contain infinitely-grand members or
even whether there are any they don't, this gets into
what's called Skolem, Louwenheim, Levy, with regards to
that there are _extensions_ and _fragments_ often enough
called "generic" extensions or fragments that though in
model theory _model the same things_.
Then, there are some theorems about the objects that
are the integers, for example only the natural integers,
that have models where the theorems are so and models
where the theorems are not so. These include, according
to model theory and proof theory about set theory, one
where there _are_ and ones where there _aren't_,
infinite members among the infinitely many. (There
are always the "{... infinitely-many ...}".)
This is often familiar as "Goedel's Incompleteness",
where he notes "by the by the number theory you know
sort of varies in the infinite with regards to whether
there is a point at infinity or not", as with regards
to the usual law of large numbers being the usual law
of small numbers, that there are other laws of large
numbers, and that Goedel _proves_, in ordinary set theory,
that there are non-standard models (extra-ordinary models)
of numbers according to a fundamental theory like set theory.
Then, a sort of result for example is that there aren't
actually, "standard" models of integers - only fragments
and extensions, then for theorems so in "fragments" and
theorems so in "extensions", then in those the "ordinary"
and "extra-ordinary", these making classications for the
multiple law(s), plural, of large numbers, and with regards
to number theory, and also analysis, with regards to fast
and slow, and weak and strong, and uniform and global,
convergence.
So, at least there's a way to be making sense of this.