Re: how (Aristotle says "potential is actual and actual is potential")

Liste des GroupesRevenir à s math 
Sujet : Re: how (Aristotle says "potential is actual and actual is potential")
De : ross.a.finlayson (at) *nospam* gmail.com (Ross Finlayson)
Groupes : sci.math
Date : 02. Sep 2024, 18:08:23
Autres entêtes
Message-ID : <53Sdnd83Y8aZbEj7nZ2dnZfqn_WdnZ2d@giganews.com>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
User-Agent : Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0
On 05/27/2024 07:10 PM, Ross Finlayson wrote:
On 05/27/2024 01:08 PM, Ross Finlayson wrote:
On 05/23/2024 10:45 AM, Jim Burns wrote:
On 5/23/2024 9:50 AM, Ross Finlayson wrote:
On 05/22/2024 11:33 AM, Jim Burns wrote:
>
[...]
>
The partitions of the rationals,
>
{P⊆ℚ:∅≠Pᣔ<ᘁPᣔ<ᣔℚ\P≠∅}
>
have a smaller cardinal, than
the powerset of the rationals, where
>
You're accepting the Continuum Hypothesis.
CH is independent of ZFC. [Gödel,Cohen]
>
More to the point,
{P⊆ℚ:∅≠Pᣔ<ᘁPᣔ<ᣔℚ\P≠∅} has
a larger cardinal than ℚ
>
Upthread:
>
Subsets of rationals in rationals are rationals.
Furthermore, the partitions of rationals are countable.
</WM>
Date: Tue, 21 May 2024 13:06:23 -0700
>
The partitionsᴿꟳ aren't countable.
>
⎛ {P⊆ℚ:∅≠Pᣔ<ᘁPᣔ<ᣔℚ\P≠∅} is the set of
⎜ partitionsᴿꟳ == open.foresplits of ℚ
⎜ P⊆ℚ  subset of rationals P
⎜ ∅≠P  nonempty P
⎜ Pᣔ<ᘁP  ⇔  ∀r ∈ P: ∃s ∈ P: r<s
⎜ Pᣔ<ᣔℚ\P  ⇔  ∀r ∈ P: ∀s ∈ ℚ\P: r<s
⎝ ℚ\P≠∅  nonempty ℚ\P
>
according to the laws of arithmetic,
a partition of the set of rationals into two,
where the partitioning's partitions are
those above a value
>
That's not {P⊆ℚ:∅≠Pᣔ<ᘁPᣔ<ᣔℚ\P≠∅}
>
Pᣔ<ᣔℚ\P  ⇔  ∀r ∈ P: ∀s ∈ ℚ\P: r<s
Each of P < each of ℚ\P
There is no requirement for
a rational point between P and ℚ\P
and examples of P without it are easy to find.
>
according to
the laws of trichotomy in arithmetic,
and below, respectively,
there aren't more partitionings than
there are rationals.
>
Pᣔ<ᣔℚ\P
>
Of course,
each partitioning is only distinct,
according to trichotomy,
by have at least one value unique for itself
not shared with all less than it,
>
No.
Pᣔ<ᘁP  ⇔  ∀r ∈ P: ∃s ∈ P: r<s
>
There are _at least_ partitionsᴿꟳ {q ∈ ℚ: q<r}
for each rational r
>
For each rational qᵢ in partitionᴿꟳ Pₖ
Pₖᣔ<ᘁPₖ
qᵢ is not max Pₖ
qᵢ < rⱼ ∈ Pₖ
>
Pⱼ = {q ∈ Q: q < rⱼ}
qᵢ ∈ Pⱼ
qᵢ is shared between Pₖ and Pⱼ among others.
>
There is no unshared element in any partitionᴿꟳ
>
That's not the same as saying that
☠ there are different partitionsᴿꟳ which
☠ have all the same elements.
>
and at least one value unique for itself
not shared all all greater than it.
>
The partitionsᴿꟳ are totally.ordered by
the subset relation ⊆
Pⱼ ≤ Pₖ  ⇔  Pⱼ ⊆ Pₖ
>
There are no elements of Pⱼ not shared with
greater Pₖ  because subset.
>
>
>
>
Not shared with both partitions..., i.e., whether
bounded-above or bounded-below it's same.
>
Dedekind cuts are partitionings of the reals by
the complete ordered field already, which eventually
does include transcendental numbers which neither
the rationals nor the algebraics do.
>
The rationals do not have the least-upper-bound property.
>
Anyways there are at least three models of continuous domains,
one of which is line-reals empty-to-full 0-to-and-through-1,
the other the familiar complete ordered field the field-reals,
and the signal-reals, which help illustrate that the rationals
altogether are huge, then as with regards to their completeness.
>
Dedekind completeness, if one only associates it with the
complete-ordered-field least-upper-bound property, and not
just otherwise any old least-upper-bound property for the
completeness or gaplessness, is just one aspect of the
more replete nature of the linear continuum its models
of continuous domains, thus in a wider sense it's insufficient
to model continuity quite properly, and Dedekind cuts are
just a halfway-sound reflection on what's otherwise courtesy
the sets dense and equidistributed, as by their values,
according to trichotomy, while no where continuous, whose
complements also have the same properties according to
the laws of arithmetic as about trichotomy.
>
Then the interesting features of topology that arrive at
doubling- and halving- the -measures and -spaces, courtesy
there existing these at least three definitions of continuity,
are fuller and better mathematics, and quite primarily together.
>
>
Then that Aristotle definitely reflects on both line-reals
and field-reals, and also reflects on there not being a
distinction the very-much-so actual and potential, infinity,
has that these three definitions of continuity all live
together consistently according to a simple-enough result
about function theory and about Cantor-Schroeder-Bernstein
theorem, and as well Cantor-Schroeder them, that there are
spaces of functions as modeled in set theory that have
some particular non-Cartesian (or, "only Cartesian")
functions, that it's nice, altogether, and primary,
thusly voiding what's otherwise the hypocrisy of
modeling line-reals and signal-reals much later in
the analysis and topology, which of course one may find
in many examples, in the applied, today, as since
laboriously built all way above rudimentary topological analysis.
>
>
Where they should be, because where they are.
>
>
It seems you've axiomatized least-upper-bound for
partitionings of rationals then made a loop.
>
Of course, least-upper-bound and measure 1.0 are
"axioms", stipulations, of the usual modern standard
formalism.
>
As is so well reflected in words. (Or terms.)
>
>
>
Ah, this goes a few hours without a reply,
I'll wait patiently all content and mirthful.
>
And right.
>
>
That Aristotle says potential is actual and actual is potential, ....

Date Sujet#  Auteur
2 Sep 24 o Re: how (Aristotle says "potential is actual and actual is potential")1Ross Finlayson

Haut de la page

Les messages affichés proviennent d'usenet.

NewsPortal