On 02/18/2024 12:32 PM, Mild Shock wrote:
> Liar Liar Liar Liar Liar Liar Liar Liar
> He nowhere said that about Rule 85, you moron.
>
> A few rules that are famous:
>
> https://en.wikipedia.org/wiki/Rule_30
>
> https://en.wikipedia.org/wiki/Rule_90
>
> https://en.wikipedia.org/wiki/Rule_110
>
> https://en.wikipedia.org/wiki/Rule_184
>
> Ross Finlayson schrieb am Sonntag, 18. Februar 2024 um 19:27:38 UTC+1:
>> At the time, Wolfram's stated opinion was "Rule 85
>> is one of the most profound things in cellular automata."
>> https://atlas.wolfram.com/01/01/85/

At the time, that was pretty much all he said
about it, after digesting his 1500 page book,
"A New Kind of Science". He was like "Rule 85
is especially profound", that was the line.

I'm not a liar so I'd kindly ask you to retract that,
as I'm not a liar. So you wouldn't end up wrong.

Wolfram of course is pretty great, for lots of
people Mathematica and Maple V are the best mathematicians
they know, then I suppose there's Macsyma and other CAS's,
any number of which make analytical, symbolic derivations,
in the infinitesimal calculus the integral analysis.

On 02/18/2024 06:23 PM, Mild Shock wrote:
> Dream on quarkus.
>
> Ross Finlayson schrieb am Montag, 19. Februar 2024 um 01:06:43 UTC+1:
>> At the time, that was pretty much all he said
>> about it, after digesting his 1500 page book,
>> "A New Kind of Science". He was like "Rule 85
>> is especially profound", that was the line.

You know zero meters/second is infinity seconds/meter.

Yeah if you look up Rule 85 in all the tabulated characteristics
of the rules, it's got about the most zeros.

It's sort of a "universal attractor" bit.

Now Rule 110 more people note it "universal", but,
for "least action" or what, having a gradient is a thing.

It's kind of like one of those existence proofs,
"Rule 110 is like a Turing machine, doesn't say how",
while, "Rule 85 is like a ruler, sort of implements a draw".
I mean, you can build something with it, for example if
you had a language like BrainFund or a Connection Machine
or were just sort of abitrarily left-to-right, or, I
suppose, right-to-left, just gating wide in silicon.

That's what I recall hearing about, Rule 85.
Of course this was about 2003, so, search results
are going to vary. This was when I leafed through
and read the words in "A New Kind of Science".

Heh Burse said "hole bunch of physics without continuity".

I don't know what a quarkus is yet in Technicolour
it's quarks, all the way down. It's continuous, you see.

On 02/19/2024 06:34 AM, Eugene Prizler wrote:
> On Monday, February 19, 2024 at 12:17:18 PM UTC+1, Mild Shock wrote:
>
>> Liar Liar Liar Liar Liar Liar
>
> Ähnlich wie bei Mückenheim, wäre wohl auch in Deinem Fall der Besuch bei einem guten Psychiater anzuraten.
>

Hallo, ich weiss nicht wieviel Worten zu lernen, aber inhalt.

"To not be like Muckenheim..."

Aber, sind ihren nicht schwer eins "sock-puppet" ansammelt? Gesammelt?

"waere wohl auch in Deinem Fall der Besuch..."

[Don't go looking for trouble the same...]

"... bei einum guten Psychiater anzuraten."

[... dieser sock-puppets sind nicht klar.]

Well, yeah, among all the "Burse-Bots", I thought JG, WM, and
the other "stupid bots" and like the various "lamer flamers" that
if there's a real WM/MW he's kind of clueless these Usenet ongoings,
it's easy to anthropomorphize bots, especially when they're
models of the incorrigeable and "flaming howlers".

Danke bische fur diener Worten, ich hoffe zu abfahren besser
Gesprechern immer mit Gelernung. (Und, "mea culpa", kein klar.)

So, if I am like a clown sometimes, it is to share being human,
behaviors and norms are rather usual, and expectations of the
demonstration of comprehension of fundamental foundations the
theory, are expected to be same for all.

Usenet is special because free association, free expression,
free press, and these kinds of things.

Not a clown. A fraud. You are constantly emitting
fake news. Just try a minimum verification before
you claim something from the stomach.

The scientific approach to make a claim about
somebody (like Wolfram) or something (his book),
is to give a relevant citation, that you think supports

your claim! You can electronically
borrow the book free of charge:

A new kind of science by Wolfram, Stephen
Publication date 2002
https://archive.org/details/newkindofscience00wolf/

Ross Finlayson schrieb:
> So, if I am like a clown sometimes, it is to share being human,
> behaviors and norms are rather usual, and expectations of the
> demonstration of comprehension of fundamental foundations the
> theory, are expected to be same for all.

On 02/19/2024 09:25 AM, Mild Shock wrote:
>
> Not a clown. A fraud. You are constantly emitting
> fake news. Just try a minimum verification before
> you claim something from the stomach.
>
> The scientific approach to make a claim about
> somebody (like Wolfram) or something (his book),
> is to give a relevant citation, that you think supports
>
> your claim! You can electronically
> borrow the book free of charge:
>
> A new kind of science by Wolfram, Stephen
> Publication date 2002
> archive.org/details/newkindofscience00wolf/
>
>
> Ross Finlayson schrieb:
>> So, if I am like a clown sometimes, it is to share being human,
>> behaviors and norms are rather usual, and expectations of the
>> demonstration of comprehension of fundamental foundations the
>> theory, are expected to be same for all.
>
>

The Burse-bot's opinion is not accurate.

"Non-sequitur: non-sequiturs are eventually expected values ..."
It's called "don't let statistics make a liar".

Thanks I've already read it when it came out,
"ANKoS", also I have a physical copy, thanks.

Kind of like, what's the guy, "beell-ions and beell-ions",
Carl Sagan, he said one time or wrote in one of his books,
"educated man has a library".

I stand by everything I say and if it's erroneous
I would want to know and correct it.

It's like in this talk when I talk about
"the interior angles of a polyon is 2pi".
Then I found in Wertheimer's "Productive Thinking"
an exposition of same.

www.youtube.com/watch?v=tODnCZvVtLg&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=15

Rule 85 got special comment, right there among other
reversible rules.

So, like I'm reading Dunham's survey of mathematics,
and he really makes a strong point for a "postulate
of continuity".

Heh, you got a problem with that.

One time I was reading this book, and one of the
characters was in a position where he was more
intelligent than his surrounds, and he explains
that this causes fear thus antagonism, so what
he explained was that by being humorous and even
sometimes self-deprecating, then what resulted is
that it sort of defused otherwise the perceived
threat of intelligence.

That's a useful little fact of applied psychology,
but here it's that humor is irrelevant, and this
is not clowning, and really in the large that clowning
is considered non-sense if amusing, garbage if not,
then here that I by default point at the canon,
and "news" really is, ..., "news".

Relevance logic is a pretty great thing these days.

This thread: "Seven deadly sins of set theory",
can end.

Because, all you have to do, is actually according to
rhetorical forensics, follow on out the arguments
like any sufficient rational agent can do, and
the last word was a long time ago.

Dear readers, please have that I'll always try
to speak to your highest intellect, while providing
a contemporary, fair, telling, modicum of wit.

Infinity, continuity, resolving the paradoxes of
mathematical logic, there's lots there.

Your posts, Rossy Boy, are as colorful as the eternal rainbow
of the smoke comming from the chimeys of a big steel mill.

The dynamics you depict when your reproduce mathematics or physics
reflects your disposition between resigned and ambitious.

Credits: Keisuke Kinoshita, 1958.

Must be the influence of WWII, that these films are so
stiff, the performers have a stick up their asses.

What influenced Rossy Boy to become similarly nauseating?

Mild Shock schrieb:
>
> You got shipwrecked, and are ashore of a lost lagoon.
> But who are the coloured ones that steal the show?
>
> Credits: Paramount Pictures, 1958.
>
> Mild Shock schrieb am Montag, 19. Februar 2024 um 20:13:38 UTC+1:
>> Your posts are full of cynicism and galloping horses.
>> Your method is mechanical and desultory. Just
>> not good enough. Only interesting as a punch face.
>>
>> Ross Finlayson schrieb am Montag, 19. Februar 2024 um 19:14:20 UTC+1:
>>> Dear readers, please have that I'll always try
>>> to speak to your highest intellect, while providing
>>> a contemporary, fair, telling, modicum of wit.

On 02/19/2024 10:32 AM, Eugene Prizler wrote:
> On Monday, February 19, 2024 at 6:07:03 PM UTC+1, Ross Finlayson wrote:
>> On 02/19/2024 06:34 AM, Eugene Prizler wrote:
>>> On Monday, February 19, 2024 at 12:17:18 PM UTC+1, Mild Shock wrote:
>>>>
>>>> Liar Liar Liar Liar Liar Liar
>>>>
>>> Ähnlich wie bei Mückenheim, wäre wohl auch in Deinem Fall der Besuch bei einem guten Psychiater anzuraten.
>
> This means: "Similar to Mückenheim's case, a visit to a good psychiatrist would also be advisable in your case." (sort of).
>

Maybe it would help if you studied psychology, also,
to help you practice medicine on yourself. If you
study the history of medicine, one of the great precepts
is "do no harm" and another is "heal thyself".
"Primum non nocere / Cura te ipsem."

Studying Jung, Freud, the Gestalt, symbology,
the passions of figures of religion and history,
studying models of cognition, communication, and behavior,
and these kinds of things.

Wenn man wird Psychologie gestudiern, also man wird
hilfen Sie gefahren Medicine fur Man-sich.

Internalization is a continuing state of change in being.

Objectivity is a usual goal of a course in meditation,
contemplation, Theoria, ....

Myers-Briggs? Transactional? Child -> Parent -> Adult?

Questionality personnaire?

Objectivity, is an idea of having one's own first-class model,
of psychology, and being critical and not hypocritical, with
regards to being judgmental, with regards to being happy.

So, aber man schriebt "Aehnlich wie bie WM/MW, waere wohl
auch in Dienem Fall der Besuch bei einem guten Psychiatur
anzuraten", man hat auch schriebt "du bist in-saaane, Herr".

"Leistungsfahigkeit unter Beweis stellen"
Show your work

Mit freundlichen Gruesse fur Lesern,
mit beschieden Entschildigung fur Lesern,
nit hoffsnungsvolle Nachricht for Lesern.

Huffman

You only make it worse!

> There are roundabout arguments that, for example,
> the FINITE ORDINALS, as a set, consequently contain
> themselves, as an element. This is a direct
> compactness result.

If you want to have ordinals that contain themselves,
you need to mention an encoding. Because per se,
we understand by ordinal an order type.

There ware various encodings for finite ordinals around:
1) von Neuman encoding, based on succ(X) = X u {X} and 0 = {}
2) Zermelo encoding, bsaed on succ(X) = {X} and 0 = {}
3) Your Powerset idea, based on succ(X) = P(X) and 0 = {}

All 3 have the property that:

/* provable */
n in n+1 and n is finite

Proof:
case 1): n+1 = n u {n}, n in n+1 because n in {n}.
further succ(X) sendes an already finite set into a finite set.
case 2): n+1 = {n}, n in n+1 because n in {n}.
further succ(X) sendes an already finite set into a finite set.
case 3): n+1 = P(n), n in n+1 because n in P(n).
further succ(X) sendes an already finite set into a finite set.
Q.E.D.

But none has the property that omega = { n } contains
itself, the proof of contradiction applies irrelevant
of the encoding, it only makes use of the

notion finite and infinite:

/* provable */
~(omega in omega) & (Y in omega => Y finite)

Proof:
(Y in omega => Y finite) follows by the claim that
omega = { n }, i.e. the least set that contains all finite
ordinals in the corresponding encoding. If it would
contain something infinite it would not be the least

set that contains all finite ordinals, would have some
extra in it. Violating the very construction of omega from
the finite ordinals.
Q.E.D.

Ross Finlayson schrieb:
> On 02/19/2024 05:01 PM, Mild Shock wrote:
>> The contradiction is very easy:
>>
>> Lets say X is the set of all finite ordinals.
>>
>> - observe that X is an infinite ordinal.
>> - observe that if Y in X, then Y is a finite ordinal.
>> - hence if X in X it would be an infinite and finite ordinal at the
same time.
>> - an X cannot be infinite and finite at the same time.
>> Q.E.D:
>>
>> Ross Finlayson schrieb am Dienstag, 20. Februar 2024 um 00:04:06 UTC+1:
>>>>>> There are roundabout arguments that, for example, the finite
ordinals,
>>>>>> as a set, consequently contain themselves, as an element. This is a
>>>>>> direct compactness result.
>>> (Maybe that's just me.)
>
> Imagine if ordinals' proper model was that the successor
> was powerset, instead of just any old ordered pair.
>
> So, those together are the "sets that don't contain themselves",
> the sets of ordinals.
>
> Quantifying over those, results the "Russell set the ordinal",
> it contains itself.
>
> So here Y isn't necessarily a finite ordinal.
>
> Q.E.R.
>

You could use an encoding of finite ordinals
into infinite objects, like:

0 = omega, 1 = omega+1, etc..

Then my proof doesn't work so easily. You can then
use the regularity axiom, to show:

/* provable */
~(omega in omega)

Axiom of regularity
https://en.wikipedia.org/wiki/Axiom_of_regularity

Is this your A "paradox" is not a set in ZF?
In non-ZF you could aim at making omega a Quine atom:
https://en.wikipedia.org/wiki/Urelement#Quine_atoms

Or any other construction and encoding where you
would sneak in a set into itself.

Mild Shock schrieb:
> You only make it worse!
>
> > There are roundabout arguments that, for example,
> > the FINITE ORDINALS, as a set, consequently contain
> > themselves, as an element. This is a direct
> > compactness result.
>
> If you want to have ordinals that contain themselves,
> you need to mention an encoding. Because per se,
> we understand by ordinal an order type.
>
> There ware various encodings for finite ordinals around:
> 1) von Neuman encoding, based on succ(X) = X u {X} and 0 = {}
> 2) Zermelo encoding, bsaed on succ(X) = {X} and 0 = {}
> 3) Your Powerset idea, based on succ(X) = P(X) and 0 = {}
>
> All 3 have the property that:
>
> /* provable */
> n in n+1 and n is finite
>
> Proof:
> case 1): n+1 = n u {n}, n in n+1 because n in {n}.
> further succ(X) sendes an already finite set into a finite set.
> case 2): n+1 = {n}, n in n+1 because n in {n}.
> further succ(X) sendes an already finite set into a finite set.
> case 3): n+1 = P(n), n in n+1 because n in P(n).
> further succ(X) sendes an already finite set into a finite set.
> Q.E.D.
>
> But none has the property that omega = { n } contains
> itself, the proof of contradiction applies irrelevant
> of the encoding, it only makes use of the
>
> notion finite and infinite:
>
> /* provable */
> ~(omega in omega) & (Y in omega => Y finite)
>
> Proof:
> (Y in omega => Y finite) follows by the claim that
> omega = { n }, i.e. the least set that contains all finite
> ordinals in the corresponding encoding. If it would
> contain something infinite it would not be the least
>
> set that contains all finite ordinals, would have some
> extra in it. Violating the very construction of omega from
> the finite ordinals.
> Q.E.D.
>
>
> Ross Finlayson schrieb:
> > On 02/19/2024 05:01 PM, Mild Shock wrote:
> >> The contradiction is very easy:
> >>
> >> Lets say X is the set of all finite ordinals.
> >>
> >> - observe that X is an infinite ordinal.
> >> - observe that if Y in X, then Y is a finite ordinal.
> >> - hence if X in X it would be an infinite and finite ordinal at the
> same time.
> >> - an X cannot be infinite and finite at the same time.
> >> Q.E.D:
> >>
> >> Ross Finlayson schrieb am Dienstag, 20. Februar 2024 um 00:04:06 UTC+1:
> >>>>>> There are roundabout arguments that, for example, the finite
> ordinals,
> >>>>>> as a set, consequently contain themselves, as an element. This is a
> >>>>>> direct compactness result.
> >>> (Maybe that's just me.)
> >
> > Imagine if ordinals' proper model was that the successor
> > was powerset, instead of just any old ordered pair.
> >
> > So, those together are the "sets that don't contain themselves",
> > the sets of ordinals.
> >
> > Quantifying over those, results the "Russell set the ordinal",
> > it contains itself.
> >
> > So here Y isn't necessarily a finite ordinal.
> >
> > Q.E.R.
> >

Mild Shock schrieb am Dienstag, 20. Februar 2024 um 20:51:19 UTC+1:

> FromTheRafters schrieb am Donnerstag, 18. Januar 2024 um 13:44:43 UTC+1:
> > WM explained :

> > > Nevertheless ***all*** unit fractions have gaps between each other. There is
>
> > > no exception.
> >
> > Yes, there are gaps in Q+ with respect to the positive reals fractional
> > parts. Unless "***all*** unit fractions" means the set of unit
> > fractions,

Nonsense. I mean all points of unit fractions. After each one there is a
gap. Hence NUF(x) can only increase by 1 in every case.

Regards, WM

On 02/13/2024 04:39 PM, Ross Finlayson wrote:
> On 02/13/2024 03:47 PM, Jim Burns wrote:
>> On 2/13/2024 4:38 PM, Ross Finlayson wrote:
>>> On 02/13/2024 01:15 PM, Jim Burns wrote:
>>>> On 2/13/2024 2:05 PM, Ross Finlayson wrote:
>>
>>>>> What I have in mind here is that
>>>>> the continuous domain has sorts of topologies, that
>>>>> make it so, that every open set is also
>>>>> a continuous domain, and, to make it so, that
>>>>> only two open sets that are "contiguous", are in
>>>>> this sort of topology.
>>>>
>>>> I vaguely recall from a topology course that
>>>> my instructor blended "contiguous" and "continuous"
>>>> into "contiuous" for some theorems.
>>>>
>>>> There might be a notion of "contiguous" that's
>>>> useful here. If there is, I'd like to know it.
>>>>
>>>> Please define "contiguous" for the context
>>>> in which you are using it here.
>>>>
>>>> Googling "contiguous" or "contiuous"
>>>> could in principle resolve my quandary,
>>>> but there are too many other uses.
>>>> I tried it and got no joy.
>>>> (For "contiuous", too many typos of continuous")
>>>
>>> Contiguous basically means "in all neighborhoods
>>> together".
>>
>> So, for 𝒪 = {∅,X}
>> all points in X are contiguous to each other?
>> Do I have that right?
>>
>> This seems related to Hausdorff space.
>> | for any two distinct points, there exist
>> | neighbourhoods of each that are disjoint from each other.
>> |
>> https://en.wikipedia.org/wiki/Hausdorff_space
>>
>> With your interest in defining ℝ,
>> that seems right up your (RF's) alley.
>> | Of the many separation axioms that can be imposed on
>> | a topological space, the "Hausdorff condition" (T2) is
>> | the most frequently used and discussed.
>> | It implies the uniqueness of limits of
>> | sequences, nets, and filters
>> |
>> ibid.
>>
>> ----
>> Side note:
>> You know what's a connected topology?
>> 𝒪 = {∅,X} is a connected topology.
>>
>> Condition #4 is vacuously satisfied.
>> 4. B ∈ 𝒪\{∅,X} ⇒ X\B ∉ 𝒪
>>
>> But 𝒪 = {∅,X} isn't Hausdorff.
>>
>> It makes me go "Hmmm."
>>
>>> Not sure about "contiuous", "conti?uous".
>>
>> I would not be surprised if "contiuous"
>> was an invention of my instructor,
>> and not particularly widely used.
>>
>>> Mathematics: both sides to anything.
>>
>> ...unless only one side exists.
>>
>> Mathematicians do agree on some things.
>> But those seem to only be things upon which
>> there is no rational alternative to agreeing.
>>
>>
>
>
>
>
> Consider for example the matrix product.
> It's well known that aligning m x p and p x n
> results an m x n matrix, its product.
>
> Yet, there are as well generalized products,
> for matrices of any dimensions, and as well
> inner and outer products, for example the
> determinant, or as with regards to bivectors,
> for examples, these fuller and scalar and
> inner and outer: "complementary duals",
> the duals of the complements.
>
>
> There's as well sort of generalized inverses,
> with regards to principal values p.v. for example,
> as with regards for example to complex numbers,
> or for example the postive and negative roots of even
> and odd powers of non-negative numbers.
>
> Again this is as with regards to "complementary duals",
> and about the usual notion of the inverse as axiomatic.
>
> Another key notion is for the deconstructivist account,
> for as what results what was elementary, for example
> atoms, has "sub-atomic particles", where for example
> the very theory of atoms is a deconstructivist account,
> and the very theory of axiomatic set theory making
> a descriptive set theory a model of models of mathematics,
> is as of deconstructive and constructivist accounts,
> for the constructivist side that it is structure, and
> for the intuitionist side that there's more than one
> way to look at things, then as with regards to the
> results of existence, distinctness, and uniqueness.
>
> Here then this "continuous topology" notion is rather
> for a sort of inner product already, if a wider and more
> general class of structures the topologies.
>
>
> Here, pair-wise contiguity is sharing all neighborhoods,
> while transitive contiguity has that contiguity is associative.
> It seems an example of a usual property like transitivity
> that happens to also model and be modeled by an other
> usual property like associativity, sort of in simile to how
> here the notion of a continuous domain's continuous topology,
> has that composing and decomposing them is just like as
> of the regions or analytical regions of the surfaces or bodies
> they represent, being together, and meeting.
>
>
> So, I think this is a good idea and part of mathematics.
> It's a bit more involved than topology which is connected
> enough to establish usual results, while it provides for
> various certain stronger and more direct results.
>
> When I think of algebraic geometry, I sort of have the
> geometry first, because it's Euclidean and there's the
> Cartesian and R^N. The algebras are a much richer
> milieu for the deconstructivist and analytical accounts,
> and models of all the products, whose images, are geometric.
>
> So, let's consider further this notion of a definition of
> "continuous topology", and why it's a central and primary
> feature in the structure of the objects of mathematics.
>
>

Here then let us consider furthermore this notion
of a "continuous topology" it being embodying contiguity,
and of the sort of least-tenuous "tenuous topology",
as with regards to the myriad kinds of "torsional topology",
that vis-a-vis the initial and final and vacuous and discrete
and indiscrete sorts of topologies, arises these particular
varieties that do and don't relay and maintain the notion
of the neighborhoods, and neighborliness, of points,
in contiguity, among most usual abstractions that represent
the surfaces of the manifolds of the geometry.

On 02/13/2024 11:46 AM, Ross Finlayson wrote:
> On 02/13/2024 11:31 AM, Ross Finlayson wrote:
>> On 02/13/2024 11:23 AM, Ross Finlayson wrote:
>>> On 02/13/2024 08:28 AM, Fritz Feldhase wrote:
>>>> On Tuesday, February 13, 2024 at 11:36:15 AM UTC+1, WM wrote:
>>>>
>>>>> not even two [unit fractions], let alone more, can lie before every x
>>>>> > 0.
>>>>
>>>> Mückenheim, NIEMAND AUSSER DIR behauptet, dass dem so wäre.
>>>>
>>>> Behauptet wird (weil es so ist): "For exery x > 0 there are infinitely
>>>> many unit fractions which are smaller than x." (1)
>>>>
>>>> Es wird aber NICHT behauptet (außer in Deiner Wahnwelt): "There are
>>>> infinitely many unit fractions that are smaller than every x > 0." (2)
>>>>
>>>> o Hinweis 1:
>>>>
>>>> (1) kann so formalisiert werden:
>>>>
>>>> (1') A x e (0, oo): E^oo u e {1/n : n e IN}: u < x. (wahr)
>>>>
>>>> (2) kann so formalisiert werden:
>>>>
>>>> (2') E^oo u e {1/n : n e IN}: A x e (0, oo): u < x. (falsch)
>>>>
>>>> Wie Du richtig beobachtet hast, gibt es NICHT EINMAL 2 unit fractions,
>>>> die kleiner sind als alle x > 0. Also erst recht nicht unendlich viele
>>>> solche unit fractions. Tatsächlich gibt es NICHT EINMAL éinen
>>>> Stammbruch, der kleiner ist als alle x > 0. (Siehe Hinweis 3.)
>>>>
>>>> o Hinweis 2: (1') does not imply (2'). Hence beware of the "quantifier
>>>> shift fallacy"!
>>>>
>>>> See: https://en.wikipedia.org/wiki/Quantifier_shift
>>>> and:
>>>> https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100357607
>>>>
>>>>
>>>>
>>>>
>>>> o Hinweis 3: Tatsächlich kann leicht gezeigt werden, dass ~E^oo u e
>>>> {1/n : n e IN}: A x e (0, oo): u < x gilt.
>>>>
>>>> [ Der einfachste Beweis geht wohl so, dass man zuerst ~E u e {1/n : n
>>>> e IN}: A x e (0, oo): u < x zeigt. Das impliziert dann ~E^oo u e {1/n
>>>> : n e IN}: A x e (0, oo): u < x. Zu zeigen, dass ~E u e {1/n : n e
>>>> IN}: A x e (0, oo): u < x gilt, ist trivial: Gäbe es einen Stammbruch
>>>> u, so dass für alle x in (0, oo) u < x gilt, dann würde daraus u < u
>>>> folgen. ]
>>>>
>>>
>>>
>>> Strecke-Reellen
>>> Koerper-Reellen
>>> Signale-Reellen
>>>
>>>
>>> Drei-viel Kontinuumen, Eins.
>>>
>>> Strecke-Reellen:
>>>
>>> ran(f), f(n) = n/d, 0 <= n <= d, d -> oo, 1 e ran(f), f(m+1) > f(m)
>>>
>>> Koerper-Reellen
>>>
>>> Weierstrass-schen, Cauchy-klassen
>>>
>>> Signale-Reellen
>>>
>>> Nyquist-schen
>>>
>>>
>>> Es gibt besser Namen, ....
>>>
>>>
>>> Die lange-linie, hier es ist duBois-Reymond-schen
>>> fur die Raume der Reellefunktionen.
>>>
>>> Sorryy, mein Wortschatz est mehr hohl als voll.
>>>
>>>
>>> 0 -> 1
>>> hohl -> voll
>>>
>>> MW/WM est un Zwerg, hier ist ein Hengst.
>>>
>>>
>>>
>>
>> Teile Zeile ad infinitum: ha, es klar!
>>
>> Das Unendlichkeit aber das Endlichkeit,
>> for ein Definitionsbereich, es fur ein
>> eundeutige vollstandigen Induktion,
>> ein eundeutige vollstandigen Deduktion,
>> fur jedes Teile Zeile von Eins.
>>
>> Fur jedes!
>>
>>
>
> Durch van de Waerden's "Moderne Algebra":
>
> Neben dem "Beweis durch vollstandige Induktion"
> is seinem beidem Formen gibt es noch die "Definition
> (oder Konstruktion) durch vollstandige Induktion".
> Man will jeder natuerlichen Zahl x ein neues
> Object \phi(x) zuordnen, und man gibt ein System
> von "rekursiven Bestimmungsrelationen vor, die den
> Funktionswert \phi(n) jeweis mit den vorangehenden
> Werten \phi(m) (m < n) verknupfen sollen. Angenommen
> wird, dass deise Relationen jeweils den Wert \phi(n)
> eindeutig bestimmen, so alle \phi(m) (m < n) gegeben
> sind und untereinander die gegebenen Relationen erfuellen.
>
> ...
>
> Nun wird behauptet: Unter den angegebenen Voraussetsungen
> gibt es eine under nur eine Funktion \phi(x), deren Werte
> die gegebenen Relationen erfuellen.
>
> -- Modern Algebra, "Copyright vested in the Alien Property Custodian, 1943"
>
>
> So, zo, zo, ..., wohin deruber das aussetsungen Deduktion?
> Hier es muss sein durch aber hat gesammelt, ....
>

Hallo, von sci.math es gibt ein Amerikaner, wohin, man wird
die Worten finden, zo, ...

n/d Kontinuumsgrenze, [0,1], "Strecke-Reellen"
n/d Aequivalenzklassen, nichnegative rational Zahlen

Hier ich have nur einmal Georg Cantor und Abzahlen
und der Zuordnungs gestudiert.

Danke Fritz - danke fur Anglishsprechern

Ross Finlayson, https youtube @rossfinlayson

On 02/23/2024 11:08 AM, Ross Finlayson wrote:
> On 02/13/2024 04:39 PM, Ross Finlayson wrote:
>> On 02/13/2024 03:47 PM, Jim Burns wrote:
>>> On 2/13/2024 4:38 PM, Ross Finlayson wrote:
>>>> On 02/13/2024 01:15 PM, Jim Burns wrote:
>>>>> On 2/13/2024 2:05 PM, Ross Finlayson wrote:
>>>
>>>>>> What I have in mind here is that
>>>>>> the continuous domain has sorts of topologies, that
>>>>>> make it so, that every open set is also
>>>>>> a continuous domain, and, to make it so, that
>>>>>> only two open sets that are "contiguous", are in
>>>>>> this sort of topology.
>>>>>
>>>>> I vaguely recall from a topology course that
>>>>> my instructor blended "contiguous" and "continuous"
>>>>> into "contiuous" for some theorems.
>>>>>
>>>>> There might be a notion of "contiguous" that's
>>>>> useful here. If there is, I'd like to know it.
>>>>>
>>>>> Please define "contiguous" for the context
>>>>> in which you are using it here.
>>>>>
>>>>> Googling "contiguous" or "contiuous"
>>>>> could in principle resolve my quandary,
>>>>> but there are too many other uses.
>>>>> I tried it and got no joy.
>>>>> (For "contiuous", too many typos of continuous")
>>>>
>>>> Contiguous basically means "in all neighborhoods
>>>> together".
>>>
>>> So, for 𝒪 = {∅,X}
>>> all points in X are contiguous to each other?
>>> Do I have that right?
>>>
>>> This seems related to Hausdorff space.
>>> | for any two distinct points, there exist
>>> | neighbourhoods of each that are disjoint from each other.
>>> |
>>> https://en.wikipedia.org/wiki/Hausdorff_space
>>>
>>> With your interest in defining ℝ,
>>> that seems right up your (RF's) alley.
>>> | Of the many separation axioms that can be imposed on
>>> | a topological space, the "Hausdorff condition" (T2) is
>>> | the most frequently used and discussed.
>>> | It implies the uniqueness of limits of
>>> | sequences, nets, and filters
>>> |
>>> ibid.
>>>
>>> ----
>>> Side note:
>>> You know what's a connected topology?
>>> 𝒪 = {∅,X} is a connected topology.
>>>
>>> Condition #4 is vacuously satisfied.
>>> 4. B ∈ 𝒪\{∅,X} ⇒ X\B ∉ 𝒪
>>>
>>> But 𝒪 = {∅,X} isn't Hausdorff.
>>>
>>> It makes me go "Hmmm."
>>>
>>>> Not sure about "contiuous", "conti?uous".
>>>
>>> I would not be surprised if "contiuous"
>>> was an invention of my instructor,
>>> and not particularly widely used.
>>>
>>>> Mathematics: both sides to anything.
>>>
>>> ...unless only one side exists.
>>>
>>> Mathematicians do agree on some things.
>>> But those seem to only be things upon which
>>> there is no rational alternative to agreeing.
>>>
>>>
>>
>>
>>
>>
>> Consider for example the matrix product.
>> It's well known that aligning m x p and p x n
>> results an m x n matrix, its product.
>>
>> Yet, there are as well generalized products,
>> for matrices of any dimensions, and as well
>> inner and outer products, for example the
>> determinant, or as with regards to bivectors,
>> for examples, these fuller and scalar and
>> inner and outer: "complementary duals",
>> the duals of the complements.
>>
>>
>> There's as well sort of generalized inverses,
>> with regards to principal values p.v. for example,
>> as with regards for example to complex numbers,
>> or for example the postive and negative roots of even
>> and odd powers of non-negative numbers.
>>
>> Again this is as with regards to "complementary duals",
>> and about the usual notion of the inverse as axiomatic.
>>
>> Another key notion is for the deconstructivist account,
>> for as what results what was elementary, for example
>> atoms, has "sub-atomic particles", where for example
>> the very theory of atoms is a deconstructivist account,
>> and the very theory of axiomatic set theory making
>> a descriptive set theory a model of models of mathematics,
>> is as of deconstructive and constructivist accounts,
>> for the constructivist side that it is structure, and
>> for the intuitionist side that there's more than one
>> way to look at things, then as with regards to the
>> results of existence, distinctness, and uniqueness.
>>
>> Here then this "continuous topology" notion is rather
>> for a sort of inner product already, if a wider and more
>> general class of structures the topologies.
>>
>>
>> Here, pair-wise contiguity is sharing all neighborhoods,
>> while transitive contiguity has that contiguity is associative.
>> It seems an example of a usual property like transitivity
>> that happens to also model and be modeled by an other
>> usual property like associativity, sort of in simile to how
>> here the notion of a continuous domain's continuous topology,
>> has that composing and decomposing them is just like as
>> of the regions or analytical regions of the surfaces or bodies
>> they represent, being together, and meeting.
>>
>>
>> So, I think this is a good idea and part of mathematics.
>> It's a bit more involved than topology which is connected
>> enough to establish usual results, while it provides for
>> various certain stronger and more direct results.
>>
>> When I think of algebraic geometry, I sort of have the
>> geometry first, because it's Euclidean and there's the
>> Cartesian and R^N. The algebras are a much richer
>> milieu for the deconstructivist and analytical accounts,
>> and models of all the products, whose images, are geometric.
>>
>> So, let's consider further this notion of a definition of
>> "continuous topology", and why it's a central and primary
>> feature in the structure of the objects of mathematics.
>>
>>
>
>
> Here then let us consider furthermore this notion
> of a "continuous topology" it being embodying contiguity,
> and of the sort of least-tenuous "tenuous topology",
> as with regards to the myriad kinds of "torsional topology",
> that vis-a-vis the initial and final and vacuous and discrete
> and indiscrete sorts of topologies, arises these particular
> varieties that do and don't relay and maintain the notion
> of the neighborhoods, and neighborliness, of points,
> in contiguity, among most usual abstractions that represent
> the surfaces of the manifolds of the geometry.
>
>

Jakubowicz points out that Aristotle introduces a notion of contiguity.
It's quite a great survey of Aristotle on continuity and the bibliography.

A Problem in Aristotle's Continuity Theory, Sammy T. Jakubowicz
The Trouble With Touching: A Problem in Aristotle's Continuity Theory

https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=de4554185846fe34baf94cbb1ce77530ebe32ec7

This is where, these days, people will often find "Aristotle must be our
master so what he says must be the same as what is our standard today",
yet that is mostly Eudoxus about the complete ordered field, that
Aristotle's continuum also includes notions of the contiguity of points,
so, it is so that both modern echoes of the coat-tailing wall-papering
sort dont' say much at all and indeed obscure the fuller dialectic, and,
such notions as the "divided, ad infinitum" follows the "infinitely
divisible" concept, explored at least since Aristotle.

So, when I say that "line-reals are as an Aristotle's continuum", it is
like so.

Then, this notion of "continuous topology" is a great complement
to the notion of "continuous domain" that we have here today.

On 04/23/2024 07:05 PM, Ross Finlayson wrote:
> On 02/23/2024 11:08 AM, Ross Finlayson wrote:
>> On 02/13/2024 04:39 PM, Ross Finlayson wrote:
>>> On 02/13/2024 03:47 PM, Jim Burns wrote:
>>>> On 2/13/2024 4:38 PM, Ross Finlayson wrote:
>>>>> On 02/13/2024 01:15 PM, Jim Burns wrote:
>>>>>> On 2/13/2024 2:05 PM, Ross Finlayson wrote:
>>>>
>>>>>>> What I have in mind here is that
>>>>>>> the continuous domain has sorts of topologies, that
>>>>>>> make it so, that every open set is also
>>>>>>> a continuous domain, and, to make it so, that
>>>>>>> only two open sets that are "contiguous", are in
>>>>>>> this sort of topology.
>>>>>>
>>>>>> I vaguely recall from a topology course that
>>>>>> my instructor blended "contiguous" and "continuous"
>>>>>> into "contiuous" for some theorems.
>>>>>>
>>>>>> There might be a notion of "contiguous" that's
>>>>>> useful here. If there is, I'd like to know it.
>>>>>>
>>>>>> Please define "contiguous" for the context
>>>>>> in which you are using it here.
>>>>>>
>>>>>> Googling "contiguous" or "contiuous"
>>>>>> could in principle resolve my quandary,
>>>>>> but there are too many other uses.
>>>>>> I tried it and got no joy.
>>>>>> (For "contiuous", too many typos of continuous")
>>>>>
>>>>> Contiguous basically means "in all neighborhoods
>>>>> together".
>>>>
>>>> So, for 𝒪 = {∅,X}
>>>> all points in X are contiguous to each other?
>>>> Do I have that right?
>>>>
>>>> This seems related to Hausdorff space.
>>>> | for any two distinct points, there exist
>>>> | neighbourhoods of each that are disjoint from each other.
>>>> |
>>>> https://en.wikipedia.org/wiki/Hausdorff_space
>>>>
>>>> With your interest in defining ℝ,
>>>> that seems right up your (RF's) alley.
>>>> | Of the many separation axioms that can be imposed on
>>>> | a topological space, the "Hausdorff condition" (T2) is
>>>> | the most frequently used and discussed.
>>>> | It implies the uniqueness of limits of
>>>> | sequences, nets, and filters
>>>> |
>>>> ibid.
>>>>
>>>> ----
>>>> Side note:
>>>> You know what's a connected topology?
>>>> 𝒪 = {∅,X} is a connected topology.
>>>>
>>>> Condition #4 is vacuously satisfied.
>>>> 4. B ∈ 𝒪\{∅,X} ⇒ X\B ∉ 𝒪
>>>>
>>>> But 𝒪 = {∅,X} isn't Hausdorff.
>>>>
>>>> It makes me go "Hmmm."
>>>>
>>>>> Not sure about "contiuous", "conti?uous".
>>>>
>>>> I would not be surprised if "contiuous"
>>>> was an invention of my instructor,
>>>> and not particularly widely used.
>>>>
>>>>> Mathematics: both sides to anything.
>>>>
>>>> ...unless only one side exists.
>>>>
>>>> Mathematicians do agree on some things.
>>>> But those seem to only be things upon which
>>>> there is no rational alternative to agreeing.
>>>>
>>>>
>>>
>>>
>>>
>>>
>>> Consider for example the matrix product.
>>> It's well known that aligning m x p and p x n
>>> results an m x n matrix, its product.
>>>
>>> Yet, there are as well generalized products,
>>> for matrices of any dimensions, and as well
>>> inner and outer products, for example the
>>> determinant, or as with regards to bivectors,
>>> for examples, these fuller and scalar and
>>> inner and outer: "complementary duals",
>>> the duals of the complements.
>>>
>>>
>>> There's as well sort of generalized inverses,
>>> with regards to principal values p.v. for example,
>>> as with regards for example to complex numbers,
>>> or for example the postive and negative roots of even
>>> and odd powers of non-negative numbers.
>>>
>>> Again this is as with regards to "complementary duals",
>>> and about the usual notion of the inverse as axiomatic.
>>>
>>> Another key notion is for the deconstructivist account,
>>> for as what results what was elementary, for example
>>> atoms, has "sub-atomic particles", where for example
>>> the very theory of atoms is a deconstructivist account,
>>> and the very theory of axiomatic set theory making
>>> a descriptive set theory a model of models of mathematics,
>>> is as of deconstructive and constructivist accounts,
>>> for the constructivist side that it is structure, and
>>> for the intuitionist side that there's more than one
>>> way to look at things, then as with regards to the
>>> results of existence, distinctness, and uniqueness.
>>>
>>> Here then this "continuous topology" notion is rather
>>> for a sort of inner product already, if a wider and more
>>> general class of structures the topologies.
>>>
>>>
>>> Here, pair-wise contiguity is sharing all neighborhoods,
>>> while transitive contiguity has that contiguity is associative.
>>> It seems an example of a usual property like transitivity
>>> that happens to also model and be modeled by an other
>>> usual property like associativity, sort of in simile to how
>>> here the notion of a continuous domain's continuous topology,
>>> has that composing and decomposing them is just like as
>>> of the regions or analytical regions of the surfaces or bodies
>>> they represent, being together, and meeting.
>>>
>>>
>>> So, I think this is a good idea and part of mathematics.
>>> It's a bit more involved than topology which is connected
>>> enough to establish usual results, while it provides for
>>> various certain stronger and more direct results.
>>>
>>> When I think of algebraic geometry, I sort of have the
>>> geometry first, because it's Euclidean and there's the
>>> Cartesian and R^N. The algebras are a much richer
>>> milieu for the deconstructivist and analytical accounts,
>>> and models of all the products, whose images, are geometric.
>>>
>>> So, let's consider further this notion of a definition of
>>> "continuous topology", and why it's a central and primary
>>> feature in the structure of the objects of mathematics.
>>>
>>>
>>
>>
>> Here then let us consider furthermore this notion
>> of a "continuous topology" it being embodying contiguity,
>> and of the sort of least-tenuous "tenuous topology",
>> as with regards to the myriad kinds of "torsional topology",
>> that vis-a-vis the initial and final and vacuous and discrete
>> and indiscrete sorts of topologies, arises these particular
>> varieties that do and don't relay and maintain the notion
>> of the neighborhoods, and neighborliness, of points,
>> in contiguity, among most usual abstractions that represent
>> the surfaces of the manifolds of the geometry.
>>
>>
>
>
>
>
> Jakubowicz points out that Aristotle introduces a notion of contiguity.
> It's quite a great survey of Aristotle on continuity and the bibliography.
>
> A Problem in Aristotle's Continuity Theory, Sammy T. Jakubowicz
> The Trouble With Touching: A Problem in Aristotle's Continuity Theory
>
> https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=de4554185846fe34baf94cbb1ce77530ebe32ec7
>
>
> This is where, these days, people will often find "Aristotle must be our
> master so what he says must be the same as what is our standard today",
> yet that is mostly Eudoxus about the complete ordered field, that
> Aristotle's continuum also includes notions of the contiguity of points,
> so, it is so that both modern echoes of the coat-tailing wall-papering
> sort dont' say much at all and indeed obscure the fuller dialectic, and,
> such notions as the "divided, ad infinitum" follows the "infinitely
> divisible" concept, explored at least since Aristotle.
>
> So, when I say that "line-reals are as an Aristotle's continuum", it is
> like so.
>
>
> Then, this notion of "continuous topology" is a great complement
> to the notion of "continuous domain" that we have here today.
>