Discontinuity of real numbers (as an irrefutable fact)

[bassam karzeddin]

I saw very clearly and since long ago the discontinuity of the so-called real number in modern mathematics (as simple as it is)

And please forget completely about the many tonnes of complete nonsense written about it in huge volumes by many cranks of alleged scientists like from the middle ages like those of (Godel, Hilbert, Cantor, Defdikined, Cauchy, Kant, ..., etc)

Since the irrefutable proof is only two lines and of middle school levels FOR SURE

Proof: Consider any true existing number saying arbitrary like sqrt(3), then ask yourself (but never ask your alleged best teachers in this particular issue) the following two questions

1) What is the greatest real number that is strictly less than sqrt(3)?

The correct answer (without your very silly opinions), it doesn't exist FOR SURE

2) What is the least real number that is strictly greater than sqrt(3)

Answer: It doesn't exist, hence real numbers are isolated and discontinuous and they are certainly discrete numbers

However, the real numbers are only described as "constructible" numbers as distinct existing distances on the real number line

Repeating those too elementary lessons for several times in many occasions for the academic mainstreams trolls in theoretical sciences and mathematics as well is mainly to shame them perpetually for their absolute (dishonesty, cowardness, in nobility, layers, severe mental retardation, ..., etc)

And purposely for a truer future natural historical record that is never oriented by the imbeciles wishes as "Donkeyoedia" anonymous writers for the sake of protecting the global ignorance about their own silly and too unnecessary business FOR SURE

Copyright (c), 2020
Bassam Karzeddin

[bassam karzeddin]

However, there is no harm from an engineering point of view to decide the real numbers as continuous for many other **NON-MATHEMATICAL** purposes for the solution of earthy little problems that never needs perfect solutions but an approximation solutions

A Proved example for YOU to start understanding the fact I strictly publish

Consider a carpenter who was asked to make a cube of two units volumes, then his modest skill is more than sufficient to make something like a cube root two number and produce his cube that certainly pleases every one

Despite the fact that we ALL know nowadays that the number representing the cube root two is not a number in fact

So to say, we need those artificial numbers despite their pure nonexistence

And very similar to the oldest case of needing the artificial number called *Pi* for approximating the circle area to a needed degree and exactly like the primitive mam had done it long ago when mathematics started

But the only very minor difference nowadays that computer engineers and programmers engineers have made it faster to approximate with more of the so-called accurate digits

Try to understand the core deep theme of my fundamental issues

Bassam King Kareddin

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

Even with many rigorous and irrefutable proofs, the astray academic mathematicians IMBECILES are still asking about published proofs and no matter if it was few tonnes of papers and no matter if they can't grasp anything of it but must be only published by the full fart of those many alleged best reputable Journals and Universities

How can the mentally retarded of the common mainstream academic mathematicians overcome this huge mental barrier? Wonders!

This needs Psychological doctors great help and many volunteers as well and FOR SURE
BKK

[Dan Christensen]

On Saturday, August 8, 2020 at 4:27:08 AM UTC-4, bassam karzeddin (aka "BKK") wrote:
> I saw very clearly and since long ago the discontinuity of the so-called real number in modern mathematics (as simple as it is)
>

From Psycho Troll BKK who also wrote here:

“Those many challenges of mine (in my posts) weren't actually designed for human beings, but for the future artificial beings that would certainly replace them not far away from now, for sure.”
-- BKK, Dec. 6, 2017

"The Devils deeds that are strictly and basically sourced from mathematicians like humans, FOR SURE!"
-- BKK, June 11, 2020 *** NEW ***

“You know certainly that I'm the man, and more specially the KING who is going to upside down most of your current false mathematics for all future generations.”
-- BKK, Nov. 22, 2018

“Despite thousands of years of continuous juggling and false definitions of what is truly the real number, they [us carbon-based lifeforms?] truly don't want to understand it as was discovered strictly by the *KING* [BKK Himself!]”
-- BKK, Nov. 28, 2019

“I don't believe even in one being a number”
-- BKK, Dec. 31, 2019

Math failure, BKK, doesn't believe in negative numbers, zero, one or numbers like pi and root 2. He doesn't even believe in 40 degree angles or circles. Simple speed-distance-time problems seem to be impossible for him. Really!

Needless to say his own goofy little system is getting nowhere and never will. As such he is insanely jealous of wildly successful mainstream mathematics. He seems to believe these super-intelligent artificial beings of his will somehow be enlisting his aid to "reform" mathematics worldwide when they take over the planet in the near future. He is truly delusional.


Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[Steve Harris]

Welcome back imbecile.

[Sergio]

On 8/9/2020 11:15 AM, Steve Harris wrote:
> Welcome back imbecile.
>

it is discontinuity in BKK thought processes, an irrefutable fact.

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

Any counter proofs for my **OLD** simplest proven claimes about the most fundamental issues that mainstream sheeple of academic mathematicians and alike usually adore and worship ... very stupidly... and so shamefully as well

Keep trying *HOPELESSLY* YOU BIG MORONS of mathematics

BKK

[bassam karzeddin]

How easy to prove that real numbers are purely discret numbers, in just a few lines or few minutes only

And forget completely about the many huge unnecessary volumes published officially by your top-most repuitabe authorities & the Donkypedia 🌎 world!

Got it? Of course not at all!
Why? Because it ruins completely our inhireted beliefs, and hence our illegal business as well that is based upon the innocent shoulders of the clueless students

What a stubborn morons of math we have globally in plenty & every where in this century?No woundrs!

BKK

[markus...@gmail.com]

Neither 1) nor 2) disproves the real numbers.

[bassam karzeddin]

Nothing would disprove a dogmatic common blind believer as the mainstream academic proffessional mathematicians
For sure

BKK

[markus...@gmail.com]

You haven't explained why real numbers are inherit inconsistent.

[mitchr...@gmail.com]

In the continuum hypothesis quantity is continuous.

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

How so elementary & very simple the very short proof about the discontinuous of real numbers?

Is it still too difficult for the mainstream academic sheeples to understand? No wonder!

Proofs must only published by allegedly top-most repuitabe Journals or Universities or well-know mathematicians in power, Isn't it 🤔?

BKK

[bassam karzeddin]

What else is true in modern mathematics?
BKK

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

Are there any believers nowadays of continiouty of real numbers?

I don't think so for sure even if hired resident Trolls keep confusing the cluless innocent minds of school students 🤔
BKK

[Kevin S]

Nothing and everything. Nothing in the sense that math is a language or game, it's not fundamentally different from Haskell, Agda, Scala, etc. There's nothing real or natural about math. Everything in the sense that we are playing in consistent with the rules up to now.

[mitchr...@gmail.com]

If it is not real what is it in the mind to be used?
A tesseract is not real. But the hypersphere manifests.

[markus...@gmail.com]

This is the formalist view, yes.

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

Yes, real numbers are purely discret numbers, where all that ill-established about continiouty of real numbers is completely false & very misleading, since simply that waa entirely established by eingineers & scientists but for other non-mathematical purposes but how do theoretical (mathematicians, philosophers, logicians, physicians..., etc) understand it? Wonder!

BKK 🔊

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

What a simple direct proof that refutes the huge accumulated volumes written about continiouty of real numbers in mathematics?

Where the proof is publicly & freely published since years!


BKK

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

So, real numbers are discrete numbers FOR SURE

BKK

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

How it was so easy task to disprove the continiouty of real numbers in mathematics?

However, many other simpler proofs are also available provided that concernd humans do understand it & appreciate it as well

But if humans generally don't like the very short proofs, then it is too easy task also to discover very lengthy proofs that would certainly please them FOR SURE

BKK

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

Since the density of constructible numbers & angles is simply unlimited, real numbers seem to us as continuous, but this is completely false as the proof is too elementary & so rigorous beside being irrefutable FOR SURE
BKK

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:
> I saw very clearly and since long ago the discontinuity of the so-called real number in modern mathematics (as simple as it is)
>
> And please forget completely about the many tonnes of complete nonsense written about it in huge volumes by many cranks of alleged scientists like from the middle ages like those of (Godel, Hilbert, Cantor, Defdikined, Cauchy, Kant, ..., etc)
>
> Since the irrefutable proof is only two lines and of middle school levels FOR SURE
>
> Proof: Consider any true existing number saying arbitrary like sqrt(3), then ask yourself (but never ask your alleged best teachers in this particular issue) the following two questions
>
> 1) What is the greatest real number that is strictly less than sqrt(3)?
>
> The correct answer (without your very silly opinions), it doesn't exist FOR SURE
>
> 2) What is the least real number that is strictly greater than sqrt(3)
>
> Answer: It doesn't exist, hence real numbers are isolated and discontinuous and they are certainly discrete numbers
>
> However, the real numbers are only described as "constructible" numbers as distinct existing distances on the real number line
>
> Repeating those too elementary lessons for several times in many occasions for the academic mainstreams trolls in theoretical sciences and mathematics as well is mainly to shame them perpetually for their absolute (dishonesty, cowardness, in nobility, layers, severe mental retardation, ..., etc)
>
> And purposely for a truer future natural historical record that is never oriented by the imbeciles wishes as "Donkeyoedia" anonymous writers for the sake of protecting the global ignorance about their own silly and too unnecessary business FOR SURE
>
> Copyright (c), 2020
> Bassam Karzeddin

Now, forget completely about the so many huge volumes written about the continiouty of real numbers by many allegedly top-most historians mathematicians & alike, since it is ALL big fart & complete waste of human knowledge & resources as well FOR SURE

BKK

[Jim Burns]

On 10/1/2023 10:07 PM, bassam karzeddin wrote:
> On Saturday, August 8, 2020
> at 11:27:08 AM UTC+3, bassam karzeddin wrote:

>> the discontinuity of the so-called
>> real number in modern mathematics

>> 1) What is the greatest real number that is
>> strictly less than sqrt(3)?

>> it doesn't exist

>>
>> 2) What is the least real number that is
>> strictly greater than sqrt(3)

>> It doesn't exist,

>> hence real numbers are
>> isolated and discontinuous
>> and they are certainly discrete numbers

You are using "isolated" as though it means
the opposite of what it actually means.

| In mathematics,
| a point x is called
| an isolated point of a subset S
| (in a topological space X)
| if
| x is an element of S and
| there exists a neighborhood of x that
| does not contain any other points of S.
|
https://en.wikipedia.org/wiki/Isolated_point

>> However,
>> the real numbers are only described as
>> "constructible" numbers as
>> distinct existing distances on
>> the real number line

Each real number is distinct.
No real number is isolated.


If a real-number function jumps,
then at least one point exists at which
that function is not continuous.

That is why,
for each split F‖H of the rationals ℚ
an irrational u ∈ ℝ\ℚ is between F and H
F ᣔ< u <ᣔ H

Consider two irrationals u v
between two splits Fᵤ‖Hᵤ Fᵥ‖Hᵥ of ℚ
Fᵤ ᣔ< u <ᣔ Hᵤ ∧ Fᵥ ᣔ< v <ᣔ Hᵥ













One of (i) (ii) (iii) is true.

(i)
Hᵤ and Fᵥ overlap
Fᵤ ᣔ< u <ᣔ Hᵤ∩Fᵥ ᣔ< v <ᣔ Hᵥ
Hᵤ∩Fᵥ ≠ ∅
p/q ∈ Hᵤ∩Fᵥ
u < p/q < v
u < v

(ii)
Hᵥ and Fᵤ overlap
Fᵥ ᣔ< v <ᣔ Hᵥ∩Fᵤ ᣔ< u <ᣔ Hᵤ
Hᵥ∩Fᵤ ≠ ∅
p/q ∈ Hᵥ∩Fᵤ
v < p/q < u
v < u

(iii)
Neither Hᵤ and Fᵥ nor Hᵥ and Fᵤ overlap
Hᵤ∩Fᵥ = Hᵥ∩Fᵤ = ∅
Fᵤ = Fᵥ ᣔ< u <ᣔ Hᵤ = Hᵥ
Fᵤ = Fᵥ ᣔ< v <ᣔ Hᵤ = Hᵥ
¬(|u-v| > 0)
u = v

Points between splits of rationals
are ordered according to
whether a rational is between them, or not.

> [...] FOR SURE

[bassam karzeddin]

The entire confusions actually are sourced from the so-called set theory where it was refuted by my modest self & few other distinguished members of sci.math where the mainstream academic sheeples usually don't accept anything rigorously proven & globally published freely from public disgusting sources like sci.math

However, the same confusion comes basically from the unlimited density of "constructible " numbers

Where between any distinct locations on the real number line & no matter however close to each others or far from each others one might think, there are always & perpetually an endless number of only other constructible numbers where it is absolutely impossible to determine all of them

Where no other types of real numbers other than the constructible numbers ever exist, FOR SURE

Any bigginer in number theory can immediately well-understand what I simply announce as perpetual truth about real numbers

Bassam karzeddin

[Jim Burns]

On 10/2/2023 3:38 PM, bassam karzeddin wrote:
> On Monday, October 2, 2023 at 8:09:44 PM UTC+3,
> Jim Burns wrote:

>> If a real-number function jumps,
>> then at least one point exists at which
>> that function is not continuous.
>>
>> That is why,
>> for each split F‖H of the rationals ℚ
>> an irrational u ∈ ℝ\ℚ is between F and H
>> F ᣔ< u <ᣔ H

> Where no other types of real numbers
> other than the constructible numbers
> ever exist, FOR SURE

[bassam karzeddin]

Believe me, all that nonsense notations belonging to set theory products are so irrelevant to any truth in mathematics but a brain wash of human mathematicians & alike

To prove my Unique true vision about the absolute discontinuity of real numbers, one should understand that a number is an EXACT thing where inexact alleged real numbers are infact No existing numbers

However, if an alleged real "non-constructible " number exists, then please state only one of them either numerically or Geometrically but EXACTLY

simply because the best description of a real existing number is an exact distance

And if a believed is number in human minds isn't an exact distance then it doesn't exist nor it is a true existing real number
So unfortunately, the most famous historical examples if non-existing real numbers are (Cubrt2 & Pi), where all humans generally believe in their existence

Assuring that the proofs are too elementary & were publically published in my old posts FOR SURE

BKK

[Ross Finlayson]

Well, and rationals too, they're both dense.

Yeah, "least upper bound" property is sort of an axiom
of the axiom what results making the "complete" of the
"complete ordered field".

Yeah, continuous functions are the usual milieu,
and the usual topology is the open topology.

Of course Jim has seen and even sat through me writing
out developments into "line continuity", "field continuity",
"signal continuity", about where it was left off was
adding a number "infinity" and having d/oo = 0,
except d = oo, d/oo = 1.

For integers, ....

Here, King Karzeddin, here's an irrefutable fact, punch yourself in the face.
When you see stars, then you'll be getting somewhere.

Or, it's more than you got, now.


Yeah, Jim, Mr. Burns or for my professorial admiration,
there was that "Correct Presentation of the Antidiagonal Argument",
a couple other long thread here, it's broken way down,
at least three definitions of continuity, by which
I mean continuous _domains_, each sort of regular and
rulial, like for Hardy makes the point-sets like lines.
(Which they, "are".)

Over on sci.logic that was I suppose, which at the
time had less spam.

[Ross Finlayson]

If you're not willing to box yourself about the face then don't be
suggesting that researchers essentially lobotomize themselves,
here it's called "retro-finitism" and considered un-conscientious.

Now, you're perfectly welcome to be "ultra-finitist", and most are,
what results for laws of form and proportion, after regular numbers,
and regular laws of small numbers, that grow, and a law of large numbers
according to small numbers, that, any largest would be out-grown.

Then there's no saying though that there's a large enough number.
Such a thing would be an"ultra-finitist's infinity".

[Ross Finlayson]

Of course, it's unfortunate that many reject 'infinity' because their first
and only fitting notion of infinity is "a largest number, on the number
line, there's a word infinity and it means arbitrarily large, and because
dividing by a larger number results a smaller number, dividing by
infinity equals zero, through it's still as of a quantity itself".

That is, most people's "infinity" is from "infinite divisibility", goes to zero.

Then, it's unfortunate that "that is apparent but also it can't be a quantity,
because in our quantities they can be in any order", which is not so, in
terms of numbers and their types and their products in their operations
and the closures in their operation or here lack thereof.

So, what's unfortunate or bad fortune is that how was found a way to make
an "infinity" or something, in mathematics, by itself, wasn't numbers but instead
just an inductive set of sets. So, there's no arithmetic.

Now why that's unfortunate is it totally doesn't fulfill what people expect and
want from the "natural" definition of infinity, according to numbers, some "scalar",
infinity.

But, it's still what there is, then, to get back into how that can be, involves a sort
of mathematics that takes function theory, and establishes that not all functions
are Cartesian, then that those are different types and can't just go together but
have to basically agree how to go together, then it's pretty easy.

Then, you might ask yourself "what's this line-continuity line-integral called
already?" and it's "called Jordan content: would be Jordan measure but would
interfere with the standing formalism called measure theory, called Jordan content",
and similarly "Dirichlet functions: on or off the analytical character, or averaged
making one half, again don't look too close", "signal-continuity".

Here the key part of continuity is the infinite divisilbity of time, never falsified.
Then, what it reflects is fundamentally for the area, and geometry, how things are.

So, anyways, it goes into "function theory" and "topology", such explanations called
"apologetics", that what we call "conscientious mathematicians" put together.

Thus, I made one to sit for me.

Really the differential is natural, everybody knows the differential is "infinitely divisible".

It's the same old dx, these day's usually a "line invariant".

[Earle]

> > an irrational u \ is between F and H
> > F < u < H

> >
> > Consider two irrationals u v

> > between two splits F H F H of
> > F < u < H F < v < H

> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> > One of (i) (ii) (iii) is true.
> >
> > (i)

> > H and F overlap
> > F < u < H F < v < H
> > H F
> > p/q H F

> > u < p/q < v
> > u < v
> >
> > (ii)

> > H and F overlap
> > F < v < H F < u < H
> > H F
> > p/q H F

> > v < p/q < u
>
>
> > v < u
> >
> > (iii)

> > Neither H and F nor H and F overlap
> > H F = H F =
> > F = F < u < H = H
> > F = F < v < H = H

> > (|u-v| > 0)
> > u = v
> >
> > Points between splits of rationals
> > are ordered according to
> > whether a rational is between them, or not.
> >
> > > [...] FOR SURE
> The entire confusions actually are sourced from the so-called set theory where it was refuted by my modest self & few other distinguished members of sci.math where the mainstream academic sheeples usually don't accept anything rigorously proven & globally published freely from public disgusting sources like sci.math
>
> However, the same confusion comes basically from the unlimited density of "constructible " numbers
>
> Where between any distinct locations on the real number line & no matter however close to each others or far from each others one might think, there are always & perpetually an endless number of only other constructible numbers where it is absolutely impossible to determine all of them
>
> Where no other types of real numbers other than the constructible numbers ever exist, FOR SURE
>
> Any bigginer in number theory can immediately well-understand what I simply announce as perpetual truth about real numbers
>
> Bassam karzeddin

*
Bassam: I believe that you know much more about mathematics than Archimedes Pluotonium.

earle
*

[Jim Burns]

On 10/2/2023 7:35 PM, bassam karzeddin wrote:
> On Monday, October 2, 2023

> at 10:59:24 PM UTC+3, Jim Burns wrote:

>> [...]

> the absolute discontinuity of real numbers,

An isolated point is roughly
the opposite of what you (bk) think it is.

> a number is an EXACT thing

For each split F|H of Q
exactly one x is between F and H

If not,
then some functions jump and also
are continuous at all points.

> the best description of
> a real existing number is
> an exact distance

If only points with descriptions exist,
then some functions jump and also
are continuous at all points.

[bassam karzeddin]

On Saturday, August 8, 2020 at 11:27:08 AM UTC+3, bassam karzeddin wrote:

> I saw very clearly and since long ago the discontinuity of the so-called real number in modern mathematics (as simple as it is)
>
> And please forget completely about the many tonnes of complete nonsense written about it in huge volumes by many cranks of alleged scientists like from the middle ages like those of (Godel, Hilbert, Cantor, Defdikined, Cauchy, Kant, ..., etc)
>
> Since the irrefutable proof is only two lines and of middle school levels FOR SURE
>
> Proof: Consider any true existing number saying arbitrary like sqrt(3), then ask yourself (but never ask your alleged best teachers in this particular issue) the following two questions
>

> 1) What is the greatest real number that is strictly less than sqrt(3)?
>

> The correct answer (without your very silly opinions), it doesn't exist FOR SURE
>

> 2) What is the least real number that is strictly greater than sqrt(3)
>

> Answer: It doesn't exist, hence real numbers are isolated and discontinuous and they are certainly discrete numbers

>
> However, the real numbers are only described as "constructible" numbers as distinct existing distances on the real number line
>

> Repeating those too elementary lessons for several times in many occasions for the academic mainstreams trolls in theoretical sciences and mathematics as well is mainly to shame them perpetually for their absolute (dishonesty, cowardness, in nobility, layers, severe mental retardation, ..., etc)
>
> And purposely for a truer future natural historical record that is never oriented by the imbeciles wishes as "Donkeyoedia" anonymous writers for the sake of protecting the global ignorance about their own silly and too unnecessary business FOR SURE
>
> Copyright (c), 2020
> Bassam Karzeddin

Note how academic Trolls 🧌 try tirelessly to prevent the clever mid-school students around the whole world not to easily well-understand such a very simple topic!

Don't be impressed at all by their too lengthy tongues & many meaningless symbols

Since they do momoreise terminologies they had learnt without the ability of understanding anything FOR SURE

BKK

[Jim Burns]

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:

>> the discontinuity of
>> the so-called real number

>> the real numbers are only described as
>> "constructible" numbers as
>> distinct existing distances on
>> the real number line

Either some function jumps and
is also continuous at each point
or
all splits of the rationals
have real points between their two sides.

There are more splits F‖H of ℚ
than elements p of ℚ

There aren't more definitions than
elements of ℚ

There are more splits F‖H of ℚ
than definitions.

If only points with definitions exist,
then
splits exist which do not have points-between
and
functions exist which jump and
are also continuous at each point.

We choose that
our continuous functions do not jump.
One consequence of that choice is that
more points than definitions exist.


For our purposes,
the real numbers are
the rational numbers and
points between splits of the rationals.

Some do not have definitions.

We are able to learn about all of them,
with or without definitions,
by describing one of them as:
| a rational number or
| a point between a split of the rationals.
That's true of each one of them,
with or without definition.

We can augment that claim about a real number
with not-first-false claims.
Each augmenting claim is true of
each real number, with or without definition.

[bassam karzeddin]

What humans generally cannot comprehend is the real number line itself, where a line is so simply an existing distance, which is the real number itself that is relevant to an existing arbitrary distance as unity distance

Where also, every existing constructible number is simply an exact existing distance

Something too elementary but so unfortunately was completely missed by ALL humans up to this moment 😢!

In short, distance is the real number where this is purely physical terminology that mathematicians mustn't define by their utter many sick imaginations


And between distinct Locations on the real number line (no matter however close or far from each others a human mind might think, there are always a countless number of only & strictly other constructible numbers between them, where no other types of real numbers ever exists between the distinct locations, FOR SURE

BKK

[bassam karzeddin]

And assuming in good faith that a real number which is not a constructible number exists, then mathematicians are globally revealed to present only one of them either numerically or Geometrically but Exactly & not foolishly as meaningless Symbolic that doesn't mean anything in true mere mathematics
..
It is also an old request for 🌎 academic proffessional mathematicians & for many years by now where they couldn't, ^ they would never be able to do so simply because they globally refuse to easily well-understand what is truly a real number?
Mainly because if they admit the truth then every thing they did achieve would be lost

So to say, they fear the truth because they are too (stupid, stubborns, , dishonests, innobles, too selfish, traitors to the science that feeds them, . ., etc ) FOR SURE

BKK

[Ross Finlayson]

Nah, "the complete ordered field's elements are the equivalence classes
of sequences that are Cauchy, then after that you can make Dedekind cuts,
identified by the members of the complete ordered field".

[Jim Burns]

There is more than one construction of
the complete ordered field ℝ

The equivalence classes of
Cauchy sequences of ℚ
is one construction of ℝ

ℚ and points between splits of ℚ
is another construction of ℝ

They both satisfy all the axioms of
the complete ordered field.

They are different mathematical objects,
but they can be mapped, point-for-point,
from one to the other.


For points between splits F‖H of ℚ
we can identify between-points x with
foresplits F of ℚ
(If x ∈ ℚ, assign it to H, not to F)

In this construction,
ℝ = {betweenless foresplit of ℚ} ⊆ 𝒫(ℚ)

Let S be a bounded non-empty set
of betweenless foresplits of ℚ

Its union ⋃S is a betweenless foresplit of ℚ
and ⋃S is the least upper bound of S

Thus,
{betweenless foresplit of ℚ} has
the least-upper-bound property.

{betweenless foresplit of ℚ}
has the other properties required of
the complete ordered field, too,
given the correct definitions of + - * / <
Those are basically inherited from ℚ


The purpose of a mathematical construction
is not to tell us
what some described object _is_
Its purpose is to tell us
what some described object _could be_

We have these axioms for
the complete ordered field ℝ

Equivalence classes of
Cauchy sequences of ℚ satisfy them.
The axioms could be referring to
the equivalence classes of
Cauchy sequences of ℚ

Betweenless foresplits of ℚ satisfy them.
The axioms could be referring to
the betweenless foresplits of ℚ

For the purpose of showing that
a contradiction cannot be proved from
the axioms for a complete ordered field,
it is enough that _something_ satisfy
those axioms, equivalence classes,
foresplits, or something else.

[FredJeffries]

On Monday, October 9, 2023 at 1:05:06 PM UTC-7, Jim Burns wrote:

> There is more than one construction of
> the complete ordered field ℝ

https://mattbaker.blog/2021/12/15/the-eudoxus-reals/

https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Explicit_constructions_of_models

[Jim Burns]

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.

[Ross Finlayson]

But, sirrah, I aver you cannot name the cut of root two,
without mentioning root two, which is not a ratio.



Furthermore, I must hope you are familiar with the pigeonhole principle,
then that if Dedekind cuts identified an irrational number, there's a
distinct and unique rational in _all_ the neighborhoods of it, if only
that for any neighborhood of it, in the cuts, each rational ticked off of
it only relates to a distinct and individual ir-rational, Dirichlet's.

With the rationals being countable and all, I don't know how you imagine
they're not, and whilst your poetry and language is becoming,
that's I suppose an altogether other concern.


You've invoked geometry, why this _needs_ be, but fully thusly,
you've simply _declared_ that it's so.

Or: "what points between the irrationals"?

Aren't then "Dedekind's ir-rational cuts" same and same, and same?

[Ross Finlayson]

Either way you've _axiomatized_ the least-upper-bound property into existence,
while, something like line-reals sees it result neatly from "next".



I got next, ..., is a traditional aspect of fair play, which is exactly taking
distinct turns, in turns, to take one's turn, and await one's next turn,
that he who claims, "next", gets next.



"Fearful symmetry", is a turn of phrase of Blake, if I thought it was Kipling,
"what frames thy fearful symmetry".




Then please excuse any overfamiliarity, here "sirrah" meant 'sir',
or a genial familiar appellative.



So, what frames the symmetry?

[bassam karzeddin]

@ Jim Burns

Believe me, you are constantly so delusional about your common conclusion which aren't at all any kind of rigorous proof, no matter if you fill up the Galaxy with your meaningless many symbols that were inserted so merelesly into your innocent skull since your early childhood

However, this isn't only your mental case but so unfortunately a global case of greatest delusions among an astray catagory of human beings believing themselves as (Logicians, Philosophers, Physicians & especially Mathematicians)

However, you were personally & freely given all the necessary lessons (Ref: older discussions ), to overcome your mind barriers & be freed completely from all the false misleading education that you had innocently inhireted like every one else

But, utterly & so abnormally humans generally deny & resist aimlessly against the superior truths & not only necessarily in mathematics but generally in all walks of life

Almost nothing of your inhireted & false logic works but fails drastically before the untought truths

Fighting ignorance is far better than discovering new dependent ignorance which is coming finally to a point of explosion where nothing can stop, FOR SURE

BKK

[Ross Finlayson]

Well that's sad, here even children know "infinity" since grade-school.

[bassam karzeddin]