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

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.

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:
>
> >> 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.

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

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:
> >
> > >> 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.
> 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
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

On Sunday, October 8, 2023 at 9:48:28 PM UTC-7, 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:
>
> >> 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.

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".

On 10/9/2023 1:16 PM, Ross Finlayson wrote:
> On Sunday, October 8, 2023
>> at 9:48:28 PM UTC-7, Jim Burns wrote:

>> 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.
>
> 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".

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.

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

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.

On Monday, October 9, 2023 at 1:56:20 PM UTC-7, 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.

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?

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:
> > 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.
> 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?

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?

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.
@ 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

On Tuesday, October 10, 2023 at 1:03:27 AM 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.
> @ 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

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

On Tuesday, October 10, 2023 at 6:26:41 PM UTC+3, Ross Finlayson wrote:
> On Tuesday, October 10, 2023 at 1:03:27 AM 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.
> > @ 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
> Well that's sad, here even children know "infinity" since grade-school.

Really!
So you know infinity ♾️, Don’t you?
So, tell us about it please!
But remember that ♾️ isn't a number nor a dynasor or a 🌳 or anything else, & tell us how it is greater than 7 for example if infinity itself isn't a number?

I mean you can't compare numbers with tree's so you can't compare numbers with no number like your infinity

And tell us 🙏 how do the mathematicians tend suddenly & miraculously to your infinity ♾️ whenever they feel that is too urgent to hide behind

Do they really approach it or all that is only imagination?

But since integers are ENDLESS chain of successive integers, then how can be there somewhere something like your infinity ♾️

Sorry, I know it wasn't yours personally but like all others you did inherit it from your grand masters, Right?

And I know that mathematicians generally cannot live without it since it is a great Paradis for them & without it they would go back thousands of years in mathematics FOR SURE

And the best rules of ♾️ are truely strickong, almost like zero logic or zero rules

For illustration:
♾️ + ♾️ =/= 2* ♾️, but only one infinity ♾️, Right ✅️?

But they say that they have discovered many other ♾️'s; which aren't the sane!

Do people want to laugh 😃 more abiut infinity ♾️ & alike?

Then go to my relevant 📫 posts and have greatest fun with ♾️

It is truly a golden milking cow 🐄 for all types if human mathematickers for sure

BKK

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.

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.

The infinitesimal sequence is continuous.
It leads first to .999 repeating and then the first integer.

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:

>> 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,

That is incorrect.
No rational is in all neighborhoods of
an irrational.

Perhaps you're thinking of
a rational closest to the irrational.

It is incorrect that there is
a rational closest to an irrational.

>> 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.

I don't see what you are trying to say.
Each rational in in each cut,
either in the foresplit or in
the hindsplit.

>> With the rationals being countable and all,
>> I don't know how you imagine they're not,

Fortunately, I don't imagine that.
ℚ is countable.
The betweenless foresplits of ℚ
{F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F}\{∅,ℚ}
are not countable.

>> and whilst your poetry and language is becoming,
>> that's I suppose an altogether other concern.

A different concern, and yet
a strong indication that you and I
only appear to be communicating.

I have faith, broadly speaking, that
misunderstandings can be worked past.
It is, perhaps,
a triumph of hope over experience.

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

The real numbers describe locations on
a geometric line.
The rational numbers are a first draft,
but there aren't enough of them.

What's missing from the rationals are
some points of intersection from
the crossing of continuous curves.

I'm sure that requiring points of intersection
is nothing new in geometry.
It's part of a common proof of the Pythagorean
theorem, in which a perpendicular is drawn
from the right angle to the hypotenuse
_where it intersects_

It is sometimes the case in mathematics
that some fact is deemed too obvious to
need stating, and so
it is not stated, maybe not for centuries.
I think that requiring points of intersection
is an example of this.

Requiring points of intersection ==
the intermediate value theorem ==
the least-upper-bound property
are more often met in a calculus course,
yes, but
it seems to me that they correct
our description of lines and curves
which we met in a geometry course.

>> Or: "what points between the irrationals"?

Once Q is augmented with
points between splits of Q
each split has its point-between.

The points between splits of
irrationals and rationals are
irrationals and rationals.

> So, what frames the symmetry?

Continuous functions should not jump.

On Tuesday, October 10, 2023 at 10:35:22 PM UTC+3, 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:
>
> >> 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,
> That is incorrect.
> No rational is in all neighborhoods of
> an irrational.
>
> Perhaps you're thinking of
> a rational closest to the irrational.
>
> It is incorrect that there is
> a rational closest to an irrational.
> >> 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.
> I don't see what you are trying to say.
> Each rational in in each cut,
> either in the foresplit or in
> the hindsplit.
> >> With the rationals being countable and all,
> >> I don't know how you imagine they're not,
> Fortunately, I don't imagine that.
> ℚ is countable.
> The betweenless foresplits of ℚ
> {F ⊆ ℚ| F ᣔ<ᘁ F ᣔ<ᣔ ℚ\F}\{∅,ℚ}
> are not countable.
> >> and whilst your poetry and language is becoming,
> >> that's I suppose an altogether other concern.
> A different concern, and yet
> a strong indication that you and I
> only appear to be communicating.
>
> I have faith, broadly speaking, that
> misunderstandings can be worked past.
> It is, perhaps,
> a triumph of hope over experience.
> >> You've invoked geometry,
> >> why this _needs_ be, but fully thusly,
> >> you've simply _declared_ that it's so.
> The real numbers describe locations on
> a geometric line.
> The rational numbers are a first draft,
> but there aren't enough of them.
>
> What's missing from the rationals are
> some points of intersection from
> the crossing of continuous curves.
>
> I'm sure that requiring points of intersection
> is nothing new in geometry.
> It's part of a common proof of the Pythagorean
> theorem, in which a perpendicular is drawn
> from the right angle to the hypotenuse
> _where it intersects_
>
> It is sometimes the case in mathematics
> that some fact is deemed too obvious to
> need stating, and so
> it is not stated, maybe not for centuries.
> I think that requiring points of intersection
> is an example of this.
>
> Requiring points of intersection ==
> the intermediate value theorem ==
> the least-upper-bound property
> are more often met in a calculus course,
> yes, but
> it seems to me that they correct
> our description of lines and curves
> which we met in a geometry course.
> >> Or: "what points between the irrationals"?
> Once Q is augmented with
> points between splits of Q
> each split has its point-between.
>
> The points between splits of
> irrationals and rationals are
> irrationals and rationals.
> > So, what frames the symmetry?
> Continuous functions should not jump.

It is so obvious that you are oriented from basic axiomatic definitions where you don't question those axioms & considering them as absolute facts, which makes almost impossible to understand the simplest truth about the discontinuity of real existing numbers on a number line

But If you think uniasedly you should discover by your own self that those definitions are actually decisions taken mostky by non-mathematicians but purely by scientists & eingineers who are actually behind our modern mathematics, mainly to fitful their own practical earthy problem solutions; where that doesn't necessarily spoil their aims, and to illustrate that for readers to understand, there is no harm at all for a skilled carpenter to consider (1 = 0.999) only as long as it satisfy their achievements

Adding to that a fact that mathematicians in older celebrities were mostly themselves as practical scientists & skilled eingineers & carpenters

Only recently a catagory of scientific stream was born & named as mathematicians

Generally speaking, the theoretical mathematicians are still the sane as scientists & eingineers, with their pure orientation to solve their own practical problems by those mathematical definitions you are usung

But , mathematics itself is something completely different & doesn't consider any scientific or eingineering calculations & approximations

For example: close enough is purely eingineering or scientific term that has no relevance to the truth in mathematics, & to understand this particular point, consider a real existing number like Sqrt2 for insurance, and observe carefully that its comparison with distinct decimal rational numbers is never ending task, where there are always & perpetually an endless chain of other comparison of the decimal rational forn that is absolutely impossible to know by any means
But this is actually not convincing for skilled carpenters where they may consider (Sqrt2 = 1.412), same for scientists & eingineers but with a bit of more digits

Can't you still comprehend & distinguish btween what is the true in mere mathematics & what is the modern eingineering mathematics

This of course applies on Dedekind cuts & Cauchy sequences, Newton's approximations, ..., etc

Where all that isn’t true math but scientific & completely eingineering math made by them FOR SURE

In short, there are rarely a few mathematicians in a century, but so many millions of skilled ( carpenters, eingineers & scientists)

I'm also quite sure that you will refuse stubbornly to understand because mathematicians of these days don't have other tools to Play with & hide behind practicality & usefulness

BKK

On 10/10/2023 4:03 AM, bassam karzeddin wrote:
> On Monday, October 9, 2023
> at 11:56:20 PM UTC+3, Jim Burns wrote:

>> [...]
>

> any kind of rigorous proof,

Claims and sheep have something in common.

In a finite sequence,
if any of them have a property, then
one of them is first to have that property.

In a finite sequence of sheep,
if any sheep is black, then
there is a first-black sheep.

In a finite sequence of claims,
if any claim is false, then
there is a first-false claim.

In some finite sequences of claims,
no claim is first-false, and so
no claim is false.

In some finite sequences of claims,
some claims are not-first-false.
Either they are true
or they are preceded by other false claims.
In no case are they first-false.

For example, consider the sequence
⟨... P∨Q ... ¬P ... Q ...⟩
Q is not-first-false in that sequence.
Either Q is true or one of P∨Q ¬P is false.

If, in a finite sequence of claims,
each claim is not-first-false,
then each claim is not-false.
There is no help for it, just as,
in a finite sequence of sheep with
each sheep not-first-black,
there is no help for
each sheep being not-black.

Mathematicians spend their lives
finding or inventing finite sequences of
only not-first-false claims,
which are called "proofs".

Typically, the conclusion of a proof
only applies to a certain subject matter.
The Pythagorean theorem only applies to
right triangles, not to just any triangles.

This restriction of subject matter
is incorporated into the proof as one or more
claims about _what the proof is about_
In a proof of the Pythagorean theorem,
there will likely be, somewhere,
a claim that ABC is a triangle and
the angle at C is a right angle,
or something like that.

That is true of each right triangle
(allowing for re-naming vertices)
If ABC refers to a right triangle,
that description and each not-first-false
claim after the description
must be true.

True about _a right triangle_
For objects _not a right triangle_
we can't say the sequence of claims
is only not-first-false.
We can't say the proof's conclusion
should apply to the not-right-triangle.

For those objects a proof says it is about,
the conclusion is inescapable.

However, for other objects about which
the proof is silent,
we should be silent about them too,
if we don't have different proofs.

Consider two proofs.
One is about constructible numbers.
One is about ℚ and points between splits of ℚ

There is a little overlap because
some points between splits are constructible,
but generally they are different proofs
about different subject matter, and
although each is very strong in
its own domain,
it is incorrect to take them outside
that domain.

On 10/10/2023 4:03 AM, bassam karzeddin wrote:
> On Monday, October 9, 2023
> at 11:56:20 PM UTC+3, Jim Burns wrote:

>> [...]
>

> any kind of rigorous proof,

Claims and sheep have something in common.

In a finite sequence,
if any of them have a property, then
one of them is first to have that property.

In a finite sequence of sheep,
if any sheep is black, then
there is a first-black sheep.

In a finite sequence of claims,
if any claim is false, then
there is a first-false claim.

In some finite sequences of claims,
no claim is first-false, and so
no claim is false.

In some finite sequences of claims,
some claims are not-first-false.
Either they are true
or they are preceded by other false claims.
In no case are they first-false.

For example, consider the sequence
⟨... P∨Q ... ¬P ... Q ...⟩
Q is not-first-false in that sequence.
Either Q is true or one of P∨Q ¬P is false.

If, in a finite sequence of claims,
each claim is not-first-false,
then each claim is not-false.
There is no help for it, just as,
in a finite sequence of sheep with
each sheep not-first-black,
there is no help for
each sheep being not-black.

Mathematicians spend their lives
finding or inventing finite sequences of
only not-first-false claims,
which are called "proofs".

Typically, the conclusion of a proof
only applies to a certain subject matter.
The Pythagorean theorem only applies to
right triangles, not to just any triangles.

This restriction of subject matter
is incorporated into the proof as one or more
claims about _what the proof is about_
In a proof of the Pythagorean theorem,
there will likely be, somewhere,
a claim that ABC is a triangle and
the angle at C is a right angle,
or something like that.

That is true of each right triangle
(allowing for re-naming vertices)
If ABC refers to a right triangle,
that description and each not-first-false
claim after the description
must be true.

True about _a right triangle_
For objects _not a right triangle_
we can't say the sequence of claims
is only not-first-false.
We can't say the proof's conclusion
should apply to the not-right-triangle.

For those objects a proof says it is about,
the conclusion is inescapable.

However, for other objects about which
the proof is silent,
we should be silent about them too,
if we don't have different proofs.

Consider two proofs.
One is about constructible numbers.
One is about ℚ and points between splits of ℚ

There is a little overlap because
some points between splits are constructible,
but generally they are different proofs
about different subject matter, and
although each is very strong in
its own domain,
it is incorrect to take them outside
that domain.

On Wednesday, October 11, 2023 at 12:21:33 AM UTC+3, Jim Burns wrote:
> On 10/10/2023 4:03 AM, bassam karzeddin wrote:
> > On Monday, October 9, 2023
> > at 11:56:20 PM UTC+3, Jim Burns wrote:
> >> [...]
> >
>
> > any kind of rigorous proof,
> Claims and sheep have something in common.
>
> In a finite sequence,

Here is a false common inhireted beileaf among logicians about the term finite and infinite as well, where humans generally think that thetmy are absolutely different & opposing each other in meaning think that finite isn't, where this comes basically from set theory, but the truth is something else, What? Yes

Then what is the largest FINITE natural number then?

Of course, it is also Infinite which doesn't exist FOR SURE, Does it?
See clearly, how are those terminologies as FINITE & Infinite have no real meaning in true mathematics, but probably in eingineering science has some usefulness

So, the problem with mathematians & logicians is deeply rooted & very serious as well as long as they are adopting the eingineering principles in finding satisfaction solutions to practical problems!

> if any of them have a property, then
> one of them is first to have that property.
>
> In a finite sequence of sheep,
> if any sheep is black, then
> there is a first-black sheep.
>
> In a finite sequence of claims,
> if any claim is false, then in science & eingineering may have applica
> there is a first-false claim.
>
> In some finite sequences of claims,
> no claim is first-false, and so
> no claim is false.
>
>
> In some finite sequences of claims,
> some claims are not-first-false.
> Either they are true
> or they are preceded by other false claims.
> In no case are they first-false.
>
> For example, consider the sequence
> ⟨... P∨Q ... ¬P ... Q ...⟩
> Q is not-first-false in that sequence.
> Either Q is true or one of P∨Q ¬P is false.
>
>
> If, in a finite sequence of claims,
> each claim is not-first-false,
> then each claim is not-false.
> There is no help for it, just as,
> in a finite sequence of sheep with
> each sheep not-first-black,
> there is no help for
> each sheep being not-black.
>
> Mathematicians spend their lives
> finding or inventing finite sequences of
> only not-first-false claims,
> which are called "proofs".
>
>
> Typically, the conclusion of a proof
> only applies to a certain subject matter.
> The Pythagorean theorem only applies to
> right triangles, not to just any triangles.
>
> This restriction of subject matter
> is incorporated into the proof as one or more
> claims about _what the proof is about_
> In a proof of the Pythagorean theorem,
> there will likely be, somewhere,
> a claim that ABC is a triangle and
> the angle at C is a right angle,
> or something like that.
>
> That is true of each right triangle
> (allowing for re-naming vertices)
> If ABC refers to a right triangle,
> that description and each not-first-false
> claim after the description
> must be true.
>
> True about _a right triangle_
> For objects _not a right triangle_
> we can't say the sequence of claims
> is only not-first-false.
> We can't say the proof's conclusion
> should apply to the not-right-triangle.
>
> For those objects a proof says it is about,
> the conclusion is inescapable.
>
> However, for other objects about which
> the proof is silent,
> we should be silent about them too,
> if we don't have different proofs.
>
>
> Consider two proofs.
> One is about constructible numbers.
> One is about ℚ and points between splits of ℚ
>
> There is a little overlap because
> some points between splits are constructible,
> but generally they are different proofs
> about different subject matter, and
> although each is very strong in
> its own domain,
> it is incorrect to take them outside
> that domain.
I don't know what to say about the rest of the logic

But what I personally think the cause of this global misunderstanding about the falsehood continiouty of real existing numbers is the endless density of constructible numbers on a real number line, where it becomes too easy for human mind cheaters to deceive others by exploiting such un imaginable density of constructible numbers

And what is a real number line? It is simply numbering of existing distances, where it should be named as number distance axies or equevelently distance number axies, since line has no meaning other than eingineering approximations

So, back to my direct proof by considering the Dedekind cuts for Sqrt2, where the greatest real number that is strictly less than Sqrt2, doesn't exist, Right?
Similarly, the least real number that is strictly greater than sqrt2 although doesn't exist, Right?

Then real existing numbers which are only constructible are infact isolated, thus discrete numbers
But, due to the high endless density of constructible numbers, they seem to our limited eyes as continuous FOR SURE

Any computer eingineer can see this truth by zooming!

So, we ought to start from the beginning without making any consideration to our eingineering science requirements

I mean mathematics must be liberated from the control of scientists & eingineers as well, simply because it is universal & not earthy

🔊 Bassam Karzeddin 🔊

On Tuesday, October 10, 2023 at 5:21:37 PM UTC-7, bassam karzeddin wrote:
> On Wednesday, October 11, 2023 at 12:21:33 AM UTC+3, Jim Burns wrote:
> > On 10/10/2023 4:03 AM, bassam karzeddin wrote:
> > > On Monday, October 9, 2023
> > > at 11:56:20 PM UTC+3, Jim Burns wrote:
> > >> [...]
> > >
> >
> > > any kind of rigorous proof,
> > Claims and sheep have something in common.
> >
> > In a finite sequence,
> Here is a false common inhireted beileaf among logicians about the term finite and infinite as well, where humans generally think that thetmy are absolutely different & opposing each other in meaning think that finite isn't, where this comes basically from set theory, but the truth is something else, What? Yes
>
> Then what is the largest FINITE natural number then?
>
> Of course, it is also Infinite which doesn't exist FOR SURE, Does it?
> See clearly, how are those terminologies as FINITE & Infinite have no real meaning in true mathematics, but probably in eingineering science has some usefulness
>
> So, the problem with mathematians & logicians is deeply rooted & very serious as well as long as they are adopting the eingineering principles in finding satisfaction solutions to practical problems!
> > if any of them have a property, then
> > one of them is first to have that property.
> >
> > In a finite sequence of sheep,
> > if any sheep is black, then
> > there is a first-black sheep.
> >
> > In a finite sequence of claims,
> > if any claim is false, then in science & eingineering may have applica
> > there is a first-false claim.
> >
> > In some finite sequences of claims,
> > no claim is first-false, and so
> > no claim is false.
> >
> >
> > In some finite sequences of claims,
> > some claims are not-first-false.
> > Either they are true
> > or they are preceded by other false claims.
> > In no case are they first-false.
> >
> > For example, consider the sequence
> > ⟨... P∨Q ... ¬P ... Q ...⟩
> > Q is not-first-false in that sequence.
> > Either Q is true or one of P∨Q ¬P is false.
> >
> >
> > If, in a finite sequence of claims,
> > each claim is not-first-false,
> > then each claim is not-false.
> > There is no help for it, just as,
> > in a finite sequence of sheep with
> > each sheep not-first-black,
> > there is no help for
> > each sheep being not-black.
> >
> > Mathematicians spend their lives
> > finding or inventing finite sequences of
> > only not-first-false claims,
> > which are called "proofs".
> >
> >
> > Typically, the conclusion of a proof
> > only applies to a certain subject matter.
> > The Pythagorean theorem only applies to
> > right triangles, not to just any triangles.
> >
> > This restriction of subject matter
> > is incorporated into the proof as one or more
> > claims about _what the proof is about_
> > In a proof of the Pythagorean theorem,
> > there will likely be, somewhere,
> > a claim that ABC is a triangle and
> > the angle at C is a right angle,
> > or something like that.
> >
> > That is true of each right triangle
> > (allowing for re-naming vertices)
> > If ABC refers to a right triangle,
> > that description and each not-first-false
> > claim after the description
> > must be true.
> >
> > True about _a right triangle_
> > For objects _not a right triangle_
> > we can't say the sequence of claims
> > is only not-first-false.
> > We can't say the proof's conclusion
> > should apply to the not-right-triangle.
> >
> > For those objects a proof says it is about,
> > the conclusion is inescapable.
> >
> > However, for other objects about which
> > the proof is silent,
> > we should be silent about them too,
> > if we don't have different proofs.
> >
> >
> > Consider two proofs.
> > One is about constructible numbers.
> > One is about ℚ and points between splits of ℚ
> >
> > There is a little overlap because
> > some points between splits are constructible,
> > but generally they are different proofs
> > about different subject matter, and
> > although each is very strong in
> > its own domain,
> > it is incorrect to take them outside
> > that domain.
> I don't know what to say about the rest of the logic
>
> But what I personally think the cause of this global misunderstanding about the falsehood continiouty of real existing numbers is the endless density of constructible numbers on a real number line, where it becomes too easy for human mind cheaters to deceive others by exploiting such un imaginable density of constructible numbers
>
> And what is a real number line? It is simply numbering of existing distances, where it should be named as number distance axies or equevelently distance number axies, since line has no meaning other than eingineering approximations
>
> So, back to my direct proof by considering the Dedekind cuts for Sqrt2, where the greatest real number that is strictly less than Sqrt2, doesn't exist, Right?
> Similarly, the least real number that is strictly greater than sqrt2 although doesn't exist, Right?
>
> Then real existing numbers which are only constructible are infact isolated, thus discrete numbers
> But, due to the high endless density of constructible numbers, they seem to our limited eyes as continuous FOR SURE
>
> Any computer eingineer can see this truth by zooming!
>
> So, we ought to start from the beginning without making any consideration to our eingineering science requirements
>
> I mean mathematics must be liberated from the control of scientists & eingineers as well, simply because it is universal & not earthy
>
> 🔊 Bassam Karzeddin 🔊

Re "Earthly", aren't we all?

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.

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 🔊

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.

Idiocy of Bassam Karzeddin (as an irrefutable fact)

On Wednesday, October 11, 2023 at 5:10:31 AM UTC+2, 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 🔊