On Wednesday, October 11, 2023 at 6:38:08 AM UTC+3, Ross Finlayson wrote:
> On Tuesday, October 10, 2023 at 8:10:31 PM UTC-7, bassam karzeddin wrote:
> > On Monday, October 9, 2023 at 11:56:20 PM UTC+3, Jim Burns wrote:
> > > On 10/9/2023 8:27 AM, bassam karzeddin wrote:
> > > > On Monday, October 9, 2023
> > > > at 1:02:10 PM UTC+3, bassam karzeddin wrote:
> > > >> On Monday, October 9, 2023
> > > >> at 7:48:28 AM UTC+3, Jim Burns wrote:
> > > >>> On 10/8/2023 12:56 PM, bassam karzeddin wrote:
> > > >>>> On Saturday, August 8, 2020
> > > >>>> at 11:27:08 AM UTC+3, bassam karzeddin wrote:
> > >
> > > > what is truly a real number?
> > > A real number x ∈ ℝ is
> > > a rational number x ∈ ℚ or
> > > a point x between a split F‖H of ℚ
> > > F ᣔ< x <ᣔ H
> > > F∪H = ℚ\{x}
> > > > And assuming in good faith that
> > > > a real number which is
> > > > not a constructible number exists,
> > > > then mathematicians are globally
> > > > revealed [requested?] to present
> > > > only one of them
> > > > either numerically or Geometrically
> > > > but Exactly [...]
> > >
> > > Points-between-splits are geometry.
> > >
> > > Without all points-between-splits,
> > > some continuous curves cross but
> > > don't intersect,
> > > which is not geometry.
> > >
> > >
> > > Points-between-splits are exact.
> > >
> > > For each split F‖H of ℚ and
> > > for each distance d > 0
> > > there exist rationals p₋ p₊
> > > p₋ ∈ F ᣔ< x₁ <ᣔ H ∋ p₊
> > > p₋ ∈ F ᣔ< x₂ <ᣔ H ∋ p₊
> > > such that p₊-p₋ < d
> > >
> > > However,
> > > if there are two between-points x₁ x₂
> > > then
> > > there is some d ≥ |x₂-x₁| > 0
> > > such that ¬(p₊-p₋ < d)
> > > for all p₋ p₊
> > > Contradiction.
> > >
> > > Thus,
> > > for each split,
> > > there is at most one between-point x
> > >
> > > | The point 2¹ᐟ³ between
> > > | F = {p ∈ ℚ| p³ < 2} and
> > > | H = {p ∈ ℚ| 2 < p³}
> > > |
> > > describes exactly one point.
> > Yes, here is exactly the whole unnoticeable point of all kinds of confusions
> >
> > P^3 < 2, you mean exactly the greatest real number for p^3 that is strictly less than 2, Right ✅️?
> >
> > Which of course doesn't exist, ✅️ Right?
> > Similarly; when you are describing the least real number for P^3 that is strictly greater than 2; Right ✅️?
> > Which of course doesn't exist, Right ✅️?
> > Then you aren't talking about a point or an exact location on the real number line, but talking about a distance that you conclude as very small distance & permit yourself to neglect its entire existence inorder to justify illegally your conclusions based entirely on an eingineering point of view whish is completely irrelevant to any theoretical mathematics
> >
> > Aren't you an eingineer or at least with eingineering thinking as if you want to solve a problem such that a carpenter can make approximately a cube with nearly two units volume
> >
> > This is exactly Doubling the cube problem which is one of the most popular historical problem raised by ancient Greeks few thousands of years back, where it was known as impossible task by unmarked straigt edge & a compass
> >
> > And Wentzel proof in 1836 is actually not any valid proof but a true conclusion only, since he couldn't understand the non-existing numbers like Cubrt2, which confirms such absolute impossibilities about all the Greeks problems
> >
> > Where,with false proof by Wentzel, and without understanding the non-existing principles He kept the doors opened widely to all skilled carpenters, scientists & mathematickers to invent new tools & many approximations methods inorder to get that number as Cubrt2 that exists only & strictly in human minds as a matter of beliefs & necessities, & still humans try tirelessly to invent new methods inorder to catch that belived number which never exists EXACTLY, but some how, humans generally believe that they are very close to it
> >
> > Of course, the three famous historical Greeks problems are absolutely impossible to solve bu any tools & by any means of human methods of endless approximations
> >
> > Those problems had been completely missed by ALL humans up to this date, & mathematics got as a result all kinds of contradictions & inconsistency that every one is talking about nowadays
> >
> > Which is why I claim that human mental problems with mathematics aren't in mathematics itself but completely with mathematians themselves for purely human desires of attention & achievements
> >
> > However, the whole world 🌎 academy of Geometry & Number theorists are requested to reinvestigaite legally the 3 Greeks insolvable problems by using the Artificial Intelligence nowadays
> >
> > 🔊 Bassam Karzeddin 🔊
> Well that's uninformed, an "Archimedean spiral" as third classical tool
> makes it pretty easy to "square the circle", for example, as it were.
>
> Then, also and besides, arithmetic and algebra and analysis are "tools",
> and pi is a well-known constant, so "squaring the circle" is pretty easy,
> as are the other notions like "dividing by three" and taking and making
> powers and roots.
>
> Oh, you mean like sticks, that we dig in the mud for grubs,
> yeah, those must be some tasty grubs.

The problem with academic mainstreams is that they don't read, also they don't like to understand anything outside of their own education especially if it is contradicting their common global education

Also, I can't bring up all my relevant public published posts with me which include ( proofs with irrefutable numerical counterexamples, illustrations, standing conjectures, ..., etc) for each annymous academic to teach online every one alone & spereatly, where I know how a common acadimic salvation mentality to what is only acceptable by them are the published math from their authorities represented in their alleged top-most repuitabe Journals & Universities

For illustration, they do believe a published proof of Fermat's last theorem despite the fact that non of them understands a section of that alleged proof since quarter a century, where they have no choice to accept or approve, but utterly they refuse anything publicly & freely published even if it is too simple to understand & even with a dozen of rigorous irrefutable proofs just because it is public mathematics

Also the mainstream can't comprehend that an official sources are in a secretive un announced war with public published mathematics as a matter of laws of wars & survivors, where they can't admit the truth & lose their nonsense products

Now we had been talking about Pi & Cubroot2 for many years & suddenly someone like you wants me to teach him every thing written, proven from thr beginning

Of course that is an impossible task with common mainstream academic sheeples, where it is becoming so obvious that many of them are fighting anonymously to protect their own published products in forms of published (papers, books, articles, ..., etc)

However, the more important discovery about academic mathematicians generally is their common unbelievable qualities, so unlike what they actually are teaching people about them as being (clever, decent, smart, honnest, noble, trustful, brave, open mind, ...., etc) humans, where all that is exactly opposite to the truth

But naturally, for each rule there is an exception.... hopefully!

So these get annoyed when they hear about an issue of non-existing angles, where they can't meet the announced challenges nor do they accept the simplist elementary proofs about my claims

They are the same inhibitors of older centuries where many of them did secretly learn the truth & keeping silent about it as long as it invalidates their common inherited false beliefs

So simply, go to my public published intellectual challenges & refute openly anyone of them gladly

Why not, they were basically designed for you , so why are you here still arguing aimlessly

Go below them & refute any of them by a single numerical counter example only

Don't be like a well-known trolls as Dan C with his DC Proof & keeps repeating his old song to diverse the attention

Good luck

🔊 Bassam Karzeddin 🔊

On Wednesday, October 11, 2023 at 6:10:31 AM UTC+3, bassam karzeddin wrote:
> On Monday, October 9, 2023 at 11:56:20 PM UTC+3, Jim Burns wrote:
> > On 10/9/2023 8:27 AM, bassam karzeddin wrote:
> > > On Monday, October 9, 2023
> > > at 1:02:10 PM UTC+3, bassam karzeddin wrote:
> > >> On Monday, October 9, 2023
> > >> at 7:48:28 AM UTC+3, Jim Burns wrote:
> > >>> On 10/8/2023 12:56 PM, bassam karzeddin wrote:
> > >>>> On Saturday, August 8, 2020
> > >>>> at 11:27:08 AM UTC+3, bassam karzeddin wrote:
> >
> > > what is truly a real number?
> > A real number x ∈ ℝ is
> > a rational number x ∈ ℚ or
> > a point x between a split F‖H of ℚ
> > F ᣔ< x <ᣔ H
> > F∪H = ℚ\{x}
> > > And assuming in good faith that
> > > a real number which is
> > > not a constructible number exists,
> > > then mathematicians are globally
> > > revealed [requested?] to present
> > > only one of them
> > > either numerically or Geometrically
> > > but Exactly [...]
> >
> > Points-between-splits are geometry.
> >
> > Without all points-between-splits,
> > some continuous curves cross but
> > don't intersect,
> > which is not geometry.
> >
> >
> > Points-between-splits are exact.
> >
> > For each split F‖H of ℚ and
> > for each distance d > 0
> > there exist rationals p₋ p₊
> > p₋ ∈ F ᣔ< x₁ <ᣔ H ∋ p₊
> > p₋ ∈ F ᣔ< x₂ <ᣔ H ∋ p₊
> > such that p₊-p₋ < d
> >
> > However,
> > if there are two between-points x₁ x₂
> > then
> > there is some d ≥ |x₂-x₁| > 0
> > such that ¬(p₊-p₋ < d)
> > for all p₋ p₊
> > Contradiction.
> >
> > Thus,
> > for each split,
> > there is at most one between-point x
> >
> > | The point 2¹ᐟ³ between
> > | F = {p ∈ ℚ| p³ < 2} and
> > | H = {p ∈ ℚ| 2 < p³}
> > |
> > describes exactly one point.
> Yes, here is exactly the whole unnoticeable point of all kinds of confusions
>
> P^3 < 2, you mean exactly the greatest real number for p^3 that is strictly less than 2, Right ✅️?
>
> Which of course doesn't exist, ✅️ Right?
> Similarly; when you are describing the least real number for P^3 that is strictly greater than 2; Right ✅️?
> Which of course doesn't exist, Right ✅️?
> Then you aren't talking about a point or an exact location on the real number line, but talking about a distance that you conclude as very small distance & permit yourself to neglect its entire existence inorder to justify illegally your conclusions based entirely on an eingineering point of view whish is completely irrelevant to any theoretical mathematics
>
> Aren't you an eingineer or at least with eingineering thinking as if you want to solve a problem such that a carpenter can make approximately a cube with nearly two units volume
>
> This is exactly Doubling the cube problem which is one of the most popular historical problem raised by ancient Greeks few thousands of years back, where it was known as impossible task by unmarked straigt edge & a compass
>
> And Wentzel proof in 1836 is actually not any valid proof but a true conclusion only, since he couldn't understand the non-existing numbers like Cubrt2, which confirms such absolute impossibilities about all the Greeks problems
>
> Where,with false proof by Wentzel, and without understanding the non-existing principles He kept the doors opened widely to all skilled carpenters, scientists & mathematickers to invent new tools & many approximations methods inorder to get that number as Cubrt2 that exists only & strictly in human minds as a matter of beliefs & necessities, & still humans try tirelessly to invent new methods inorder to catch that belived number which never exists EXACTLY, but some how, humans generally believe that they are very close to it
>
> Of course, the three famous historical Greeks problems are absolutely impossible to solve bu any tools & by any means of human methods of endless approximations
>
> Those problems had been completely missed by ALL humans up to this date, & mathematics got as a result all kinds of contradictions & inconsistency that every one is talking about nowadays
>
> Which is why I claim that human mental problems with mathematics aren't in mathematics itself but completely with mathematians themselves for purely human desires of attention & achievements
>
> However, the whole world 🌎 academy of Geometry & Number theorists are requested to reinvestigaite legally the 3 Greeks insolvable problems by using the Artificial Intelligence nowadays
>
> 🔊 Bassam Karzeddin

@ Jim Burns, with all neck names like Mild Shock & many others he uses, where YOU personally were given hundreds of free chances to simply understand what is truly an exciting real number since you are a very argunanrmtaive person & since many years as I remember, but so unfortunately didn't work with you except in a opposite way exactly, & here I did show you your main source of confusions inorder to upraise the truth above all lies that had been accumulated since the start of human mere knowledge in mathematics, where also I know that wasn't at all your own guilts & mistakes but they were simply inherited from first manhood by heritage & not by Genatic disorder

And my true aim was never to desestinate other human abilities or to insult their peculiar intellectuality but truly to enlight them about the dangerous way of their inhereted thinking about logic & philosophy of mathematics & science as well

In fact, I wish that I'm the only person who is completely so delusional & mistaken but so unfortunately nobody so far could ever convince me so & on the contrary, I used to immediately discover where they had gone astray & exactly in the opposite direction twords the truth, as if a Devil 😈 is always there preventing them to see the very obvious truth they don't like to hear & see it clearly even secretly between themselves!
And why all that fear from seeing the shining truth?

Simply to protect others human towers achievements which is the whole story about!

But so unfortunately for all humans, a new era of Artificial Intelligence with no human traits have arrived & nobody would ever be able to stand firmly against the truth

BKK

On Tuesday, October 10, 2023 at 6:51:41 PM UTC-7, Ross Finlayson wrote:
> On Tuesday, October 10, 2023 at 10:45:49 AM UTC-7, Jim Burns wrote:
> > On 10/9/2023 10:23 PM, Ross Finlayson wrote:
> > > On Monday, October 9, 2023
> > > at 6:39:43 PM UTC-7, Ross Finlayson wrote:
> > >> On Monday, October 9, 2023
> > >> at 1:56:20 PM UTC-7, Jim Burns wrote:
> >
> > >>> | The point 2¹ᐟ³ between
> > >>> | F = {p ∈ ℚ| p³ < 2} and
> > >>> | H = {p ∈ ℚ| 2 < p³}
> > >>> |
> > >>> describes exactly one point.
> > >>
> > >> But, sirrah,
> > >> I aver you cannot name
> > >> the cut of root two,
> > >> without mentioning root two,
> > >> which is not a ratio.
> > You see that I have named
> > the cut of cube root of two
> > without mentioning
> > the cube root of two.
> >
> > Yes, I gave that point the name 2¹ᐟ³
> > but the name was a practical necessity,
> > not a logical necessity.
> > Everywhere I write 2¹ᐟ³
> > I could instead write
> > | The point between
> > | F = {p ∈ ℚ| p³ < 2} and
> > | H = {p ∈ ℚ| 2 < p³}
> > |
> > I will not ask you to wade through that,
> > but they mean the same.
> >
> > | The point between
> > | F = {p∈ℚ| p²<2} ∪ {p∈ℚ| p<0} and
> > | H = {p∈ℚ| 2<p²} \ {p∈ℚ| p<0}
> > |
> > names the cut of +2¹ᐟ²
> >
> > To identify which rationals are on
> > which side of an irrational
> > is
> > to uniquely identify that irrational.
> >
> > That follows inevitably from
> > any of the equivalent statements
> > -- continuous curves which cross intersect
> > -- ℝ is ℚ with points between splits of ℚ
> > -- for each bounded non-empty set of
> > real numbers, a least upper bound exists.
> >
> >
> > We can say what
> > a betweenless foresplit of ℚ
> > is without saying which
> > betweenless foresplit of ℚ
> > is referred to.
> >
> > For example, we can say
> > F is a betweenless foresplit of ℚ
> > in this way
> > F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F ∧
> > ℚ\F ≠ ∅ ≠ F ⊆ ℚ
> >
> > F ᣔ<ᘁ F means
> > ∀p₋ ∈ F, ∃p′₋ ∈ F: p₋ < p′₋
> >
> > F ᣔ<ᣔ ℚ\F means
> > ∀p₋ ∈ F, ∀p₊ ∈ ℚ\F: p₋ < p₊
> >
> > We can say
> > ℝ is
> > the set of betweenless foresplits of ℚ
> > in this way
> > ℝ := {F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F}\{∅,ℚ}
> >
> > We can prove the existence of
> > {F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F}\{∅,ℚ}
> > from
> > the existence of ℚ
> > the existence of its powerset 𝒫(ℚ)
> > the existence of subsets by separation.
> >
> > We can prove the existence of
> > the least upper bound ⋃S of
> > a bounded non-empty subset S of
> > {F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F}\{∅,ℚ}
> > from
> > the existence of set unions.
> > > Either way you've _axiomatized_
> > > the least-upper-bound property
> > > into existence,
> > I'm pretty sure that I have not done that.
> > The existence of
> > {F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F}\{∅,ℚ} and
> > the existence of least-upper-bounds
> > follow from axioms.
> > But not that one.
> > > while,
> > > something like line-reals sees it
> > > result neatly from "next".
> > Please remind me what line-reals and
> > the rest of their phylum are.
> >
> >
> > If there is a split without a point-between
> > then there is function which jumps but
> > is not discontinuous anywhere.
> >
> > It is my desire to avoid
> > jumping continuous functions which is
> > the source of my interest in
> > {F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F}\{∅,ℚ}
> >
> > It would be nice if it turned out that
> > you also have this desire.
> > Then you and I would not be in
> > two different non-overlapping conversations.
> You know there are algebraics and they're irrationals and there are
> transcendentals and they're irrationals, and algebraics aren't transcendental.
>
> So, just saying, you haven't shown that between or "not greater than"
> a given rational, isn't an indistinguishable, unordered pair of algebraic
> and transcendental. I.e. it's not necessarily "a point", between two rationals,
> i.e. less than a given rational and greater than all rationals less than it.
>
> Gaplessness, my friend, is a usual notion meaning the same thing as completeness
> topologically, here as it's what's to be arrived at for the character of a point-set
> having the character of a line or a continuous curve.
>
> Of course the line reals, are as like Aristotle's continuum, and much like the differential,
> as of path elements or line elements which are infinitesimal but have analytical character
> that's non-zero, and sum, if only altogether and only altogether, to a finite quantity,
> or here exactly that it's n/d as d goes to infinity and n goes to d, in the limit, or in the
> infinite, is established "extent, density, completeness, measure", these being the
> qualities of a continuous domain.
>
> It's so that least-upper-bound is a usual axiom added to make the field-reals,
> the standard reals, possible, and another is "measure 1.0", to establish the
> analytical character, of their magnitudes.
>
> I have a very proper curriculum of standard analysis and the field reals,
> and it's the one there is.

You see: the rationals and ir-rationals are each dense, everywhere discontinuous,
and whose complement is each other, but the algebraics and transcendentals are also.

Then you might say "well after the rationals I'll define algebraics, and then make cuts of those".

But, there are more ways to make "NCD2" or "nowhere continuous, dense, and equi-distributed",
these points on this line, any one of which belies any other of which being only the gaps of
the other, and only points, the gaps.

Then, this also gets into the "vague fugue" of the real numbers, where they're so close
that trichotomy only arises later, lots of these kinds of issues.

Now, whether NCD2 sets of points basically reflect for a weaving and a frequency, and
thusly, a signal construction like Nyquist's, speaks directly to the notion of 'signal-continuity',
a third definition of completeness of a continuous domain or gaplessness.

line-continuity: line-drawing's
field-continuity: limits from the left, and, limits from the right
signal-continuity: the smoothness of a grain
long-line-continuity: space of all real functions intersecting zero

Thusly are at least _four_ definitions of completeness/gaplessness, and not just
"conservative extensions, but incomplete and nicht richtiges".

I hope this is a gentle remonstration, but any can build each of those, then to
reflect why there's a real first-class treatment of "function theory", for something
like "Zermelo said set theory was about orders, numbers, and functions".

Reading Kepler's "system of the world", Kepler makes some really good points that
basically match those of the sublime of Kant, that, our minds, make an object-sense,
that allows for reason, making the facility, of "zero, one, and infinity".

On 10/10/2023 11:10 PM, bassam karzeddin wrote:
> On Monday, October 9, 2023
> at 11:56:20 PM UTC+3, Jim Burns wrote:

>> Points-between-splits are exact.
>>
>> For each split F‖H of ℚ and
>> for each distance d > 0
>> there exist rationals p₋ p₊
>> p₋ ∈ F ᣔ< x₁ <ᣔ H ∋ p₊
>> p₋ ∈ F ᣔ< x₂ <ᣔ H ∋ p₊
>> such that p₊-p₋ < d
>>
>> However,
>> if there are two between-points x₁ x₂
>> then
>> there is some d ≥ |x₂-x₁| > 0
>> such that ¬(p₊-p₋ < d)
>> for all p₋ p₊
>> Contradiction.
>>
>> Thus,
>> for each split,
>> there is at most one between-point x
>>
>> | The point 2¹ᐟ³ between
>> | F = {p ∈ ℚ| p³ < 2} and
>> | H = {p ∈ ℚ| 2 < p³}
>> |
>> describes exactly one point.
>
> Yes, here is exactly
> the whole unnoticeable point of
> all kinds of confusions
>
> P^3 < 2, you mean exactly
> the greatest real number for p^3
> that is strictly less than 2,
> Right ✅️?

Wrong.

'p' is a variable.
https://en.wikipedia.org/wiki/Variable_(mathematics)

The role of 'p' is similar to
the role of "it" in English:
a way to talk about something without
needing to say _which thing_
we are talking about.

There is a game, "Twenty Questions".
One player, the answerer, picks something,
anything, and
the other players, the askers, try to
find out what it is.
What _it_ is.
| Is it animal, vegetable, or mineral?
| Is it a means of transportation?
| Is it manufactured in Europe?
And so on.

Pronouns/variables are valuable for
discussing
whether that nest at the top of the
pine tree out back is being used
to raise elephantlets, or
whether there is more than one
point between the sides of
a split of the rationals.

Pronouns/variables make available for
consideration claims like
| if _it_ is an elephant's nest
| then an elephant couldn't use it,
| way up in those little branches.
and
| if there are two between-points,
| then there are rationals on either side
| closer than the two between-points.

We can discuss _all the elephants_
at the same time. Likewise all the rationals
and all the splits and all the between-points.

'p' is "it".
Is p a rational number? Yes, for now.
We write p ∈ ℚ
In later or earlier "games",
'p' could refer to something different.

There are two sets,
{p ∈ ℚ| p³ < 2} and {p ∈ ℚ| 2 < p³}
If p ∈ {p ∈ ℚ| p³ < 2}, then p³ < 2
If p ∈ {p ∈ ℚ| 2 < p³}, then 2 < p³

Consider two rationals, p₋ p₊
p₋ ∈ {p ∈ ℚ| p³ < 2}
p₊ ∈ {p ∈ ℚ| 2 < p³}
p₋ < 2¹ᐟ³ < p₊
p₋ and p₊ bookend 2¹ᐟ³

There is a midpoint pₘ
pₘ = (p₋+p₊)/2
pₘ is rational
Either
pₘ ∈ {p ∈ ℚ| p³ < 2} or
pₘ ∈ {p ∈ ℚ| 2 < p³}

If pₘ ∈ {p ∈ ℚ| p³ < 2}
then pₘ < 2¹ᐟ³ < p₊
If pₘ ∈ {p ∈ ℚ| 2 < p³}
then p₋ < 2¹ᐟ³ < pₘ

Either
pₘ and p₊ bookend 2¹ᐟ³ or
p₋ and pₘ bookend 2¹ᐟ³
and
p₊-pₘ = pₘ-p₋ = (p₊-p₋)/2

For any two rationals p₋ p₊
which bookend 2¹ᐟ³
there are two rationals, pₘ p₊ or p₋ pₘ
half as far apart which bookend 2¹ᐟ³

Points-between-splits are exact.

| Assume otherwise.
| Assume two points-between-splits exist.
| There are no bookends closer than
| those two points.
| | But then,
| there are bookends for which
| _there aren't_ bookends half as
| far apart.
| | However,
| We see above that that can't be.
| For each p₋ p₊: pₘ p₊ or p₋ pₘ
| Contradiction.

Therefore,
points-between-splits are exact.

> Then you aren't talking about
> a point or
> an exact location on the real number line,
> but talking about a distance that
> you conclude as very small distance

I am talking about a point.
One point
between all the rationals with cubes < 2
and all the rationals with cubes > 2.

There aren't two such points-between
in one split
because,
however close two would be,
there are rationals on either side
closer than the two.

On 10/11/2023 11:28 AM, Ross Finlayson wrote:
> On Tuesday, October 10, 2023
> at 6:51:41 PM UTC-7, Ross Finlayson wrote:
>> On Tuesday, October 10, 2023
>> at 10:45:49 AM UTC-7, Jim Burns wrote:

>>> [...]
>> [...]
>

> Then you might say
> "well after the rationals
> I'll define algebraics, and
> then make cuts of those".

You could say that.
The points and between-points are the same
in both cases.

Defining rationals and algebraics
is more work than defining rationals,
for no advantage that I can see.

I don't see all, though.
Do that, then, if you think you should.
I'll continue with the rationals.

We don't get more points-between by
the more laborious route.
Each Cauchy sequence of rationals
or of algebraics converges to
a point in the reals.
It is the great claim to fame of
the real numbers.

> Then, this also gets into
> the "vague fugue" of the real numbers,
> where they're so close that
> trichotomy only arises later,

That is incorrect.

Tricotomy arises but not later for
points-between-splits.

The betweenless foresplits have trichotomy
by inclusion.
Therefore,
their points-between have trichotomy.

For
F₁ ᣔ< x₁ <ᣔ H₁
F₂ ᣔ< x₂ <ᣔ H₂

F₁ ⊂≠⊅ F₂ ∨ F₁ ⊄=⊅ F₂ ∨ F₁ ⊄≠⊃ F₂
thus
x₁ <≠≯ x₂ ∨ x₁ ≮=≯ x₂ ∨ x₁ ≮≠> x₂

Theorem.
F₁ ⊆ F₂ ∨ F₁ ⊇ F₂

Proof.
F₁ ⊆ F₂ ∨ F₁ ⊈ F₂

(i)
F₁ ⊆ F₂
F₁ ⊆ F₂ ∨ F₁ ⊇ F₂

(ii)
F₁ ⊈ F₂

∃p₀ ∈ F₁: p₀ ∉ F₂ ∧
∀p₊ ∈ ℚ: p₀ ≤ p₊ ⟹ p₊ ∉ F₂ ∧
∀p₋ ∈ ℚ: p₋ ∈ F₂ ⟹ p₋ < p₀ ∧
∀p₋ ∈ ℚ: p₋ ≤ p₀ ⟹ p₋ ∈ F₁

∀p₋ ∈ ℚ: p₋ ∈ F₂ ⟹ p₋ ∈ F₁

F₁ ⊇ F₂
F₁ ⊆ F₂ ∨ F₁ ⊇ F₂

Therefore,
whether (i) or (ii)
F₁ ⊆ F₂ ∨ F₁ ⊇ F₂

Therefore,
the betweenless foresplits of Q
have trichotomy, and
their between-points have trichotomy.

> line-continuity:
> field-continuity:
> signal-continuity:
> long-line-continuity:
>
> Thusly are at least _four_ definitions of
> completeness/gaplessness, and not just
> "conservative extensions,
> but incomplete and nicht richtiges".

There is one complete ordered field,
up to isometry.

Rationals map to rationals′
Bounded-nonempty-sets of rationals map to
bounded-nonempty-sets′ of rationals′
Least-upper-bounds of
bounded-non-empty-sets of rationals map to
to least-upper-bounds′ of
bounded-nonempty-sets′ of rationals′
That's all.

The complete ordered field includes ℚ and
its continuous functions do not jump.
That's what we very often want.

What purposes do your three(?) other
continuities serve?

On Wednesday, October 11, 2023 at 8:44:05 PM UTC+3, Jim Burns wrote:
> On 10/10/2023 11:10 PM, bassam karzeddin wrote:
> > On Monday, October 9, 2023
> > at 11:56:20 PM UTC+3, Jim Burns wrote:
>
> >> Points-between-splits are exact.
> >>
> >> For each split F‖H of ℚ and
> >> for each distance d > 0
> >> there exist rationals p₋ p₊
> >> p₋ ∈ F ᣔ< x₁ <ᣔ H ∋ p₊
> >> p₋ ∈ F ᣔ< x₂ <ᣔ H ∋ p₊
> >> such that p₊-p₋ < d
> >>
> >> However,
> >> if there are two between-points x₁ x₂
> >> then
> >> there is some d ≥ |x₂-x₁| > 0
> >> such that ¬(p₊-p₋ < d)
> >> for all p₋ p₊
> >> Contradiction.
> >>
> >> Thus,
> >> for each split,
> >> there is at most one between-point x
> >>
> >> | The point 2¹ᐟ³ between
> >> | F = {p ∈ ℚ| p³ < 2} and
> >> | H = {p ∈ ℚ| 2 < p³}
> >> |
> >> describes exactly one point.
> >
> > Yes, here is exactly
> > the whole unnoticeable point of
> > all kinds of confusions
> >
> > P^3 < 2, you mean exactly
> > the greatest real number for p^3
> > that is strictly less than 2,
> > Right ✅️?
> Wrong.
>
> 'p' is a variable.
> https://en.wikipedia.org/wiki/Variable_(mathematics)
>
Isn't that reference yours since any one can write there in Donkypedia

> The role of 'p' is similar to
> the role of "it" in English: there in Donk.
> a way to talk about something without
> needing to say _which thing_
> we are talking about.
>
> There is a game, "Twenty Questions".
> One player, the answerer, picks something,
> anything, and
> the other players, the askers, try to
> find out what it is.
> What _it_ is.
> | Is it animal, vegetable, or mineral?
> | Is it a means of transportation?
> | Is it manufactured in Europe?
> And so on.
>
> Pronouns/variables are valuable for
> discussing
> whether that nest at the top of the
> pine tree out back is being used
> to raise elephantlets, or
> whether there is more than one
> point between the sides of
> a split of the rationals.
> Pronouns/variables make available for
> consideration claims like
> | if _it_ is an elephant's nest
> | then an elephant couldn't use it,
> | way up in those little branches.
> and
> | if there are two between-points,
> | then there are rationals on either side
> | closer than the two between-points.
>
> We can discuss _all the elephants_
> at the same time. Likewise all the rationals
> and all the splits and all the between-points.
>
> 'p' is "it".
> Is p a rational number? Yes, for now.
> We write p ∈ ℚ
> In later or earlier "games",
> 'p' could refer to something different.
>
> There are two sets,
> {p ∈ ℚ| p³ < 2} and {p ∈ ℚ| 2 < p³}
> If p ∈ {p ∈ ℚ| p³ < 2}, then p³ < 2
> If p ∈ {p ∈ ℚ| 2 < p³}, then 2 < p³
>
> Consider two rationals, p₋ p₊
> p₋ ∈ {p ∈ ℚ| p³ < 2}
> p₊ ∈ {p ∈ ℚ| 2 < p³}
> p₋ < 2¹ᐟ³ < p₊
> p₋ and p₊ bookend 2¹ᐟ³
>
> There is a midpoint pₘ
> pₘ = (p₋+p₊)/2
> pₘ is rational
> Either
> pₘ ∈ {p ∈ ℚ| p³ < 2} or
> pₘ ∈ {p ∈ ℚ| 2 < p³}
>
> If pₘ ∈ {p ∈ ℚ| p³ < 2}
> then pₘ < 2¹ᐟ³ < p₊
> If pₘ ∈ {p ∈ ℚ| 2 < p³}
> then p₋ < 2¹ᐟ³ < pₘ
>
> Either
> pₘ and p₊ bookend 2¹ᐟ³ or
> p₋ and pₘ bookend 2¹ᐟ³
> and
> p₊-pₘ = pₘ-p₋ = (p₊-p₋)/2
>
> For any two rationals p₋ p₊
> which bookend 2¹ᐟ³
> there are two rationals, pₘ p₊ or p₋ pₘ
> half as far apart which bookend 2¹ᐟ³
>

>
> Points-between-splits are exact.
>
> | Assume otherwise.
> | Assume two points-between-splits exist.
> | There are no bookends closer than
> | those two points.
> |
> | But then,
> | there are bookends for which
> | _there aren't_ bookends half as
> | far apart.
> |
> | However,
> | We see above that that can't be.
> | For each p₋ p₊: pₘ p₊ or p₋ pₘ
> | Contradiction.
>
> Therefore,
> points-between-splits are exact.
> > Then you aren't talking about
> > a point or
> > an exact location on the real number line,
> > but talking about a distance that
> > you conclude as very small distance
> I am talking about a point.
> One point
> between all the rationals with cubes < 2
> and all the rationals with cubes > 2.
>
> There aren't two such points-between
> in one split
> because,
> however close two would be,
> there are rationals on either side
> closer than the two.

That is not mathematics at all but purely eingineering only & comparison between rational & irrational numbers belived in advance existing

Why must a Cuberoot for a prime number exists?

It is well understood why a prime number must have a Sqrt(9), & never ftom Dedekind cuts or Cauchy sequences but only & strictly from the Phythagorus theorem & since few thousands years

But which theorem in mathematics (if existing) would legalize the existence of a Cuberoot of any prime number?

On the contrary, Fermat’s last theorem disprove the existence of Cuberoot of any prom numbers

Ref. My relevant published posts

And without any effort, just express your number as Cubrt2 either numerically or Geometrically but Exactly, since number is an exact thing

Or express it numerically without using the decimal notation (at least once in your life time, where you can't FOR SURE
Many more too elementary proofs were publically published in my remaining & unstolen posts

The true irrational existing numbers were discovered & proven after the Pythagorean era, which are only constructible numbers where no other types of real numbers ever exist, SURE

BKK

On 10/11/2023 8:25 PM, bassam karzeddin wrote:
> On Wednesday, October 11, 2023
> at 8:44:05 PM UTC+3, Jim Burns wrote:
>> On 10/10/2023 11:10 PM, bassam karzeddin wrote:
>>> On Monday, October 9, 2023
>>> at 11:56:20 PM UTC+3, Jim Burns wrote:

>>>> | The point 2¹ᐟ³ between
>>>> | F = {p ∈ ℚ| p³ < 2} and
>>>> | H = {p ∈ ℚ| 2 < p³}
>>>> |
>>>> describes exactly one point.
>>>
>>> Yes, here is exactly
>>> the whole unnoticeable point of
>>> all kinds of confusions
>>>
>>> P^3 < 2, you mean exactly
>>> the greatest real number for p^3
>>> that is strictly less than 2,
>>> Right ✅️?
>>
>> Wrong.
>>
>> 'p' is a variable.
>> https://en.wikipedia.org/wiki/Variable_(mathematics)
>
> Isn't that reference yours
> since any one can write there in Donkypedia

Wikipedia doesn't say that 'p' is a variable.
I say that 'p' is a variable.

Wikipedia say what it is to be a variable.

Variables are useful.

Also, no, I don't mean
the greatest real number for p³ that is
strictly less than 2

> But which theorem in mathematics (if existing)
> would legalize the existence of
> a Cuberoot of any prime number?

The intermediate value theorem.
https://en.wikipedia.org/wiki/Intermediate_value_theorem

On Thursday, October 12, 2023 at 4:07:45 AM UTC+3, Jim Burns wrote:
> On 10/11/2023 8:25 PM, bassam karzeddin wrote:
> > On Wednesday, October 11, 2023
> > at 8:44:05 PM UTC+3, Jim Burns wrote:
> >> On 10/10/2023 11:10 PM, bassam karzeddin wrote:
> >>> On Monday, October 9, 2023
> >>> at 11:56:20 PM UTC+3, Jim Burns wrote:
>
> >>>> | The point 2¹ᐟ³ between
> >>>> | F = {p ∈ ℚ| p³ < 2} and
> >>>> | H = {p ∈ ℚ| 2 < p³}
> >>>> |
> >>>> describes exactly one point.
> >>>
> >>> Yes, here is exactly
> >>> the whole unnoticeable point of
> >>> all kinds of confusions
> >>>
> >>> P^3 < 2, you mean exactly
> >>> the greatest real number for p^3
> >>> that is strictly less than 2,
> >>> Right ✅️?
> >>
> >> Wrong.
> >>
> >> 'p' is a variable.
> >> https://en.wikipedia.org/wiki/Variable_(mathematics)
> >
> > Isn't that reference yours
> > since any one can write there in Donkypedia
> Wikipedia doesn't say that 'p' is a variable.
> I say that 'p' is a variable.
>
> Wikipedia say what it is to be a variable.
>
> Variables are useful.
>
> Also, no, I don't mean
> the greatest real number for p³ that is
> strictly less than 2
> > But which theorem in mathematics (if existing)
> > would legalize the existence of
> > a Cuberoot of any prime number?
> The intermediate value theorem.
> https://en.wikipedia.org/wiki/Intermediate_value_theorem

Intermediate theorem is an artificial, I mean not real theorem nor any true theorem, for which, humans made the negative & functions beside the continiouty which comes finally as a conclusion such that it is an artificial unreal fabrications

Famous Example : a real transcendental believed number among humans as Pi, lies perpetually between two constructible numbers, which makes humans search for it forever.... hopelessly & so foolishly as a matter of non-ending numerical game

It was never occurred to human minds that they were searching for something actually non-existing (except only in their minds)
And the search for Pi would continue for ever.... without being able to catch it exactly FOR SURE.

BKK

On 10/12/2023 3:08 AM, bassam karzeddin wrote:
> On Thursday, October 12, 2023
> at 4:07:45 AM UTC+3, Jim Burns wrote:
>> On 10/11/2023 8:25 PM, bassam karzeddin wrote:

>>> But which theorem in mathematics (if existing)
>>> would legalize the existence of
>>> a Cuberoot of any prime number?
>>
>> The intermediate value theorem.
>> https://en.wikipedia.org/wiki/Intermediate_value_theorem
>
> Intermediate [value] theorem is
> an artificial, I mean not real theorem
> nor any true theorem, for which,
> humans made the negative &
> functions beside the [continuity]
> which comes finally as a conclusion such that
> it is an artificial unreal fabrications

The intermediate value theorem says that,
for the continuous function f(x)
no value y between y₁=f(x₁) and y₂=f(x₂)
is skipped between x₁ and x₂

For example,
cube roots of primes are not skipped.

That's a theorem about ℝ the real numbers.

You're saying some of ℝ doesn't exist.
That π doesn't exist, for example.

Suppose all of ℚ the rational numbers
exists.
Suppose all of 𝒫(ℚ) the subsets of ℚ
exists.

Those subsets of ℚ which are
| a first part F of a split of ℚ which
| doesn't hold a point between F and
| its last part ℚ\F
behave in the way
in which points of ℝ behave.

All those subsets F, the betweenless foresplits,
all exist,
because all of the subsets of ℚ exist

Those betweenless foresplits do not skip.
For each split of the betweenless foresplits,
a betweenless foresplit exists which
is between the fore and hind parts of
that split.

{F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F}\{∅,ℚ} is
the set of all betweenless foresplits of ℚ

Suppose
ℝ = {F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F}\{∅,ℚ}

The _assumptions_ we make about ℝ
have been _proven_ about
{F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F}\{∅,ℚ}

> Famous Example :
> a real transcendental believed number
> among humans as Pi, lies perpetually
> between two constructible numbers,
> which makes humans search for it forever....
> hopelessly & so foolishly
> as a matter of non-ending numerical game

π/4 = ⅟1 - ⅟3 + ⅟5 - ⅟7 + ⅟9 - ...

π/4 is the one and only one point
π/4 < each of
⅟1
⅟1 - ⅟3 + ⅟5
⅟1 - ⅟3 + ⅟5 - ⅟7 + ⅟9
....
and π/4 > each of
⅟1 - ⅟3
⅟1 - ⅟3 + ⅟5 - ⅟7
⅟1 - ⅟3 + ⅟5 - ⅟7 + ⅟9 - ⅟11
....

We can define a function f(x) such that
f(x) = 1 if x > one of
⅟1
⅟1 - ⅟3 + ⅟5
⅟1 - ⅟3 + ⅟5 - ⅟7 + ⅟9
....
and
f(x) = 0 if x < one of
⅟1 - ⅟3
⅟1 - ⅟3 + ⅟5 - ⅟7
⅟1 - ⅟3 + ⅟5 - ⅟7 + ⅟9 - ⅟11
....

If no point exists between each of
⅟1
⅟1 - ⅟3 + ⅟5
⅟1 - ⅟3 + ⅟5 - ⅟7 + ⅟9
....
and each of
⅟1 - ⅟3
⅟1 - ⅟3 + ⅟5 - ⅟7
⅟1 - ⅟3 + ⅟5 - ⅟7 + ⅟9 - ⅟11
....
then f(x) is defined for all points,
and f(x) is continuous at all
all points which aren't π/4

If π/4 doesn't exist,
then f(x) is continuous at all points.
But,
whether π/4 exists or not,
f(x) jumps.

What we mean by "continuous"
does not accept functions which jump
as continuous at all points.

Therefore,
we do not accept a line without π/4
any more than we accept a square without
four corners.

> It was never occurred to human minds that
> they were searching for
> something actually non-existing
> (except only in their minds)
> And the search for Pi
> would continue for ever....
> without being able to catch it exactly
> FOR SURE.

We do not search for π/4
We do not catch π/4
However,
we can learn about π/4
by describing what we mean by π/4
and augmenting that description with
only not-first-false claims.

On Wednesday, October 11, 2023 at 12:48:05 PM UTC-7, Jim Burns wrote:
> On 10/11/2023 11:28 AM, Ross Finlayson wrote:
> > On Tuesday, October 10, 2023
> > at 6:51:41 PM UTC-7, Ross Finlayson wrote:
> >> On Tuesday, October 10, 2023
> >> at 10:45:49 AM UTC-7, Jim Burns wrote:
> >>> [...]
> >> [...]
> >
>
> > Then you might say
> > "well after the rationals
> > I'll define algebraics, and
> > then make cuts of those".
> You could say that.
> The points and between-points are the same
> in both cases.
>
> Defining rationals and algebraics
> is more work than defining rationals,
> for no advantage that I can see.
>
> I don't see all, though.
> Do that, then, if you think you should.
> I'll continue with the rationals.
>
> We don't get more points-between by
> the more laborious route.
> Each Cauchy sequence of rationals
> or of algebraics converges to
> a point in the reals.
> It is the great claim to fame of
> the real numbers.
> > Then, this also gets into
> > the "vague fugue" of the real numbers,
> > where they're so close that
> > trichotomy only arises later,
> That is incorrect.
>
> Tricotomy arises but not later for
> points-between-splits.
>
> The betweenless foresplits have trichotomy
> by inclusion.
> Therefore,
> their points-between have trichotomy.
>
> For
> F₁ ᣔ< x₁ <ᣔ H₁
> F₂ ᣔ< x₂ <ᣔ H₂
>
> F₁ ⊂≠⊅ F₂ ∨ F₁ ⊄=⊅ F₂ ∨ F₁ ⊄≠⊃ F₂
> thus
> x₁ <≠≯ x₂ ∨ x₁ ≮=≯ x₂ ∨ x₁ ≮≠> x₂
>
> Theorem.
> F₁ ⊆ F₂ ∨ F₁ ⊇ F₂
>
> Proof.
> F₁ ⊆ F₂ ∨ F₁ ⊈ F₂
>
> (i)
> F₁ ⊆ F₂
> F₁ ⊆ F₂ ∨ F₁ ⊇ F₂
>
>
> (ii)
> F₁ ⊈ F₂
>
> ∃p₀ ∈ F₁: p₀ ∉ F₂ ∧
> ∀p₊ ∈ ℚ: p₀ ≤ p₊ ⟹ p₊ ∉ F₂ ∧
> ∀p₋ ∈ ℚ: p₋ ∈ F₂ ⟹ p₋ < p₀ ∧
> ∀p₋ ∈ ℚ: p₋ ≤ p₀ ⟹ p₋ ∈ F₁
>
> ∀p₋ ∈ ℚ: p₋ ∈ F₂ ⟹ p₋ ∈ F₁
>
> F₁ ⊇ F₂
> F₁ ⊆ F₂ ∨ F₁ ⊇ F₂
>
> Therefore,
> whether (i) or (ii)
> F₁ ⊆ F₂ ∨ F₁ ⊇ F₂
>
> Therefore,
> the betweenless foresplits of Q
> have trichotomy, and
> their between-points have trichotomy.
>
> > line-continuity:
> > field-continuity:
> > signal-continuity:
> > long-line-continuity:
> >
> > Thusly are at least _four_ definitions of
> > completeness/gaplessness, and not just
> > "conservative extensions,
> > but incomplete and nicht richtiges".
> There is one complete ordered field,
> up to isometry.
>
> Rationals map to rationals′
> Bounded-nonempty-sets of rationals map to
> bounded-nonempty-sets′ of rationals′
> Least-upper-bounds of
> bounded-non-empty-sets of rationals map to
> to least-upper-bounds′ of
> bounded-nonempty-sets′ of rationals′
> That's all.
>
> The complete ordered field includes ℚ and
> its continuous functions do not jump.
> That's what we very often want.
>
> What purposes do your three(?) other
> continuities serve?

It's great to have further definitions of continuity, and it's rather required,
because each makes for the gaplessness and thusly an intermediate value
theorem, and making for a milieu where the fundamental theorems of calculus
hold up, that as well geometry for these "point-sets" where they are "lines",
is richer, stronger, more together, and less missing: truths, about continuous domains.

It's really quite a matter for those who study continuity and mathematics,
because they want to be "conscientious mathematicians" and avoid "hypocrisy",
because there's quite a strong sense where rigor, intuition, and formalism,
come together to make for that deduction rules that completeness, Dedekind's,
is a distinctness, not a uniqueness.

Then where something like "sweep the natural/unit equivalency function"
is for probability theory a distribution of a distribution with multiple pdf's,
remarkable, or the fact that it's its own anti-derivative, remarkable, as well
quite fully enrich mathematics with "more and better" vs. "less and none".

About the "uniqueness of the complete ordered field up to isomorphism",
once I wrote some field relations for [-1, 1], thus equipping another field..

https://groups.google.com/g/sci.math/c/4RBNLj-Q4Mo/m/Nn3T7v_rCnIJ

Consider a function taking [-1,1] to (-1,1) as so.

f_n(x) = nx, x e [-1,1], n-> oo, ran(f_oo(x)) = (-oo, oo)
g_n(y) = ny, x e (-1, 1), n-> oo, dom(g^-1_oo(g_n(y)) = (-oo, oo)

The properties of those functions is that they're linear.

Then, with j = g^-1 o f (or as composed): dom(j) = [-1,1] and ran(j)
= (-1,1).

With the composition of linear properties then j(x>y) > j(y), and
etcetera.

Then, let h_n(a) = na, n->oo, dom(h(a)) = (-1,1), ran(h(a)) = (-oo,
oo), h is linear as above.

Then a + b is h^-1( h(a) + h(b) ). And, a * b is h^-1( h(a) *
h(b) ).

Here a, b e dom(h) with +, * defined s above, h(b), h(a) e ran(h),
with +, * as in R.

c * (a + b)
= h'( h(c) * h( h'(h(a) + h(b)) ))
= h' ( h(c) * (h(a) + h(b)) )
= h' ( h(c) * h(a) + h(c) * h(b) )
= h' ( h(c) * h(a) ) + h' ( h(c) * h(b) )
= c * a + c * b

Multiplication distributes over addition, and addition and
multiplication are closed as sums and products of h(x) are in (-oo,
oo).

Via symmetry, additive inverses exist and there's an additive
identity, zero.

Then, is there a multiplicative identity and inverse. There is a
mulitplicative inverse of h(a) in [0,oo) in R with the natural
definition of multiplication, and, with the continuous and order-
preserving mapping from [0,1), for any a with 0 < a < 1, there exists
1/h(a) in R+, in ran(h), and: h' ( h(a) * 1/h(a) ) e R[0,1).

Basically: "A definition of field operations on the unit interval."

The "vague fugue" then really is a reflection on numbers and precision,
and lack thereof, about how they spin along, in the course of passage,
from zero, on to one, on to infinity. In this sense a number "infinity"
is most always a usual member of a projective space what's "numbers".

On 10/12/2023 10:16 PM, Ross Finlayson wrote:
> On Wednesday, October 11, 2023
> at 12:48:05 PM UTC-7, Jim Burns wrote:
>> On 10/11/2023 11:28 AM, Ross Finlayson wrote:

>>> line-continuity:
>>> field-continuity:
>>> signal-continuity:
>>> long-line-continuity:
>>>
>>> Thusly are at least _four_ definitions of
>>> completeness/gaplessness, and not just
>>> "conservative extensions,
>>> but incomplete and nicht richtiges".

>> What purposes do your three(?) other
>> continuities serve?

> It's great to have
> further definitions of continuity,
> and it's rather required, because
> each makes for the gaplessness and thusly
> an intermediate value theorem, and
> making for a milieu where
> the fundamental theorems of calculus hold up,
> that as well geometry for these "point-sets"
> where they are "lines",
> is richer, stronger, more together,
> and less missing:
> truths, about continuous domains.

Do you intend different continuities
to apply to different objects?

What are the definitions of
these different continuities?

> It's really quite a matter for those who
> study continuity [AKA topologists -- JB] and
> mathematics, because they want to be
> "conscientious mathematicians" and
> avoid "hypocrisy",

I am strongly of the opinion that
the class of mathematicians is
neither more nor less hypocritical than
the class of non-mathematicians,
neither more nor less _human_

I suspect that there is less hypocrisy
in mathematics, but the reason I give for that
is the reason I give (without research) for
there being fewer pickpockets in nudist camps:
considerably fewer opportunities.

> because there's quite a strong sense
> where rigor, intuition, and formalism,
> come together to make for that
> deduction rules that completeness, Dedekind's,
> is a distinctness, not a uniqueness.
>
> Then where something like
> "sweep the natural/unit equivalency function"
> is for probability theory
> a distribution of a distribution
> with multiple pdf's, remarkable,
> or the fact that it's its own anti-derivative,

I had gathered the distinct impression that
the sweep function/distribution/distribution²
was NOT the exponential function exp(x)

It is the exponential function which is
its own anti-derivative. ∫eˣdx = eˣ+C

> remarkable, as well
> quite fully enrich mathematics with
> "more and better" vs. "less and none".

I'm sorry.
All I got from that is
the other continuities do other stuff.

>> What purposes do your three(?) other
>> continuities serve?

On Thursday, October 12, 2023 at 10:07:25 PM UTC-7, Jim Burns wrote:
> On 10/12/2023 10:16 PM, Ross Finlayson wrote:
> > On Wednesday, October 11, 2023
> > at 12:48:05 PM UTC-7, Jim Burns wrote:
> >> On 10/11/2023 11:28 AM, Ross Finlayson wrote:
>
> >>> line-continuity:
> >>> field-continuity:
> >>> signal-continuity:
> >>> long-line-continuity:
> >>>
> >>> Thusly are at least _four_ definitions of
> >>> completeness/gaplessness, and not just
> >>> "conservative extensions,
> >>> but incomplete and nicht richtiges".
> >> What purposes do your three(?) other
> >> continuities serve?
>
> > It's great to have
> > further definitions of continuity,
> > and it's rather required, because
> > each makes for the gaplessness and thusly
> > an intermediate value theorem, and
> > making for a milieu where
> > the fundamental theorems of calculus hold up,
> > that as well geometry for these "point-sets"
> > where they are "lines",
> > is richer, stronger, more together,
> > and less missing:
> > truths, about continuous domains.
> Do you intend different continuities
> to apply to different objects?
>
> What are the definitions of
> these different continuities?
> > It's really quite a matter for those who
> > study continuity [AKA topologists -- JB] and
> > mathematics, because they want to be
> > "conscientious mathematicians" and
> > avoid "hypocrisy",
> I am strongly of the opinion that
> the class of mathematicians is
> neither more nor less hypocritical than
> the class of non-mathematicians,
> neither more nor less _human_
>
> I suspect that there is less hypocrisy
> in mathematics, but the reason I give for that
> is the reason I give (without research) for
> there being fewer pickpockets in nudist camps:
> considerably fewer opportunities.
> > because there's quite a strong sense
> > where rigor, intuition, and formalism,
> > come together to make for that
> > deduction rules that completeness, Dedekind's,
> > is a distinctness, not a uniqueness.
> >
> > Then where something like
> > "sweep the natural/unit equivalency function"
> > is for probability theory
> > a distribution of a distribution
> > with multiple pdf's, remarkable,
> > or the fact that it's its own anti-derivative,
> I had gathered the distinct impression that
> the sweep function/distribution/distribution²
> was NOT the exponential function exp(x)
>
> It is the exponential function which is
> its own anti-derivative. ∫eˣdx = eˣ+C
> > remarkable, as well
> > quite fully enrich mathematics with
> > "more and better" vs. "less and none".
> I'm sorry.
> All I got from that is
> the other continuities do other stuff.
> >> What purposes do your three(?) other
> >> continuities serve?

Indeed, it's topology and function theory that over time see the most
variety in definition, then about how to fit together continuity, continuity,
and continuity.

Of course everybody knows the usual formalism, limits and delta-epsilonics,
after the usual formalism of the operations of arithmetic and equalities and
inequalities, and that the objects of arithmetic (or "numbers") aren't indivisible.

Then, most people's notions of uniform continuous motion, in time, are advised
by the thought experiment of Zeno of the Eliatics, about induction over halving,
illustrating both the application of meter and that time is as divisible as anything else.

Of course, because a usual course of instruction requires everybody having sat
(and written) the same derivations, it's as follows delta-epsilonics and the couched
concept of limit, for what arrives at analysis, of course as usually over a schooling
in algebra, geometry, and most often trigonometry, as the common setting.
So, delta-epsilonics and what later also fulfills being the abstract algebra's complete
ordered field, is the usual thing.

Still, continuous uniform motion, in time, follows a course of passage through
all the real numbers between zero and one, for example, and in order, there is
a course-of-passage, of time, through the unit, just as there is of the integers,
through infinity.

So, topologists, and function theorists, for example those in physics, might see
mention, for example, Jordan measure, for line elements and path elements,
which are the analytical character of line elements and points, exactly so described
by their correlation to a continuous domain, time, between 0 and 1 or usually a and b.

Thusly there's line-continuity, then quite all in the middle field-continuity, then
when getting to these notions like the field if not the complete ordered field,
being equi-distributed but nowhere continuous, results that their complement
makes their union, as indistinguishably so for each other, a continuous domain,
signal-continuity. This is usually called a Dirichlet function, then that among various
notions of constructs in topology, that are ultrafilters, and complete ultrafilters,
and complete metrizing ultrafilters, topology's high-level hypocritical conceit of
what are unresolved developments of continuous domains, each.

So, how to sort this out results the axiomatic set theory and descriptive set theory,
then needs amends in its treatment of the consideration of the existence of functions,
where "line-continuity's sweep" or the prototype of a function from natural, to the unit,
in order, filling it, first must fall all out of set theory's that the complete ordered field
is uncountable, about Cantor space, and representations of each of the infinite sequences,
between zero and one, in binary. It does, sweep falls out of all the developments of uncountability,
as a counterexample, if of course, "not a real function", that it's a function, and not a Cartesian
function, that it's a function, about also necessary amends or retrofit or simply enough results
that follow, in axiomatic set theory, how there are those both, line-continuity and field-continuity,
and those being "continuous domains", in the space of real values (or, [0,1], then for a lattice
of integers and integer parts and non-integer parts).

So, "other continuities do not do other stuff", they are different things, that have the
same properties, or attributes, of being continuous domains. And, various models they
effect, like as of physics, are various, for what they do and what they are and how they
don't necessarily so connect, except about integer points and the bridges or ponts of
where extensionality the property of their having the same range of real values,
exist in the space of real values.

Then, this sweep function is pretty special, it is its own anti-derivative, which makes
that now the exponential is special with it in being distinct, if not unique, that way,
and, it illustrates doubling in the doubling space, making for example why Vitali's
result is re-written, as an example in the effect in doubling spaces, in real analytical
character, that it's central, primary, and the most relevant fixture in line-drawing
and measurement of time, in objects of mathematics.

So, I reduced this to two slates, one, about uncountability, why sweep is not a Cartesian function,
thusly then as it's shown that there's uncountability but also this special countable continuous
domain, and, one about paradox, writing out and down why there are none.

On Thursday, October 12, 2023 at 11:40:33 PM UTC-7, Ross Finlayson wrote:
> On Thursday, October 12, 2023 at 10:07:25 PM UTC-7, Jim Burns wrote:
> > On 10/12/2023 10:16 PM, Ross Finlayson wrote:
> > > On Wednesday, October 11, 2023
> > > at 12:48:05 PM UTC-7, Jim Burns wrote:
> > >> On 10/11/2023 11:28 AM, Ross Finlayson wrote:
> >
> > >>> line-continuity:
> > >>> field-continuity:
> > >>> signal-continuity:
> > >>> long-line-continuity:
> > >>>
> > >>> Thusly are at least _four_ definitions of
> > >>> completeness/gaplessness, and not just
> > >>> "conservative extensions,
> > >>> but incomplete and nicht richtiges".
> > >> What purposes do your three(?) other
> > >> continuities serve?
> >
> > > It's great to have
> > > further definitions of continuity,
> > > and it's rather required, because
> > > each makes for the gaplessness and thusly
> > > an intermediate value theorem, and
> > > making for a milieu where
> > > the fundamental theorems of calculus hold up,
> > > that as well geometry for these "point-sets"
> > > where they are "lines",
> > > is richer, stronger, more together,
> > > and less missing:
> > > truths, about continuous domains.
> > Do you intend different continuities
> > to apply to different objects?
> >
> > What are the definitions of
> > these different continuities?
> > > It's really quite a matter for those who
> > > study continuity [AKA topologists -- JB] and
> > > mathematics, because they want to be
> > > "conscientious mathematicians" and
> > > avoid "hypocrisy",
> > I am strongly of the opinion that
> > the class of mathematicians is
> > neither more nor less hypocritical than
> > the class of non-mathematicians,
> > neither more nor less _human_
> >
> > I suspect that there is less hypocrisy
> > in mathematics, but the reason I give for that
> > is the reason I give (without research) for
> > there being fewer pickpockets in nudist camps:
> > considerably fewer opportunities.
> > > because there's quite a strong sense
> > > where rigor, intuition, and formalism,
> > > come together to make for that
> > > deduction rules that completeness, Dedekind's,
> > > is a distinctness, not a uniqueness.
> > >
> > > Then where something like
> > > "sweep the natural/unit equivalency function"
> > > is for probability theory
> > > a distribution of a distribution
> > > with multiple pdf's, remarkable,
> > > or the fact that it's its own anti-derivative,
> > I had gathered the distinct impression that
> > the sweep function/distribution/distribution²
> > was NOT the exponential function exp(x)
> >
> > It is the exponential function which is
> > its own anti-derivative. ∫eˣdx = eˣ+C
> > > remarkable, as well
> > > quite fully enrich mathematics with
> > > "more and better" vs. "less and none".
> > I'm sorry.
> > All I got from that is
> > the other continuities do other stuff.
> > >> What purposes do your three(?) other
> > >> continuities serve?
> Indeed, it's topology and function theory that over time see the most
> variety in definition, then about how to fit together continuity, continuity,
> and continuity.
>
> Of course everybody knows the usual formalism, limits and delta-epsilonics,
> after the usual formalism of the operations of arithmetic and equalities and
> inequalities, and that the objects of arithmetic (or "numbers") aren't indivisible.
>
> Then, most people's notions of uniform continuous motion, in time, are advised
> by the thought experiment of Zeno of the Eliatics, about induction over halving,
> illustrating both the application of meter and that time is as divisible as anything else.
>
> Of course, because a usual course of instruction requires everybody having sat
> (and written) the same derivations, it's as follows delta-epsilonics and the couched
> concept of limit, for what arrives at analysis, of course as usually over a schooling
> in algebra, geometry, and most often trigonometry, as the common setting.
> So, delta-epsilonics and what later also fulfills being the abstract algebra's complete
> ordered field, is the usual thing.
>
> Still, continuous uniform motion, in time, follows a course of passage through
> all the real numbers between zero and one, for example, and in order, there is
> a course-of-passage, of time, through the unit, just as there is of the integers,
> through infinity.
>
>
> So, topologists, and function theorists, for example those in physics, might see
> mention, for example, Jordan measure, for line elements and path elements,
> which are the analytical character of line elements and points, exactly so described
> by their correlation to a continuous domain, time, between 0 and 1 or usually a and b.
>
> Thusly there's line-continuity, then quite all in the middle field-continuity, then
> when getting to these notions like the field if not the complete ordered field,
> being equi-distributed but nowhere continuous, results that their complement
> makes their union, as indistinguishably so for each other, a continuous domain,
> signal-continuity. This is usually called a Dirichlet function, then that among various
> notions of constructs in topology, that are ultrafilters, and complete ultrafilters,
> and complete metrizing ultrafilters, topology's high-level hypocritical conceit of
> what are unresolved developments of continuous domains, each.
>
> So, how to sort this out results the axiomatic set theory and descriptive set theory,
> then needs amends in its treatment of the consideration of the existence of functions,
> where "line-continuity's sweep" or the prototype of a function from natural, to the unit,
> in order, filling it, first must fall all out of set theory's that the complete ordered field
> is uncountable, about Cantor space, and representations of each of the infinite sequences,
> between zero and one, in binary. It does, sweep falls out of all the developments of uncountability,
> as a counterexample, if of course, "not a real function", that it's a function, and not a Cartesian
> function, that it's a function, about also necessary amends or retrofit or simply enough results
> that follow, in axiomatic set theory, how there are those both, line-continuity and field-continuity,
> and those being "continuous domains", in the space of real values (or, [0,1], then for a lattice
> of integers and integer parts and non-integer parts).
>
> So, "other continuities do not do other stuff", they are different things, that have the
> same properties, or attributes, of being continuous domains. And, various models they
> effect, like as of physics, are various, for what they do and what they are and how they
> don't necessarily so connect, except about integer points and the bridges or ponts of
> where extensionality the property of their having the same range of real values,
> exist in the space of real values.
>
>
>
> Then, this sweep function is pretty special, it is its own anti-derivative, which makes
> that now the exponential is special with it in being distinct, if not unique, that way,
> and, it illustrates doubling in the doubling space, making for example why Vitali's
> result is re-written, as an example in the effect in doubling spaces, in real analytical
> character, that it's central, primary, and the most relevant fixture in line-drawing
> and measurement of time, in objects of mathematics.
>
>
>
> So, I reduced this to two slates, one, about uncountability, why sweep is not a Cartesian function,
> thusly then as it's shown that there's uncountability but also this special countable continuous
> domain, and, one about paradox, writing out and down why there are none.

As thusly "irrefutable, mathematical, fact".

On 10/12/2023 10:16 PM, Ross Finlayson wrote:
> On Wednesday, October 11, 2023
> at 12:48:05 PM UTC-7, Jim Burns wrote:
>> On 10/11/2023 11:28 AM, Ross Finlayson wrote:

>>> line-continuity:
>>> field-continuity:
>>> signal-continuity:
>>> long-line-continuity:
>>>
>>> Thusly are at least _four_ definitions of
>>> completeness/gaplessness, and not just
>>> "conservative extensions,
>>> but incomplete and nicht richtiges".

I would appreciate the favor of
one sentence about line-continuity
one sentence about field-continuity
one sentence about signal-continuity
one sentence about long-line-continuity
intended to convey what they mean
to someone who doesn't already know
what they mean.

Thank you in advance.

The single notion of continuity with which
I am familiar gives rise to
the delta-epsilonic definition.
It is the widely-used notion, I think.
Call it function-continuity.

Function f: X ⟶ Y is continuous at x
iff
outputs y′=f(x′) can be kept near y=f(x)
by keeping inputs x′ near x

How near x′ needs to be
depends on how near y′ wants to be.
Thus we have "for each ε>0 exists δ>0"

In the negation of function-continuity,
there is some ε>0 which y′=f(x′) stays
farther than, for each δ>0
f jumps at x, by at least ε
f is not function-continuous at x

There is a related notion, not about
functions, but about domains, which you
seem to be pointing at: connectedness.
I'll call that domain-continuity.

S is connected
iff
S cannot be divided into
two non-overlapping _open_ sets.

For example,
ℝ can be divided into (-∞,π] (π,+∞)
(-∞,π] isn't open, though.

ℝ\{π} can be divided into (-∞,π) (π,+∞)
ℝ\{π} is not connected.

ℝ is connected
(-∞,π) is connected
(π,+∞) is connected
as a result of Dedekind completeness.

In order to rule out jumps,
there must be both function-continuity
and domain-continuity.

Consider a fly buzzing around your room.
Its path is a function from (3,4) to ℝ³
The function is continuous.
At no point does the fly teleport.

Consider two flies buzzing around.
Their paths are two continuous functions,
one from (3,π) to ℝ³ and
the other from (π,4) to ℝ³

At no point does either fly teleport.
But the flies could be in different rooms,
on different planets, in different galaxies.

Without π
without domain-continuity,
we can still have function-continuity,
but it won't mean what we want it to mean.

I am guessing that
line-continuity
field-continuity
signal-continuity
long-line-continuity
are intended to construct ℝ
and, in that way,
construct domain-continuity.

I am guessing that

line-continuity fills in the rationals
between the integers.

field-continuity extends + - * / >
from the integers to the rationals

signal-continuity plugs
the one-point gaps in the rationals,
giving domain-continuity

long-line-continuity extends the
line- field- signal-continuity toward ±∞

Even if you only say is "yes" or "no"
at this point, I think that would be
a help toward toward understanding you.

On Friday, October 13, 2023 at 11:44:33 AM UTC-7, Jim Burns wrote:
> On 10/12/2023 10:16 PM, Ross Finlayson wrote:
> > On Wednesday, October 11, 2023
> > at 12:48:05 PM UTC-7, Jim Burns wrote:
> >> On 10/11/2023 11:28 AM, Ross Finlayson wrote:
>
> >>> line-continuity:
> >>> field-continuity:
> >>> signal-continuity:
> >>> long-line-continuity:
> >>>
> >>> Thusly are at least _four_ definitions of
> >>> completeness/gaplessness, and not just
> >>> "conservative extensions,
> >>> but incomplete and nicht richtiges".
> I would appreciate the favor of
> one sentence about line-continuity
> one sentence about field-continuity
> one sentence about signal-continuity
> one sentence about long-line-continuity
> intended to convey what they mean
> to someone who doesn't already know
> what they mean.
>
> Thank you in advance.
>
>
> The single notion of continuity with which
> I am familiar gives rise to
> the delta-epsilonic definition.
> It is the widely-used notion, I think.
> Call it function-continuity.
>
> Function f: X ⟶ Y is continuous at x
> iff
> outputs y′=f(x′) can be kept near y=f(x)
> by keeping inputs x′ near x
>
> How near x′ needs to be
> depends on how near y′ wants to be.
> Thus we have "for each ε>0 exists δ>0"
>
> In the negation of function-continuity,
> there is some ε>0 which y′=f(x′) stays
> farther than, for each δ>0
> f jumps at x, by at least ε
> f is not function-continuous at x
>
>
> There is a related notion, not about
> functions, but about domains, which you
> seem to be pointing at: connectedness.
> I'll call that domain-continuity.
>
> S is connected
> iff
> S cannot be divided into
> two non-overlapping _open_ sets.
>
> For example,
> ℝ can be divided into (-∞,π] (π,+∞)
> (-∞,π] isn't open, though.
>
> ℝ\{π} can be divided into (-∞,π) (π,+∞)
> ℝ\{π} is not connected.
>
> ℝ is connected
> (-∞,π) is connected
> (π,+∞) is connected
> as a result of Dedekind completeness.
>
> In order to rule out jumps,
> there must be both function-continuity
> and domain-continuity.
>
> Consider a fly buzzing around your room.
> Its path is a function from (3,4) to ℝ³
> The function is continuous.
> At no point does the fly teleport.
>
> Consider two flies buzzing around.
> Their paths are two continuous functions,
> one from (3,π) to ℝ³ and
> the other from (π,4) to ℝ³
>
> At no point does either fly teleport.
> But the flies could be in different rooms,
> on different planets, in different galaxies.
>
> Without π
> without domain-continuity,
> we can still have function-continuity,
> but it won't mean what we want it to mean.
>
>
> I am guessing that
> line-continuity
> field-continuity
> signal-continuity
> long-line-continuity
> are intended to construct ℝ
> and, in that way,
> construct domain-continuity.
>
> I am guessing that
>
> line-continuity fills in the rationals
> between the integers.
>
> field-continuity extends + - * / >
> from the integers to the rationals
>
> signal-continuity plugs
> the one-point gaps in the rationals,
> giving domain-continuity
>
> long-line-continuity extends the
> line- field- signal-continuity toward ±∞
>
> Even if you only say is "yes" or "no"
> at this point, I think that would be
> a help toward toward understanding you.

How does math quantize physical volume?

On 10/13/2023 2:48 PM, mitchr...@gmail.com wrote:
> On Friday, October 13, 2023
> at 11:44:33 AM UTC-7, Jim Burns wrote:

> [...]

> How does math quantize physical volume?

https://en.wikipedia.org/wiki/Volume

| Volume in general is a Jordan content,
| so the volume would satisfy these axioms:
| | For all S in M, a(S) ≥ 0.
| | If S and T are in M
| then so are S ∪ T and S ∩ T,
| and also a(S∪T) = a(S) + a(T) − a(S ∩ T).
| | If S and T are in M with S ⊆ T
| then T − S is in M
| and a(T−S) = a(T) − a(S).
| | If a set S is in M and
| S is congruent to T
| then T is also in M and a(S) = a(T).
| | Every cuboid R is in M.
| If the rectangle has
| length a, breadth b, and height c
| then V(R) = abc.
| | Let Q be a set enclosed between
| two step regions S and T.
| A step region is formed from
| a finite union of adjacent cuboid
| resting on a common surface,
| i.e. S ⊆ Q ⊆ T.
| If there is a unique number c such that
| a(S) ≤ c ≤ a(T)
| for all such step regions S and T,
| then a(Q) = c.

On Friday, October 13, 2023 at 11:44:33 AM UTC-7, Jim Burns wrote:
> On 10/12/2023 10:16 PM, Ross Finlayson wrote:
> > On Wednesday, October 11, 2023
> > at 12:48:05 PM UTC-7, Jim Burns wrote:
> >> On 10/11/2023 11:28 AM, Ross Finlayson wrote:
>
> >>> line-continuity:
> >>> field-continuity:
> >>> signal-continuity:
> >>> long-line-continuity:
> >>>
> >>> Thusly are at least _four_ definitions of
> >>> completeness/gaplessness, and not just
> >>> "conservative extensions,
> >>> but incomplete and nicht richtiges".
> I would appreciate the favor of
> one sentence about line-continuity
> one sentence about field-continuity
> one sentence about signal-continuity
> one sentence about long-line-continuity
> intended to convey what they mean
> to someone who doesn't already know
> what they mean.
>
> Thank you in advance.
>
>
> The single notion of continuity with which
> I am familiar gives rise to
> the delta-epsilonic definition.
> It is the widely-used notion, I think.
> Call it function-continuity.
>
> Function f: X ⟶ Y is continuous at x
> iff
> outputs y′=f(x′) can be kept near y=f(x)
> by keeping inputs x′ near x
>
> How near x′ needs to be
> depends on how near y′ wants to be.
> Thus we have "for each ε>0 exists δ>0"
>
> In the negation of function-continuity,
> there is some ε>0 which y′=f(x′) stays
> farther than, for each δ>0
> f jumps at x, by at least ε
> f is not function-continuous at x
>
>
> There is a related notion, not about
> functions, but about domains, which you
> seem to be pointing at: connectedness.
> I'll call that domain-continuity.
>
> S is connected
> iff
> S cannot be divided into
> two non-overlapping _open_ sets.
>
> For example,
> ℝ can be divided into (-∞,π] (π,+∞)
> (-∞,π] isn't open, though.
>
> ℝ\{π} can be divided into (-∞,π) (π,+∞)
> ℝ\{π} is not connected.
>
> ℝ is connected
> (-∞,π) is connected
> (π,+∞) is connected
> as a result of Dedekind completeness.
>
> In order to rule out jumps,
> there must be both function-continuity
> and domain-continuity.
>
> Consider a fly buzzing around your room.
> Its path is a function from (3,4) to ℝ³
> The function is continuous.
> At no point does the fly teleport.
>
> Consider two flies buzzing around.
> Their paths are two continuous functions,
> one from (3,π) to ℝ³ and
> the other from (π,4) to ℝ³
>
> At no point does either fly teleport.
> But the flies could be in different rooms,
> on different planets, in different galaxies.
>
> Without π
> without domain-continuity,
> we can still have function-continuity,
> but it won't mean what we want it to mean.
>
>
> I am guessing that
> line-continuity
> field-continuity
> signal-continuity
> long-line-continuity
> are intended to construct ℝ
> and, in that way,
> construct domain-continuity.
>
> I am guessing that
>
> line-continuity fills in the rationals
> between the integers.
>
> field-continuity extends + - * / >
> from the integers to the rationals
>
> signal-continuity plugs
> the one-point gaps in the rationals,
> giving domain-continuity
>
> long-line-continuity extends the
> line- field- signal-continuity toward ±∞
>
> Even if you only say is "yes" or "no"
> at this point, I think that would be
> a help toward toward understanding you.

It is commendable, that continuous domains, like time,
make for continuous functions the properties of continuous functions.

Identity's a continuous function on a continuous domain.

The Cantor space, is the set of, all the sequences, infinite,
of the 0's and 1's.

The square Cantor space: is those in order, their natural order,
as that the series are expansions.

The other use of powerset representation of the Cantor space
is to represent each of a countable domain's, members, presence,
in any of the powersets or "sets of all subsets", the set.

So, Cantor space is uncountable, but square Cantor space is "countable",
after a geometrization and length assignment the space, line-continuity's,
"including the diagonal".

This way a "Cantor space", and a, "_Square_" Cantor space, "Square Cantor Space",
is from it results that there's a set with all the elements in order, thus sweep crosses it.

It's called "Square" rather as of that it does result, "square", because Cantor space is
infinitely long while "Square Cantor space" is "square", countable down and across,
and for continuous identity reflects that in Cantor space, the diagonal never exists,
while in Square Cantor space, the diagonal always exists.

The "anti-diagonal" argument, thusly is named this way instead of the usual "diagonal"
argument, with respect to that usual "diagonalizations", reflect a bit opposite the sense,
of "down and across, the diagonal", why since some time Cantor's "Diagonal" argument
is called 'Cantor's _Anti_ Diagonal" argument.

This is where "diagonalization" as it were is usually opposite the intent, "anti-diagonalization",
when it really means to satisfy what's there (that countability naturally leaves out).

It's the first nonstandard function "what is it? It's drawing a line and saying it was points in
order, as by integers in order, must be infinite".

Then that being a modulus, is that it's an example of what's called "clock arithmetic",
as for that in continuous quantities it's as if governed by time, a continuous domain,
"line-drawing", drawing a finite line segment and calling the first point zero and last
infinity, and that it's "0, 1, ..., infinity", if as in reverse "infinity - 0, infinity - 1, ..., zero".
I.e. they're the same numbers, opposite order.

I came looking for mathematics and when I found out that there were missing formalisms
for what I knew, I was like "where's the entire world of naming how this all fits together
these modes of continuity", "modes", and it was like "make one yourself" and thusly
it would be unconscionable to not say "here let me give this an opinion and an argument
for itself", "must be so", and I was like "really? I get to do all this myself?" and it's like
"I guess it depends if they want to go along".

So, I mostly made it for myself to have no reason why not.

It's all the 1's and 0's.

It's not called "modes", though, continuity, here it is still only getting to "domains",
"continuous domains".

It's all the 1's and 0's, ....

So, continuous domains, have that it's continuous, functions on them identity,
with no jumps or gaps, which when continuous domains of different areas
are put together, if there's no jumps or gaps is called "piece-wise" continuity,
unless smooth.

So, continuity and smooth continuity and piece-wise continuity, are the usual terms.

To your point, then,

"I [JB] am guessing that

1) line-continuity fills in the rationals
between the integers.

2) field-continuity extends + - * / >
from the integers to the rationals

3) signal-continuity plugs
the one-point gaps in the rationals,
giving domain-continuity

4) long-line-continuity extends the
line- field- signal-continuity toward ±∞

Even if you only say is "yes" or "no"
at this point, I think that would be
a help toward toward understanding you.

1: "line-continuity" is a "[0,1] unit line segment"
real numbers are integer part and non-integer part
it is the non-integer part, of otherwise integers
it's countable

2: "field-continuity" is "the complete ordered field", or "least-upper-bound of complete ordered field"
it is integers, and their ratios
integers are reduced fractions
fractions are ratios of integers
it's uncountable

3: "signal-continuity" is "was everywhere dense and discontinuous, now dense and continuous",
it's doubling from "dense" to "continuity"
it's the same cardinality

4) long-line-continuity
the space of real functions is an expression in real values, they're all ordered as asymptotically
it's uncountable to field-continuity, "twice uncountable"

On Friday, October 13, 2023 at 12:37:44 PM UTC-7, Jim Burns wrote:
> On 10/13/2023 2:48 PM, mitchr...@gmail.com wrote:
> > On Friday, October 13, 2023
> > at 11:44:33 AM UTC-7, Jim Burns wrote:
> > [...]
> > How does math quantize physical volume?
> https://en.wikipedia.org/wiki/Volume

Quantization is at the infinitely small for particles in silly string theory...
How is a volume quantized in the same way?

On Wednesday, October 11, 2023 at 6:38:08 AM UTC+3, Ross Finlayson wrote:
> On Tuesday, October 10, 2023 at 8:10:31 PM UTC-7, bassam karzeddin wrote:
> > On Monday, October 9, 2023 at 11:56:20 PM UTC+3, Jim Burns wrote:
> > > On 10/9/2023 8:27 AM, bassam karzeddin wrote:
> > > > On Monday, October 9, 2023
> > > > at 1:02:10 PM UTC+3, bassam karzeddin wrote:
> > > >> On Monday, October 9, 2023
> > > >> at 7:48:28 AM UTC+3, Jim Burns wrote:
> > > >>> On 10/8/2023 12:56 PM, bassam karzeddin wrote:
> > > >>>> On Saturday, August 8, 2020
> > > >>>> at 11:27:08 AM UTC+3, bassam karzeddin wrote:
> > >
> > > > what is truly a real number?
> > > A real number x ∈ ℝ is
> > > a rational number x ∈ ℚ or
> > > a point x between a split F‖H of ℚ
> > > F ᣔ< x <ᣔ H
> > > F∪H = ℚ\{x}
> > > > And assuming in good faith that
> > > > a real number which is
> > > > not a constructible number exists,
> > > > then mathematicians are globally
> > > > revealed [requested?] to present
> > > > only one of them
> > > > either numerically or Geometrically
> > > > but Exactly [...]
> > >
> > > Points-between-splits are geometry.
> > >
> > > Without all points-between-splits,
> > > some continuous curves cross but
> > > don't intersect,
> > > which is not geometry.
> > >
> > >
> > > Points-between-splits are exact.
> > >
> > > For each split F‖H of ℚ and
> > > for each distance d > 0
> > > there exist rationals p₋ p₊
> > > p₋ ∈ F ᣔ< x₁ <ᣔ H ∋ p₊
> > > p₋ ∈ F ᣔ< x₂ <ᣔ H ∋ p₊
> > > such that p₊-p₋ < d
> > >
> > > However,
> > > if there are two between-points x₁ x₂
> > > then
> > > there is some d ≥ |x₂-x₁| > 0
> > > such that ¬(p₊-p₋ < d)
> > > for all p₋ p₊
> > > Contradiction.
> > >
> > > Thus,
> > > for each split,
> > > there is at most one between-point x
> > >
> > > | The point 2¹ᐟ³ between
> > > | F = {p ∈ ℚ| p³ < 2} and
> > > | H = {p ∈ ℚ| 2 < p³}
> > > |
> > > describes exactly one point.
> > Yes, here is exactly the whole unnoticeable point of all kinds of confusions
> >
> > P^3 < 2, you mean exactly the greatest real number for p^3 that is strictly less than 2, Right ✅️?
> >
> > Which of course doesn't exist, ✅️ Right?
> > Similarly; when you are describing the least real number for P^3 that is strictly greater than 2; Right ✅️?
> > Which of course doesn't exist, Right ✅️?
> > Then you aren't talking about a point or an exact location on the real number line, but talking about a distance that you conclude as very small distance & permit yourself to neglect its entire existence inorder to justify illegally your conclusions based entirely on an eingineering point of view whish is completely irrelevant to any theoretical mathematics
> >
> > Aren't you an eingineer or at least with eingineering thinking as if you want to solve a problem such that a carpenter can make approximately a cube with nearly two units volume
> >
> > This is exactly Doubling the cube problem which is one of the most popular historical problem raised by ancient Greeks few thousands of years back, where it was known as impossible task by unmarked straigt edge & a compass
> >
> > And Wentzel proof in 1836 is actually not any valid proof but a true conclusion only, since he couldn't understand the non-existing numbers like Cubrt2, which confirms such absolute impossibilities about all the Greeks problems
> >
> > Where,with false proof by Wentzel, and without understanding the non-existing principles He kept the doors opened widely to all skilled carpenters, scientists & mathematickers to invent new tools & many approximations methods inorder to get that number as Cubrt2 that exists only & strictly in human minds as a matter of beliefs & necessities, & still humans try tirelessly to invent new methods inorder to catch that belived number which never exists EXACTLY, but some how, humans generally believe that they are very close to it
> >
> > Of course, the three famous historical Greeks problems are absolutely impossible to solve bu any tools & by any means of human methods of endless approximations
> >
> > Those problems had been completely missed by ALL humans up to this date, & mathematics got as a result all kinds of contradictions & inconsistency that every one is talking about nowadays
> >
> > Which is why I claim that human mental problems with mathematics aren't in mathematics itself but completely with mathematians themselves for purely human desires of attention & achievements
> >
> > However, the whole world 🌎 academy of Geometry & Number theorists are requested to reinvestigaite legally the 3 Greeks insolvable problems by using the Artificial Intelligence nowadays
> >
> > 🔊 Bassam Karzeddin 🔊
> Well that's uninformed, an "Archimedean spiral" as third classical tool
> makes it pretty easy to "square the circle", for example, as it were.
>
> Then, also and besides, arithmetic and algebra and analysis are "tools",
> and pi is a well-known constant, so "squaring the circle" is pretty easy,
> as are the other notions like "dividing by three" and taking and making
> powers and roots.
>
> Oh, you mean like sticks, that we dig in the mud for grubs,
> yeah, those must be some tasty grubs.

The main misunderstood problem in mathematics & science as well was quite old since the start of human mathematical knowledge by the deep deception of the existence of the so-called circle 🔵 in all human languages, where so unfortunately, the object circle isn't any existing object in true mathematics, despite being visible to human limited eyes & visibility, which was the only true reason behind the absolute impossibility of squaring the circle by any tools or means of human eye marking or any other method of endless approximations

However, what do exist actually the regular constructible polygons only which seem like a circle to human eyes whenever the regular polygons are relatively with many sides, where a side distance is too small to be visible by human eyes with their extreme disability of visibility

Not only the circle which is a non-existing object in true mathematics but also the vast majorities of regular polygons as well, where the regular heptagon is also a non- existing object in true mathematics

But, I know that any skilled carpenter even before BC, can easily make a regular polygon as heptagon to a degree of accuracy that can please ALL humans on earth 🌎 & the skies as well, but definitely, never EXACTLY like the regular Pentagon for example which Ecluid could make few thousands years back

However, one cannot bring all his public published posts regarding this issue to convince others about such a rarest historical claims, where the issue is really too big & beyond the human normal capacity to understand it so easily despite the fact that the whole issue is truly too simple to understand

But since humans had already established so many towers of false underlying concepts; then it becomes impossible to understand many more things like the non-existing angles for example, where this needs urgent verifications & strictly by using the so advanced & independent Artificial Intelligence inorder to uncover the untought truths that are in full contradictionS with human illegal desires & behaviors to achieve many illegal discoveries since thousands of years
Many remaining & relevant public published posts are there in my profiles that well-explain & proves my unique point of view that seems so frustrating to people not necessarily only mathematicians

BKK

On 10/13/2023 10:58 PM, mitchr...@gmail.com wrote:
> On Friday, October 13, 2023
> at 12:37:44 PM UTC-7, Jim Burns wrote:
>> On 10/13/2023 2:48 PM, mitchr...@gmail.com wrote:
>>> On Friday, October 13, 2023
>>> at 11:44:33 AM UTC-7, Jim Burns wrote:

>>>> [...]
>>>
>>> How does math quantize physical volume?
>>
>> https://en.wikipedia.org/wiki/Volume
>
> Quantization is at the infinitely small
> for particles in silly string theory...

No,
quantization is at the f$#%ing-small,
not at the infinitely-small.

"Infinite" has a meaning.
It's not a way to say f$#%ing in Mathtalk.

The main reason string theory is
considered seriously is that
strings are not infinitely-small.

Because strings are only f$#%ing-small,
calculations avoid the silly results which
plague old-fashioned general relativity with
quantum mechanics at the infinitely-small.

> How is a volume quantized in the same way?

I don't know.
I think no one knows, so far.
I hope I would have heard, otherwise.

All of this is still science-at-work,
not settled-science.

Any science you would have seen in
a classroom is settled-science.
but earlier, it was science-at-work.
Hundreds of years earlier, for
some science.

It is settled-science that
SOMETHING MUST BE DONE about
general relativity with quantum mechanics
because of calculations with silly results.
What the SOMETHING which MUST BE DONE is
is not yet settled.

Quantizing volume might well be
that SOMETHING when it's settled.
Ask again in a hundred years.

On Saturday, October 14, 2023 at 8:49:37 AM UTC-7, Jim Burns wrote:
> On 10/13/2023 10:58 PM, mitchr...@gmail.com wrote:
> > On Friday, October 13, 2023
> > at 12:37:44 PM UTC-7, Jim Burns wrote:
> >> On 10/13/2023 2:48 PM, mitchr...@gmail.com wrote:
> >>> On Friday, October 13, 2023
> >>> at 11:44:33 AM UTC-7, Jim Burns wrote:
>
> >>>> [...]
> >>>
> >>> How does math quantize physical volume?
> >>
> >> https://en.wikipedia.org/wiki/Volume
> >
> > Quantization is at the infinitely small
> > for particles in silly string theory...
> No,
> quantization is at the f$#%ing-small,
> not at the infinitely-small.
>
> "Infinite" has a meaning.
> It's not a way to say f$#%ing in Mathtalk.
>
> The main reason string theory is
> considered seriously is that
> strings are not infinitely-small.
>
> Because strings are only f$#%ing-small,
> calculations avoid the silly results which
> plague old-fashioned general relativity with
> quantum mechanics at the infinitely-small.
> > How is a volume quantized in the same way?
> I don't know.
> I think no one knows, so far.
> I hope I would have heard, otherwise.
>
> All of this is still science-at-work,
> not settled-science.
>
> Any science you would have seen in
> a classroom is settled-science.
> but earlier, it was science-at-work.
> Hundreds of years earlier, for
> some science.
>
> It is settled-science that
> SOMETHING MUST BE DONE about
> general relativity with quantum mechanics
> because of calculations with silly results.
> What the SOMETHING which MUST BE DONE is
> is not yet settled.
>
> Quantizing volume might well be
> that SOMETHING when it's settled.
> Ask again in a hundred years.

The idea of string theory actually is that
atoms are about 10^ -25 times smaller,
superstrings are about 10^ -50 times smaller,
that 50 is about 2 times 25, that superstrings
are grainy to what to atoms are smooth.

The whole idea of superstrings is just
"arbitrarily small lengths to what we've
found according to electron physics to be
the ratios defining Avogadro's number
about 10^23".

It might help if you know that every few years,
the Particle Data Group or NIST CODATA, releases
new values of these smallest constants, and,
over time, they always get smaller, not just more precise ,
but smaller, it's due what's called a "regime of running constants",
according to energy and configuration of experiment,
and quantization and atomic and sub-atomic theory,
has much involved for it the theory of "renormalization".

On Saturday, October 14, 2023 at 10:49:12 AM UTC-7, Ross Finlayson wrote:
> On Saturday, October 14, 2023 at 8:49:37 AM UTC-7, Jim Burns wrote:
> > On 10/13/2023 10:58 PM, mitchr...@gmail.com wrote:
> > > On Friday, October 13, 2023
> > > at 12:37:44 PM UTC-7, Jim Burns wrote:
> > >> On 10/13/2023 2:48 PM, mitchr...@gmail.com wrote:
> > >>> On Friday, October 13, 2023
> > >>> at 11:44:33 AM UTC-7, Jim Burns wrote:
> >
> > >>>> [...]
> > >>>
> > >>> How does math quantize physical volume?
> > >>
> > >> https://en.wikipedia.org/wiki/Volume
> > >
> > > Quantization is at the infinitely small
> > > for particles in silly string theory...
> > No,
> > quantization is at the f$#%ing-small,
> > not at the infinitely-small.
> >
> > "Infinite" has a meaning.
> > It's not a way to say f$#%ing in Mathtalk.
> >
> > The main reason string theory is
> > considered seriously is that
> > strings are not infinitely-small.
> >
> > Because strings are only f$#%ing-small,
> > calculations avoid the silly results which
> > plague old-fashioned general relativity with
> > quantum mechanics at the infinitely-small.
> > > How is a volume quantized in the same way?
> > I don't know.
> > I think no one knows, so far.
> > I hope I would have heard, otherwise.
> >
> > All of this is still science-at-work,
> > not settled-science.
> >
> > Any science you would have seen in
> > a classroom is settled-science.
> > but earlier, it was science-at-work.
> > Hundreds of years earlier, for
> > some science.
> >
> > It is settled-science that
> > SOMETHING MUST BE DONE about
> > general relativity with quantum mechanics
> > because of calculations with silly results.
> > What the SOMETHING which MUST BE DONE is
> > is not yet settled.
> >
> > Quantizing volume might well be
> > that SOMETHING when it's settled.
> > Ask again in a hundred years.
> The idea of string theory actually is that
> atoms are about 10^ -25 times smaller,
> superstrings are about 10^ -50 times smaller,
> that 50 is about 2 times 25, that superstrings
> are grainy to what to atoms are smooth.
>
> The whole idea of superstrings is just
> "arbitrarily small lengths to what we've
> found according to electron physics to be
> the ratios defining Avogadro's number
> about 10^23".
>
> It might help if you know that every few years,
> the Particle Data Group or NIST CODATA, releases
> new values of these smallest constants, and,
> over time, they always get smaller, not just more precise ,
> but smaller, it's due what's called a "regime of running constants",
> according to energy and configuration of experiment,
> and quantization and atomic and sub-atomic theory,
> has much involved for it the theory of "renormalization".

Replacing order by a higher dimension that vibrates
is an embarrassment of science... it does not work that way...

On 10/13/2023 10:00 PM, Ross Finlayson wrote:
> On Friday, October 13, 2023
> at 11:44:33 AM UTC-7, Jim Burns wrote:

>>
> To your point, then,
>
>> I [JB] am guessing that
>>
>> 1) line-continuity fills in the rationals
>> between the integers.
>>
>> 2) field-continuity extends + - * / >
>> from the integers to the rationals
>>
>> 3) signal-continuity plugs
>> the one-point gaps in the rationals,
>> giving domain-continuity
>>
>> 4) long-line-continuity extends the
>> line- field- signal-continuity toward ±∞
>>
>> Even if you only say is "yes" or "no"
>> at this point, I think that would be
>> a help toward toward understanding you.
>
> 1: "line-continuity" is a "[0,1] unit line segment"
> real numbers are integer part and non-integer part
> it is the non-integer part, of otherwise integers
> it's countable

Thank you.

The real segment [0,1],
rationals and
points between splits of rationals,
is uncountable.

> 2: "field-continuity" is
> "the complete ordered field", or
> "least-upper-bound of complete ordered field"
> it is integers, and their ratios
> integers are reduced fractions
> fractions are ratios of integers
> it's uncountable

Thank you.

I know what is usually meant by
"complete ordered field"
It is rationals and
points between splits of rationals.

How is it that we haven't
done this already at (1)?

Integers and their ratios are countable.

The complete ordered field and
the segment [0,1] of it
are each uncountable, and
are not only ratios of integers.

> 3: "signal-continuity" is
> "was everywhere dense and discontinuous,
> now dense and continuous",

Thank you.

I read "continuous" as
"domain-continuous" which is
"connected" which is
"can't divide into two non-∅ open sets"
which is least-upper-bound-propertied
which is Dedekind complete.

The rationals are countable.
Indeed, the algebraics and the definables
are countable.

The completed, domain-continuified,
connectated, one-point-gaps-plugged
real numbers are uncountable,
even only those in [0,1]

> it's doubling from "dense" to "continuity"
> it's the same cardinality

I suspect that, whatever you're saying,
it is wrong.

Delete any countable set from
any uncountable set.
The remainder has
the same uncountable cardinality.

Uncountable is not countable-doubled.
You're drawing your intuition from
the finite.

> 4) long-line-continuity
> the space of real functions
> is an expression in real values,
>
> they're all ordered as asymptotically
> it's uncountable to field-continuity,
> "twice uncountable"

Thank you.

|{f:ℝ→ℝ}| > |ℝ|
because
|{f:ℝ→ℝ}| ≥ |{f:ℝ→{0,1}}|
|{f:ℝ→{0,1}}| = |𝒫(ℝ)|
|𝒫(ℝ)| > |ℝ|
[Cantor]

There are more real functions than reals,
"twice uncountable", I suppose.
However, it sounds like you're talking about
only some of the real functions.

|{expressible f:ℝ→ℝ}| = |ℕ|
because
|{expressions}| = |ℕ|

|{continuous f:ℝ→ℝ}| = |ℝ|
because

|{continuous f:ℝ→ℝ}| = |{f:ℚ→ℝ}|
because values for ℝ\ℚ are fixed
by continuity.

|{f:ℚ→ℝ}| = |{f:ℕ→ℝ}|
because |ℚ| = |ℕ|

|{f:ℕ→ℝ}| = |ℝ|
because each function ℕ→ℝ
1 → 0.d¹₁d¹₂d¹₃...
2 → 0.d²₁d²₂d²₃...
3 → 0.d³₁d³₂d³₃...
....
can be encoded "diagonally" as one real
0.d¹₁d¹₂d²₁d¹₃d²₂d³₁d¹₄d²₃d³₂d⁴₁...

Thus
|{continuous f:ℝ→ℝ}| = |{f:ℚ→ℝ}|
|{f:ℚ→ℝ}| = |{f:ℕ→ℝ}|
|{f:ℕ→ℝ}| = |ℝ|and
|{continuous f:ℝ→ℝ}| = |ℝ|

> they're all ordered as asymptotically

Real functions which can be
ordered asymptotically are not
all the real functions or
all the expressible real functions or
all the continuous real functions.

Which leaves me wondering which functions
are long-line-continuous,
if that's even a thing.

> Then,
> "line-continuity" is
> the range of a special function,
> f = n/d, n -> d, d -> oo, with n, d in N.

I don't think we're meaning the same
by "limit"

If that limit exists, it is a single point.
I think it's unlikely that
that's what you intend.

Consider
lim ⋃⁰ᑉᵈᣟᑉᵈ{n/d′|0≤n≤d′}, d ⟶ ∞

Those are all the points which n/d hits
on the way to whatever limit there may be.

I think that's much more likely to be
what you intend,
but you are noticeably lacking in
the being-Jim-Burns property.
If that's what you mean,
you need to be the one to say it.

The set
lim ⋃⁰ᑉᵈᣟᑉᵈ{n/d′|0≤n≤d′}, d ⟶ ∞
is the set of all rationals n/d in [0,1]

Whatever line-continuity might be,
that set does not have domain-continuity.
That set can be split into
two non-overlapping open sets.

> So, it's a function defined on
> the discrete integers N, but,
> behaves as a continuous function,
> the "f is integrable over N,
> not a continuous domain",
> where it's usually of course only so
> that "functions with analytical character
> are integrable over their continuous
> domains",

I'm trying to keep straight domains and
functions.

Define
[0,1]ᴶᴮ =
lim ⋃⁰ᑉᵈᣟᑉᵈ{n/d′|0≤n≤d′}, d → ∞

fᴶᴮ(x) = 1 if x ∈ [0,1]ᴶᴮ
fᴶᴮ(x) = 0 otherwise

Is fᴶᴮ(x) what you intend?

fᴶᴮ(x) is integrable on [0,1] but
∫₀¹fᴶᴮ(x)dx = 0
It seems unlikely that's what you intend.

∫₀¹fᴶᴮ(x)dx = 0

If
fᴶᴮ(x) ≠ 0 and
x ∈ [0,1]ᴶᴮ
then
Ed ∈ N+:
En ∈ N:
n/d ∈ {n/d|0≤n≤d} &
x = n/d

The points in [0,1]ᴶᴮ can be sequenced.
0/1 1/1 0/2 1/2 2/2 0/3 1/3 2/3 3/3 ...

Each point in [0,1]ᴶᴮ can be indexed.
xⱼ = nⱼ/dⱼ
j = dⱼ(dⱼ+1)/2+nⱼ

Each point xⱼ in [0,1]ᴶᴮ can be embedded in
geometrically-shrinking but non-0-length
intervals
xⱼ ∈ [xⱼ-ε/2ʲ⁺¹,xⱼ+ε/2ʲ⁺¹]

The sum ε/2 + ε/4 + ε/8 + ... of
the lengths of those intervals is ε

Define
fᵋ(x) = 1 if x ∈ one of those intervals
fᵋ(x) = 0 otherwise

∫₀¹fᵋ(x)dx ≤ ε

fᴶᴮ(x) ≤ fᵋ(x)

0 ≤ ∫₀¹fᴶᴮ(x)dx ≤ ∫₀¹fᵋ(x)dx ≤ ε

¬(0 < ∫₀¹fᴶᴮ(x)dx)
otherwise, there would be some ε
0 < ε < ∫₀¹fᴶᴮ(x)dx
But there isn't.

0 = ∫₀¹fᴶᴮ(x)dx

On Saturday, October 14, 2023 at 11:31:22 PM UTC+3, Jim Burns wrote:
> On 10/13/2023 10:00 PM, Ross Finlayson wrote:
> > On Friday, October 13, 2023
> > at 11:44:33 AM UTC-7, Jim Burns wrote:
>
> >>
> > To your point, then,
> >
> >> I [JB] am guessing that
> >>
> >> 1) line-continuity fills in the rationals
> >> between the integers.
> >>
> >> 2) field-continuity extends + - * / >
> >> from the integers to the rationals
> >>
> >> 3) signal-continuity plugs
> >> the one-point gaps in the rationals,
> >> giving domain-continuity
> >>
> >> 4) long-line-continuity extends the
> >> line- field- signal-continuity toward ±∞
> >>
> >> Even if you only say is "yes" or "no"
> >> at this point, I think that would be
> >> a help toward toward understanding you.
> >
> > 1: "line-continuity" is a "[0,1] unit line segment"
> > real numbers are integer part and non-integer part
> > it is the non-integer part, of otherwise integers
> > it's countable
> Thank you.
>
> The real segment [0,1],
> rationals and
> points between splits of rationals,
> is uncountable.
> > 2: "field-continuity" is
> > "the complete ordered field", or
> > "least-upper-bound of complete ordered field"
> > it is integers, and their ratios
> > integers are reduced fractions
> > fractions are ratios of integers
> > it's uncountable
> Thank you.
>
> I know what is usually meant by
> "complete ordered field"
> It is rationals and
> points between splits of rationals.
>
> How is it that we haven't
> done this already at (1)?
>
> Integers and their ratios are countable.
>
> The complete ordered field and
> the segment [0,1] of it
> are each uncountable, and
> are not only ratios of integers.
> > 3: "signal-continuity" is
> > "was everywhere dense and discontinuous,
> > now dense and continuous",
> Thank you.
>
> I read "continuous" as
> "domain-continuous" which is
> "connected" which is
> "can't divide into two non-∅ open sets"
> which is least-upper-bound-propertied
> which is Dedekind complete.
>
> The rationals are countable.
> Indeed, the algebraics and the definables
> are countable.
>
> The completed, domain-continuified,
> connectated, one-point-gaps-plugged
> real numbers are uncountable,
> even only those in [0,1]
> > it's doubling from "dense" to "continuity"
> > it's the same cardinality
> I suspect that, whatever you're saying,
> it is wrong.
>
> Delete any countable set from
> any uncountable set.
> The remainder has
> the same uncountable cardinality.
>
> Uncountable is not countable-doubled.
> You're drawing your intuition from
> the finite.
> > 4) long-line-continuity
> > the space of real functions
> > is an expression in real values,
> >
> > they're all ordered as asymptotically
> > it's uncountable to field-continuity,
> > "twice uncountable"
> Thank you.
>
> |{f:ℝ→ℝ}| > |ℝ|
> because
> |{f:ℝ→ℝ}| ≥ |{f:ℝ→{0,1}}|
> |{f:ℝ→{0,1}}| = |𝒫(ℝ)|
> |𝒫(ℝ)| > |ℝ|
> [Cantor]
>
> There are more real functions than reals,
> "twice uncountable", I suppose.
> However, it sounds like you're talking about
> only some of the real functions.
>
> |{expressible f:ℝ→ℝ}| = |ℕ|
> because
> |{expressions}| = |ℕ|
>
> |{continuous f:ℝ→ℝ}| = |ℝ|
> because
>
> |{continuous f:ℝ→ℝ}| = |{f:ℚ→ℝ}|
> because values for ℝ\ℚ are fixed
> by continuity.
>
> |{f:ℚ→ℝ}| = |{f:ℕ→ℝ}|
> because |ℚ| = |ℕ|
>
> |{f:ℕ→ℝ}| = |ℝ|
> because each function ℕ→ℝ
> 1 → 0.d¹₁d¹₂d¹₃...
> 2 → 0.d²₁d²₂d²₃...
> 3 → 0.d³₁d³₂d³₃...
> ...
> can be encoded "diagonally" as one real
> 0.d¹₁d¹₂d²₁d¹₃d²₂d³₁d¹₄d²₃d³₂d⁴₁...
>
> Thus
> |{continuous f:ℝ→ℝ}| = |{f:ℚ→ℝ}|
> |{f:ℚ→ℝ}| = |{f:ℕ→ℝ}|
> |{f:ℕ→ℝ}| = |ℝ|and
> |{continuous f:ℝ→ℝ}| = |ℝ|
> > they're all ordered as asymptotically
> Real functions which can be
> ordered asymptotically are not
> all the real functions or
> all the expressible real functions or
> all the continuous real functions.
>
> Which leaves me wondering which functions
> are long-line-continuous,
> if that's even a thing.
> > Then,
> > "line-continuity" is
> > the range of a special function,
> > f = n/d, n -> d, d -> oo, with n, d in N.
> I don't think we're meaning the same
> by "limit"
>
> If that limit exists, it is a single point.
> I think it's unlikely that
> that's what you intend.
>
> Consider
> lim ⋃⁰ᑉᵈᣟᑉᵈ{n/d′|0≤n≤d′}, d ⟶ ∞
>
> Those are all the points which n/d hits
> on the way to whatever limit there may be.
>
> I think that's much more likely to be
> what you intend,
> but you are noticeably lacking in
> the being-Jim-Burns property.
> If that's what you mean,
> you need to be the one to say it.
>
> The set
> lim ⋃⁰ᑉᵈᣟᑉᵈ{n/d′|0≤n≤d′}, d ⟶ ∞
> is the set of all rationals n/d in [0,1]
>
> Whatever line-continuity might be,
> that set does not have domain-continuity.
> That set can be split into
> two non-overlapping open sets.
> > So, it's a function defined on
> > the discrete integers N, but,
> > behaves as a continuous function,
> > the "f is integrable over N,
> > not a continuous domain",
> > where it's usually of course only so
> > that "functions with analytical character
> > are integrable over their continuous
> > domains",
> I'm trying to keep straight domains and
> functions.
>
> Define
> [0,1]ᴶᴮ =
> lim ⋃⁰ᑉᵈᣟᑉᵈ{n/d′|0≤n≤d′}, d → ∞
>
> fᴶᴮ(x) = 1 if x ∈ [0,1]ᴶᴮ
> fᴶᴮ(x) = 0 otherwise
>
> Is fᴶᴮ(x) what you intend?
>
> fᴶᴮ(x) is integrable on [0,1] but
> ∫₀¹fᴶᴮ(x)dx = 0
> It seems unlikely that's what you intend.
>
>
> ∫₀¹fᴶᴮ(x)dx = 0
>
> If
> fᴶᴮ(x) ≠ 0 and
> x ∈ [0,1]ᴶᴮ
> then
> Ed ∈ N+:
> En ∈ N:
> n/d ∈ {n/d|0≤n≤d} &
> x = n/d
>
> The points in [0,1]ᴶᴮ can be sequenced.
> 0/1 1/1 0/2 1/2 2/2 0/3 1/3 2/3 3/3 ...
>
> Each point in [0,1]ᴶᴮ can be indexed.
> xⱼ = nⱼ/dⱼ
> j = dⱼ(dⱼ+1)/2+nⱼ
>
> Each point xⱼ in [0,1]ᴶᴮ can be embedded in
> geometrically-shrinking but non-0-length
> intervals
> xⱼ ∈ [xⱼ-ε/2ʲ⁺¹,xⱼ+ε/2ʲ⁺¹]
>
> The sum ε/2 + ε/4 + ε/8 + ... of
> the lengths of those intervals is ε
>
> Define
> fᵋ(x) = 1 if x ∈ one of those intervals
> fᵋ(x) = 0 otherwise
>
> ∫₀¹fᵋ(x)dx ≤ ε
>
> fᴶᴮ(x) ≤ fᵋ(x)
>
> 0 ≤ ∫₀¹fᴶᴮ(x)dx ≤ ∫₀¹fᵋ(x)dx ≤ ε
>
> ¬(0 < ∫₀¹fᴶᴮ(x)dx)
> otherwise, there would be some ε
> 0 < ε < ∫₀¹fᴶᴮ(x)dx
> But there isn't.
>
> 0 = ∫₀¹fᴶᴮ(x)dx

Believe me, all you do know would never help you to justify your false inhireted beileafs...FOR SURE

If real numbers are only positive constructible numbers (I.e distinct distances on the real number line), which are of course discrete numbers, then how would be there a continiouty? IMPOSSIBLE FOR SURE

But never mind for skilled carpenters to make an artificial continiouty to get their work done, so like for eingineers & scientists but never for (logicians, philosophers, physicians & especially Mathematicians)

On 10/14/2023 5:21 PM, bassam karzeddin wrote:
> On Saturday, October 14, 2023
> at 11:31:22 PM UTC+3, Jim Burns wrote:

> [...]

> If real numbers are
> only positive constructible numbers
> (I.e distinct distances on the real number line),
> which are of course discrete numbers,
> then how would be there a continiouty?

If
real numbers are
only positive constructible numbers
then
there are continuous curves which
cross but do not intersect.

Curves which cross but do not intersect
are not continuous.

Real numbers are not
only positive constructible numbers.