Re: Replacement of Cardinality (real-valued)

On 08/04/2024 08:54 AM, Ross Finlayson wrote:

On 08/04/2024 08:46 AM, Ross Finlayson wrote:

On 08/04/2024 07:59 AM, Ross Finlayson wrote:

On 08/04/2024 07:52 AM, Ross Finlayson wrote:

On 08/04/2024 03:41 AM, FromTheRafters wrote:

Ross Finlayson formulated the question :

On 08/03/2024 02:59 PM, FromTheRafters wrote:

Chris M. Thomasson formulated on Saturday :

On 8/3/2024 7:25 AM, WM wrote:

Le 02/08/2024 à 19:31, Moebius a écrit :

For each and every of these points [here referred to with the
variable "x"]: NUF(x) = ℵ₀ .

>
I recognized lately that you use the wrong definition of NUF.
Here is the correct definition:
There exist NUF(x) unit fractions u, such that for all y >= x: u
< y.
Note that the order is ∃ u ∀ y.
NUF(x) = ℵ₀ for all x > 0 is wrong. NUF(x) = 1 for all x > 0
already
is wrong since there is no unit fraction smaller than all unit
fractions.
ℵ₀ unit fractions need ℵ₀*2ℵ₀ points above zero.

>
0->(...)->(1/1)
>
Contains infinite unit fractions.
>
0->(...)->(1/2)->(1/1)
>
Contains infinite unit fractions.
>
0->(...)->(1/3)->(1/2)->(1/1)
>
Contains infinite unit fractions.
>
However, (1/3)->(1/1) is finite and only has three unit fractions
expanded to:
>
(1/3)->(1/2)->(1/1)
>
Just like the following has four of them:
>
(1/4)->(1/3)->(1/2)->(1/1)
>
>
(0/1) is not a unit fraction. There is no smallest unit fraction.
However, the is a largest one at 1/1.
>
A interesting part that breaks the ordering is say well:
>
(1/4)->(1/2)
>
has two unit fractions. Then we can make it more fine grain:
>
(1/4)->(1/2) = ((1/8)+(1/8))->(1/4+1/4)
>
;^)

>
Unit fractions are ordered pairs, not infinite. :)

>
Real numbers are equivalence classes of sequences that are Cauchy,
and cardinals are equivalence classes of sets under
Cantor-Schroeder-Bernstein.

>
He didn't use real intervals this time, so I will treat this as
dealing
with a subset of rationals. He often uses a term like 'infinite unit
fractions' when he means 'infinitely many unit fractions' instead.
>

Rationals are equivalence classes of reduced fractions.

>
Need they be reduced, or are the reduced and/or proper fractions
chosen
from all of the proper and improper fractions?
>

In ZF's usual standard descriptive set theory, ....
>
>
Then, a common way to talk about this is the "real values",
that, the real-valued of course makes sure that there are
equivalence classes of integers, their values as rationals,
and their values as real numbers, keeping trichotomy or
otherwise the usual laws of arithmetic all among them,
where they're totally different sets of, you know, classes,
that though in the "real-valued" it's said that extensionality
is free and in fact given.
>
It's necessary to book-keep and disambiguate these things
in case the ignorant stop at a definition that though is
supported way above in the rest of the usual model assignment.

>
My view is that the rationals as embedded in the reals should act like
the rationals in Q, so why not use Q's ordered pairs instead of R to
reduce complications. It's like simplification in chess.

>
Yeah, the reduced fractions is a bit contrived, thanks.
>
Here "Dedekind cuts" or "partitions of rationals by reals"
don't exist except as "partitions of rationals by reals",
as with regards to the rationals being HUGE and all.
>
The other day I was reading about Cantor at Halle and Dirichlet
and the formulation and formalism of the Fourier series in the
Fourier-style analysis, where right before the Mengenlehre or
set theory, Cantor arrived at a way to show that the coefficients
of a Fourier series are unique. Then though the other day I was
reading a collection from a symposium after the '50's and '60's
in turbulence theory, where it's suggested that Phythian provides
a counterexample.
>
After Cauchy-Weierstrass then the Riemann then Lebesgue "what is
integrable" or measure theory and the measure problem and the
Dirichlet function (1 at rationals, 0 at irrationals, content?)
then there are lots of developments in the measure, the content,
the analytical character.
>
What's of interest of formalism is to provide rigor to derivations,
here it's so that the standard reals are equivalence classes of
sequences that are Cauchy, and that about the HUGE rationals and
that their real-values are trichtomous and dense in the reals,
they yet do not have the least-upper-bound property, which
the real numbers, of the linear continuum, do.
>
>

>
(Apocryphally there was already a development with regards to
the uniqueness of the coefficients of Fourier series.  Also
the anti-diagonal was discovered by du Bois-Reymond and various
other turns of thought in combinatorics and quantification were
already known.)
>
>

>
>
It's kind of like when people say "hey you know the
initial ordinal assignment is what we can say 'are'
cardinals", then it's like, "with the Continuum Hypothesis
being undecide-able and all, then there are and aren't
ordinals between what would be those cardinals by
their cardinals the ordinals", sort of establishing that
such a definition does and doesn't keep itself non-contradictory,
then that's more or template boiler-plate lines to
add to "rule 1: stop thinking and forget".
>
So, cardinals are equivalence classes of sets according
to function theory, which itself is a bit loose, here though
that it's battened down that there's always the Cartesian
courtesy comprehension, except a sort of special non-Cartesian
example, then that above that again is the long-line of
duBois-Reymond of all the expressions of real functions.
>
... Which only has the "complete" linear continuum to sit
on, these line-reals, field-reals, and signal-reals, "real-valued".
>

>
Of course Eudoxus is really great about the field and complete
ordered field, in terms of Aristotle's line-reals and field-reals.
>
Zeno's theories, ....
>
>

Have you studied Cohen's "Independence of the
Continuum Hypothesis"? It's an exercise in model theory,
in the mold of Skolem, who gave us extensions and
countable models of ZF, where in it, Paul Cohen makes
an argument for ordinals in model theory, thusly, that
M the model, is extra-ordinary, giving that the Continuum
Hypothesis, is independent, ZF and ZFC.
Now, you might know otherwise that he could give
contradictions either way (both ways).
So, you might thank him for invoking Mirimanoff and
the extra-ordinary, and with respect to Goedelian the
"incompleteness", of the theory, which instead intends
to reflect the "extra-ordinariness", of the theory, keeping
it from being closed, and not truly complete, regardless
that Russell's realm is just a fixed-point analysis with
not just implicitly yet explicitly a compactification of the naturals,
or with respect to its most naive model an inductive set,
why Cohen's "Independence of the Continuum Hypothesis"
is a complement to Skolem's generic extensions and collapses,
and Goedel's incompleteness theorems,
re-framed as acknowledging the extra-ordinary in theory,
and then, why an axiomless or universal theory, is these
ubiquitous ordinals and complete for Goedel.
Yeah, "Independence of the Continuum Hypothesis"
has sort of a surprise ending, to keep it from spoiling itself.
See rule zero, ....