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

> [...]

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

It's a good thing to remember.

I think that that is why
the real numbers are so beloved by physicists,
even though they use almost none of them.
It is the continuous functions which are
their true love.
Real numbers are a means to that end.

> Identity's a continuous function on
> a continuous domain.

Yes,
identity is not one of the functions
which we intend to rule out for jumping.

Identity is exceptionally well-behaved.

There are other functions less well-behaved,
which are only piecewise well-behaved.
A function may be continuous on
each of the internally-connected components of
of its domain, but it may jump
from one disconnected component to another.

Consider
H(x) = [0≤x]
H(x) = 1 if 0≤x
H(x) = 0 if x<0

H(x) is continuous on
the disconnected domain ℝ\{0}
H(x) is discontinuous at 0
on the connected domain ℝ

We can't rule out H: ℝ\{0} -> {0,1}
by requiring that
there is no point at which it jumps.
There isn't such a point.
0 is not in its domain.

We can rule out the domain ℝ\{0} of H(x)
by requiring that
there are no two disjoint open sets
into which its domain can be split.
There are.

No jumps in a connected domain
gives us no jumps, period,
our Holy Grail.

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

{x:ℕ→{0,1}}

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

∀x,y ∈ {x:ℕ→{0,1}}:
x < y :⇔
∃j ∈ ℕ: x(j) < y(j) ∧ ∀i<j: x(i) = y(i)

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

The square Cantor space is uncountable.
Order doesn't change cardinality.

> after a geometrization and length assignment
> the space, line-continuity's,
> "including the diagonal".

{x:ℕ→{0,1}} has a natural order.
That order is not the 1×1 order of a sequence.

In a sequence,
there is 'successor' and 'predecessor'.
Nothing else from the sequence is between
successor and predecessor.

However,
between each two x,y ∈ {x:ℕ→{0,1}}:
there are z ∈ {x:ℕ→{0,1}}: x < z < y

-- mostly there are z between.
Suppose
∀i<j: x(i) = y(i)
x(j) = 0 ∧ y(j) = 1
∀i>j: x(i) = 1 ∧ y(i) = 0

Still, {x:ℕ→{0,1}} is mostly not 1×1, not a sequence.

> [...] while "Square Cantor space" is "square",
> countable down and across,

{x:ℕ→{0,1}} is countable across,
uncountable down in any order.

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

The anti-diagonal is not an entry.
There is always an anti-diagonal,
so there is no order in which
the list does not miss some.

> It's the first nonstandard function
> "what is it?
> It's drawing a line and saying
> it was points in order,

Yes.
Trichotomous and transitive.

> as by integers in order,

No.
Mostly not 1×1

> must be infinite".

must not be listable.

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

Mathematicians are notoriously unwilling
to just go along with the crowd.

When they agree, it is because
there is no alternative.

There is certainly a place for
thinking outside the box in mathematics.

But being outside the box in itself
does not confer honor or legitimacy.
Remember what they are notorious for.
That would be like expecting applause
for getting up in the morning.

Showing that _there is no alternative_
to leaving the box, ah, now we're talking
big legitimacy, big honor.

If All The Mathematicians
go along with the crowd
by disagreeing with you,
it is very likely that
there is no alternative.

On Sunday, October 15, 2023 at 2:25:16 AM UTC+3, Jim Burns wrote:
> On 10/14/2023 5:21 PM, bassam karzeddin wrote:
> > On Saturday, October 14, 2023
> > at 11:31:22 PM UTC+3, Jim Burns wrote:
> > [...]
> > If real numbers are
> > only positive constructible numbers
> > (I.e distinct distances on the real number line),
> > which are of course discrete numbers,
> > then how would be there a continiouty?
>
> If
> real numbers are

I meat, real numbers are only & strictly positive constructible numbers (as I did confirm & proved earlier in public published posts ) where other alleged real numbers are non-existing & artificials (introduced by non-true mathematicians ) FOR SURE
> only positive constructible numbers
> then
> there are continuous curves which
> cross but do not intersect.

No curve exists, even though they do appear so for reason of unlimited density of constructible numbers

>
> Curves which cross but do not intersect
> are not continuous.

False, Even two unparalleled lines on the same plan don't intersect forming an angle EXACTLY like Pi/7 FOR SURE

> Real numbers are not
> only positive constructible numbers.

Real Existing number are Exact distances on the real distance line, where they are EXACT itentity, & for many years by now, I'm only asking the whole 🌎 of mathematics to bring one real number which isn't a constructible number (either numerically or Geometrically, but EXACTLY), where No humans on earth & the skies as well could do it, & they can't do it (simply because they globally refuse to understand the most simple & too elementary truth about what is truly a real number, for many reasons that are in full contradictionS with their incurable physiological inhireted problems besides their illegal orientations & fake intellectuality, FOR SURE

BASSAM KARZEDDIN

On 10/15/2023 9:24 PM, bassam karzeddin wrote:
> On Sunday, October 15, 2023
> at 2:25:16 AM UTC+3, Jim Burns wrote:

>> Curves which cross but do not intersect
>> are not continuous.
>
> False,
> Even two unparalleled lines on the same plan
> don't intersect
> forming an angle EXACTLY like Pi/7
> FOR SURE

Two non-parallel co-planar lines cross
and intersect.

The angle at at which they cross
doesn't change that.

On Monday, October 16, 2023 at 7:59:47 AM UTC+3, Jim Burns wrote:
> On 10/15/2023 9:24 PM, bassam karzeddin wrote:
> > On Sunday, October 15, 2023
> > at 2:25:16 AM UTC+3, Jim Burns wrote:
>
> >> Curves which cross but do not intersect
> >> are not continuous.
> >
> > False,
> > Even two unparalleled lines on the same plan
> > don't intersect
> > forming an angle EXACTLY like Pi/7
> > FOR SURE
> Two non-parallel co-planar lines cross
> and intersect.
That is absolutely true only with true existing angles that are described as constructible angles even in standard mathematics, but absolutely impossible with the vast majority of non-existing angles that ALL humans usually believe in their mere existence
>
> The angle at at which they cross
> doesn't change that.

The point is to reprove the new issue of non-existing angles since it is the most important key to understand most of the existing contradictions in mathematics among mathematicians themselves & all other relevant theoretical scientists

So, provide only one angle of intesection lines like (1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, ..., 89) degrees or any of angles like (Pi/7, Pi/9, Pi/11, Pi/13, Pi/14, Pi/18, ...., ) as angle of two intersection lines!

But alase; this is absolutely impossible task FOR SURE

BKK

On Monday, October 16, 2023 at 10:07:55 AM UTC+3, bassam karzeddin wrote:
> On Monday, October 16, 2023 at 7:59:47 AM UTC+3, Jim Burns wrote:
> > On 10/15/2023 9:24 PM, bassam karzeddin wrote:
> > > On Sunday, October 15, 2023
> > > at 2:25:16 AM UTC+3, Jim Burns wrote:
> >
> > >> Curves which cross but do not intersect
> > >> are not continuous.
> > >
> > > False,
> > > Even two unparalleled lines on the same plan
> > > don't intersect
> > > forming an angle EXACTLY like Pi/7
> > > FOR SURE
> > Two non-parallel co-planar lines cross
> > and intersect.
> That is absolutely true only with true existing angles that are described as constructible angles even in standard mathematics, but absolutely impossible with the vast majority of non-existing angles that ALL humans usually believe in their mere existence
> >
> > The angle at at which they cross
> > doesn't change that.
> The point is to reprove the new issue of non-existing angles since it is the most important key to understand most of the existing contradictions in mathematics among mathematicians themselves & all other relevant theoretical scientists
>
> So, provide only one angle of intesection lines like (1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, ..., 89) degrees or any of angles like (Pi/7, Pi/9, Pi/11, Pi/13, Pi/14, Pi/18, ...., ) as angle of two intersection lines!
>
> But alase; this is absolutely impossible task FOR SURE
>
> BKK

Here, & whenever 🌎 mathematicians are intellectualy challenged for mathematical tasks they used to think it was too easy & whenever they realize their complete disability to do the task, they should review secretly their way of thinking & revise it to understand where did they go wrong & established purely false beliefs without being aware of it

You must ask yourself secretly (as an academic Genius mathematician, Logicians, Philosophers, Physicians, etc)

Why can't I get any of these angles ( tought only by the KING ) in a triangle with exactly known sides?

You have also to ask the whole world 🌎 of mathematics about it!

And not to ignore the issue so shamefully & cowordly as well!

There are many things in too elementary foundations of true unknown mathematics that you certainly hate to know & nobody wants you to learn anything about it for reasons that they only know

They want you personally ignorant as they usually are.... FOR SURE

BKK

On Sunday, October 15, 2023 at 3:35:07 PM UTC-7, Jim Burns wrote:
> On 10/13/2023 10:00 PM, Ross Finlayson wrote:
> > On Friday, October 13, 2023
> > at 11:44:33 AM UTC-7, Jim Burns wrote:
> > [...]
> > It is commendable, that
> > continuous domains, like time, make for
> > continuous functions
> > the properties of continuous functions.
> It's a good thing to remember.
>
> I think that that is why
> the real numbers are so beloved by physicists,
> even though they use almost none of them.
> It is the continuous functions which are
> their true love.
> Real numbers are a means to that end.
> > Identity's a continuous function on
> > a continuous domain.
> Yes,
> identity is not one of the functions
> which we intend to rule out for jumping.
>
> Identity is exceptionally well-behaved.
>
> There are other functions less well-behaved,
> which are only piecewise well-behaved.
> A function may be continuous on
> each of the internally-connected components of
> of its domain, but it may jump
> from one disconnected component to another.
>
> Consider
> H(x) = [0≤x]
> H(x) = 1 if 0≤x
> H(x) = 0 if x<0
>
> H(x) is continuous on
> the disconnected domain ℝ\{0}
> H(x) is discontinuous at 0
> on the connected domain ℝ
>
> We can't rule out H: ℝ\{0} -> {0,1}
> by requiring that
> there is no point at which it jumps.
> There isn't such a point.
> 0 is not in its domain.
>
> We can rule out the domain ℝ\{0} of H(x)
> by requiring that
> there are no two disjoint open sets
> into which its domain can be split.
> There are.
>
> No jumps in a connected domain
> gives us no jumps, period,
> our Holy Grail.
> > The Cantor space, is the set of,
> > all the sequences, infinite, of the 0's and 1's.
> {x:ℕ→{0,1}}
> > The square Cantor space:
> > is those in order, their natural order,
> > as that the series are expansions.
> ∀x,y ∈ {x:ℕ→{0,1}}:
> x < y :⇔
> ∃j ∈ ℕ: x(j) < y(j) ∧ ∀i<j: x(i) = y(i)
> > The other use of powerset representation of
> > the Cantor space is to represent each of
> > a countable domain's, members, presence, in
> > any of the powersets or "sets of all subsets",
> > the set.
> >
> > So, Cantor space is uncountable,
> > but square Cantor space is "countable",
> The square Cantor space is uncountable.
> Order doesn't change cardinality.
> > after a geometrization and length assignment
> > the space, line-continuity's,
> > "including the diagonal".
> {x:ℕ→{0,1}} has a natural order.
> That order is not the 1×1 order of a sequence.
>
> In a sequence,
> there is 'successor' and 'predecessor'.
> Nothing else from the sequence is between
> successor and predecessor.
>
> However,
> between each two x,y ∈ {x:ℕ→{0,1}}:
> there are z ∈ {x:ℕ→{0,1}}: x < z < y
>
> -- mostly there are z between.
> Suppose
> ∀i<j: x(i) = y(i)
> x(j) = 0 ∧ y(j) = 1
> ∀i>j: x(i) = 1 ∧ y(i) = 0
>
> Still, {x:ℕ→{0,1}} is mostly not 1×1, not a sequence..
>
> > [...] while "Square Cantor space" is "square",
> > countable down and across,
>
> {x:ℕ→{0,1}} is countable across,
> uncountable down in any order.
> > This is where "diagonalization" as it were
> > is usually opposite the intent,
> > "anti-diagonalization",
> > when it really means to satisfy what's there
> > (that countability naturally leaves out).
> The anti-diagonal is not an entry.
> There is always an anti-diagonal,
> so there is no order in which
> the list does not miss some.
> > It's the first nonstandard function
> > "what is it?
> > It's drawing a line and saying
> > it was points in order,
> Yes.
> Trichotomous and transitive.
> > as by integers in order,
> No.
> Mostly not 1×1
>
> > must be infinite".
>
> must not be listable.
> > I came looking for mathematics and
> > when I found out that there were missing
> > formalisms for what I knew, I was like
> > "where's the entire world of naming
> > how this all fits together
> > these modes of continuity", "modes",
> > and it was like "make one yourself"
> > and thusly it would be unconscionable
> > to not say
> > "here let me give this an opinion and
> > an argument for itself",
> > "must be so",
> > and I was like
> > "really? I get to do all this myself?"
> > and it's like
> > "I guess it depends if they want to go along".
> Mathematicians are notoriously unwilling
> to just go along with the crowd.
>
> When they agree, it is because
> there is no alternative.
>
> There is certainly a place for
> thinking outside the box in mathematics.
>
> But being outside the box in itself
> does not confer honor or legitimacy.
> Remember what they are notorious for.
> That would be like expecting applause
> for getting up in the morning.
>
> Showing that _there is no alternative_
> to leaving the box, ah, now we're talking
> big legitimacy, big honor.
>
> If All The Mathematicians
> go along with the crowd
> by disagreeing with you,
> it is very likely that
> there is no alternative.

Wonderful, I'll write a fuller reply later, about the
"modern notions of a continuous domain", then
as specifically about the theory of functions, and,
a non-standard function, though with a particular character,
of its in-decomposability, and, the establishment from its
countable and integer domain, to a subset of [0,1] as
magnitudes in real values, the extent, density, completeness,
and measure, properties of a continuous domain in real-valued
variables, in real-valued quantites, for the purposes of mathematics.

Mathematics is discovered not invented /
can't be ignored.

On 10/16/2023 3:07 AM, bassam karzeddin wrote:
> On Monday, October 16, 2023
> at 7:59:47 AM UTC+3, Jim Burns wrote:
>> On 10/15/2023 9:24 PM, bassam karzeddin wrote:
>>> On Sunday, October 15, 2023
>>> at 2:25:16 AM UTC+3, Jim Burns wrote:

>>>> Curves which cross but do not intersect
>>>> are not continuous.
>>>
>>> False,
>>> Even two unparalleled lines on the same plan
>>> don't intersect
>>> forming an angle EXACTLY like Pi/7
>>> FOR SURE
>>
>> Two non-parallel co-planar lines cross
>> and intersect.
>
> That is absolutely true only
> with true existing angles that are
> described as constructible angles
> even in standard mathematics,

No.

| If a line segment intersects two straight
| lines forming two interior angles on the same
| side that are less than two right angles,
| then the two lines, if extended indefinitely,
| meet on that side on which the angles sum to
| less than two right angles.
| https://en.wikipedia.org/wiki/Parallel_postulate

Curves which cross but do not intersect
are not continuous.

----
Achilles chases the tortoise, and passes it.
His path must contain more than only
constructible points.

Each constructible point has
a finite-length construction.

Ordered by length of construction,
each point is preceded by
only finitely-many points.

Ordered by length of construction,
each point is followed by
more constructible points.
That would be true even if we only
considered midpoints of preceding points.
In [0,1]
0 1 ¹/₂ ¹/₄ ³/₄ ¹/₈ ³/₈ ⁵/₈ ⁷/₈ ¹/₁₆ …

The details of which points precede
which points don't matter here.
What matters is that
the path-points xₖ have a sequence-order '≪'
each xₖ finitely-preceded and not-last
in that sequence.
x₁ ≪ x₂ ≪ x₃ ≪ x₄ ≪ …

Also, the path-points have a path-order '<'
trichotomous and transitive
x <≠≯ x′ ∨ x ≮=≯ x′ ∨ x ≮≠> x′
x < x′ ∨ x′ < x″ ⟹ x < x″

There is an infinite nested sub-sequence
of pairs xⱼ₁,xⱼ₂
x₁ < x₂₁ < x₃₁ < … < x₃₂ < x₂₂ < x₂
x₁,x₂ ≪ x₂₁,x₂₂ ≪ x₃₁,x₃₂ ≪ …

No sequence-point xₓ exists such that
xⱼ₁ < xₓ < xⱼ₂ for each pair xⱼ₁,xⱼ₂
[1]

However,
a function f(x) can be defined which is
f(x) = 0 before any xⱼ₁
f(x) = 1 after any xⱼ₂

If there is no point between each xⱼ₁,xⱼ₂
f(x) is continuous everywhere.
And f(x) jumps.

Because jumping functions are not
continuous everywhere,
a point xₓ exists between all xⱼ₁,xⱼ₂
but xₓ is not in the sequence.

All the constructible points are
in the sequence,
therefore
xₓ which exists not-in the sequence,
is not a constructible point.

[1]
No sequence-point xₓ exists such that
xⱼ₁ < xₓ < xⱼ₂ for each pair xⱼ₁,xⱼ₂

| Assume otherwise.
| Assume, for sequence-point xₓ
| x₁ < x₂₁ < x₃₁ < … < xx < … < x₃₂ < x₂₂ < x₂
| x₁,x₂ ≪ x₂₁,x₂₂ ≪ x₃₁,x₃₂ ≪ … ≪ xₓ
| | xₓ is preceded by infinitely-many endpoints
| | However,
| each point xₖ is finitely-preceded
| in the sequence x₁ x₂ x₃ …
| Contradiction.

xₓ is not in the sequence,
which contains all constructible points
xₓ which exists, is not constructible.

On Monday, October 16, 2023 at 12:04:19 PM UTC-7, Ross Finlayson wrote:
> On Sunday, October 15, 2023 at 3:35:07 PM UTC-7, Jim Burns wrote:
> > On 10/13/2023 10:00 PM, Ross Finlayson wrote:
> > > On Friday, October 13, 2023
> > > at 11:44:33 AM UTC-7, Jim Burns wrote:
> > > [...]
> > > It is commendable, that
> > > continuous domains, like time, make for
> > > continuous functions
> > > the properties of continuous functions.
> > It's a good thing to remember.
> >
> > I think that that is why
> > the real numbers are so beloved by physicists,
> > even though they use almost none of them.
> > It is the continuous functions which are
> > their true love.
> > Real numbers are a means to that end.
> > > Identity's a continuous function on
> > > a continuous domain.
> > Yes,
> > identity is not one of the functions
> > which we intend to rule out for jumping.
> >
> > Identity is exceptionally well-behaved.
> >
> > There are other functions less well-behaved,
> > which are only piecewise well-behaved.
> > A function may be continuous on
> > each of the internally-connected components of
> > of its domain, but it may jump
> > from one disconnected component to another.
> >
> > Consider
> > H(x) = [0≤x]
> > H(x) = 1 if 0≤x
> > H(x) = 0 if x<0
> >
> > H(x) is continuous on
> > the disconnected domain ℝ\{0}
> > H(x) is discontinuous at 0
> > on the connected domain ℝ
> >
> > We can't rule out H: ℝ\{0} -> {0,1}
> > by requiring that
> > there is no point at which it jumps.
> > There isn't such a point.
> > 0 is not in its domain.
> >
> > We can rule out the domain ℝ\{0} of H(x)
> > by requiring that
> > there are no two disjoint open sets
> > into which its domain can be split.
> > There are.
> >
> > No jumps in a connected domain
> > gives us no jumps, period,
> > our Holy Grail.
> > > The Cantor space, is the set of,
> > > all the sequences, infinite, of the 0's and 1's.
> > {x:ℕ→{0,1}}
> > > The square Cantor space:
> > > is those in order, their natural order,
> > > as that the series are expansions.
> > ∀x,y ∈ {x:ℕ→{0,1}}:
> > x < y :⇔
> > ∃j ∈ ℕ: x(j) < y(j) ∧ ∀i<j: x(i) = y(i)
> > > The other use of powerset representation of
> > > the Cantor space is to represent each of
> > > a countable domain's, members, presence, in
> > > any of the powersets or "sets of all subsets",
> > > the set.
> > >
> > > So, Cantor space is uncountable,
> > > but square Cantor space is "countable",
> > The square Cantor space is uncountable.
> > Order doesn't change cardinality.
> > > after a geometrization and length assignment
> > > the space, line-continuity's,
> > > "including the diagonal".
> > {x:ℕ→{0,1}} has a natural order.
> > That order is not the 1×1 order of a sequence.
> >
> > In a sequence,
> > there is 'successor' and 'predecessor'.
> > Nothing else from the sequence is between
> > successor and predecessor.
> >
> > However,
> > between each two x,y ∈ {x:ℕ→{0,1}}:
> > there are z ∈ {x:ℕ→{0,1}}: x < z < y
> >
> > -- mostly there are z between.
> > Suppose
> > ∀i<j: x(i) = y(i)
> > x(j) = 0 ∧ y(j) = 1
> > ∀i>j: x(i) = 1 ∧ y(i) = 0
> >
> > Still, {x:ℕ→{0,1}} is mostly not 1×1, not a sequence.
> >
> > > [...] while "Square Cantor space" is "square",
> > > countable down and across,
> >
> > {x:ℕ→{0,1}} is countable across,
> > uncountable down in any order.
> > > This is where "diagonalization" as it were
> > > is usually opposite the intent,
> > > "anti-diagonalization",
> > > when it really means to satisfy what's there
> > > (that countability naturally leaves out).
> > The anti-diagonal is not an entry.
> > There is always an anti-diagonal,
> > so there is no order in which
> > the list does not miss some.
> > > It's the first nonstandard function
> > > "what is it?
> > > It's drawing a line and saying
> > > it was points in order,
> > Yes.
> > Trichotomous and transitive.
> > > as by integers in order,
> > No.
> > Mostly not 1×1
> >
> > > must be infinite".
> >
> > must not be listable.
> > > I came looking for mathematics and
> > > when I found out that there were missing
> > > formalisms for what I knew, I was like
> > > "where's the entire world of naming
> > > how this all fits together
> > > these modes of continuity", "modes",
> > > and it was like "make one yourself"
> > > and thusly it would be unconscionable
> > > to not say
> > > "here let me give this an opinion and
> > > an argument for itself",
> > > "must be so",
> > > and I was like
> > > "really? I get to do all this myself?"
> > > and it's like
> > > "I guess it depends if they want to go along".
> > Mathematicians are notoriously unwilling
> > to just go along with the crowd.
> >
> > When they agree, it is because
> > there is no alternative.
> >
> > There is certainly a place for
> > thinking outside the box in mathematics.
> >
> > But being outside the box in itself
> > does not confer honor or legitimacy.
> > Remember what they are notorious for.
> > That would be like expecting applause
> > for getting up in the morning.
> >
> > Showing that _there is no alternative_
> > to leaving the box, ah, now we're talking
> > big legitimacy, big honor.
> >
> > If All The Mathematicians
> > go along with the crowd
> > by disagreeing with you,
> > it is very likely that
> > there is no alternative.
> Wonderful, I'll write a fuller reply later, about the
> "modern notions of a continuous domain", then
> as specifically about the theory of functions, and,
> a non-standard function, though with a particular character,
> of its in-decomposability, and, the establishment from its
> countable and integer domain, to a subset of [0,1] as
> magnitudes in real values, the extent, density, completeness,
> and measure, properties of a continuous domain in real-valued
> variables, in real-valued quantites, for the purposes of mathematics.
>
> Mathematics is discovered not invented /
> can't be ignored.

Well, a more fuller reply, here is not so much. Everyone here has had to sit through
my pointed writings which basically result an extended expression.

This is to include that mathematics is a common language, there's that "mathematics
is the common language". Symbols are the common language, often words.

So, first, I axiomatize the entire notion of "0 to 1 in the real numbers is zero to infinity
in the integers", "must be infinite", "includes infinity".

Then, any unit line segment is that, one of those, each one a different copy of an integers,
an infinitude of integers.

By what end it is, either end, it's the integers up and down, or up and down.

I.e., the "Equivalency Function", "natural/unit", of course is a "discrete/continuous" function,
a function, one side actually discrete, if infinite, and the other side continuous "[0,1] real numbers".

Then, "sweep", is the principle, for example axiomatized above.

It's like I'm reading Kepler and Courant, and Courant says "of course we must implicitly axiomatize
_away_ any conflicts existence _not_ contradicting, sweep doesn't exist".

Then of course it sits perfectly satisfied in defining accumulation and limit and critical points,
continuous and uniform continuous, Courant's.

Then, where it does, for example is "though it uses otherwise the singleton copy of the integers,
in any space its its own book-keeping for geometry in the same sense as an integer axis in an integer
lattice, also is".

On Monday, October 16, 2023 at 10:24:05 PM UTC-7, Ross Finlayson wrote:
> On Monday, October 16, 2023 at 12:04:19 PM UTC-7, Ross Finlayson wrote:
> > On Sunday, October 15, 2023 at 3:35:07 PM UTC-7, Jim Burns wrote:
> > > On 10/13/2023 10:00 PM, Ross Finlayson wrote:
> > > > On Friday, October 13, 2023
> > > > at 11:44:33 AM UTC-7, Jim Burns wrote:
> > > > [...]
> > > > It is commendable, that
> > > > continuous domains, like time, make for
> > > > continuous functions
> > > > the properties of continuous functions.
> > > It's a good thing to remember.
> > >
> > > I think that that is why
> > > the real numbers are so beloved by physicists,
> > > even though they use almost none of them.
> > > It is the continuous functions which are
> > > their true love.
> > > Real numbers are a means to that end.
> > > > Identity's a continuous function on
> > > > a continuous domain.
> > > Yes,
> > > identity is not one of the functions
> > > which we intend to rule out for jumping.
> > >
> > > Identity is exceptionally well-behaved.
> > >
> > > There are other functions less well-behaved,
> > > which are only piecewise well-behaved.
> > > A function may be continuous on
> > > each of the internally-connected components of
> > > of its domain, but it may jump
> > > from one disconnected component to another.
> > >
> > > Consider
> > > H(x) = [0≤x]
> > > H(x) = 1 if 0≤x
> > > H(x) = 0 if x<0
> > >
> > > H(x) is continuous on
> > > the disconnected domain ℝ\{0}
> > > H(x) is discontinuous at 0
> > > on the connected domain ℝ
> > >
> > > We can't rule out H: ℝ\{0} -> {0,1}
> > > by requiring that
> > > there is no point at which it jumps.
> > > There isn't such a point.
> > > 0 is not in its domain.
> > >
> > > We can rule out the domain ℝ\{0} of H(x)
> > > by requiring that
> > > there are no two disjoint open sets
> > > into which its domain can be split.
> > > There are.
> > >
> > > No jumps in a connected domain
> > > gives us no jumps, period,
> > > our Holy Grail.
> > > > The Cantor space, is the set of,
> > > > all the sequences, infinite, of the 0's and 1's.
> > > {x:ℕ→{0,1}}
> > > > The square Cantor space:
> > > > is those in order, their natural order,
> > > > as that the series are expansions.
> > > ∀x,y ∈ {x:ℕ→{0,1}}:
> > > x < y :⇔
> > > ∃j ∈ ℕ: x(j) < y(j) ∧ ∀i<j: x(i) = y(i)
> > > > The other use of powerset representation of
> > > > the Cantor space is to represent each of
> > > > a countable domain's, members, presence, in
> > > > any of the powersets or "sets of all subsets",
> > > > the set.
> > > >
> > > > So, Cantor space is uncountable,
> > > > but square Cantor space is "countable",
> > > The square Cantor space is uncountable.
> > > Order doesn't change cardinality.
> > > > after a geometrization and length assignment
> > > > the space, line-continuity's,
> > > > "including the diagonal".
> > > {x:ℕ→{0,1}} has a natural order.
> > > That order is not the 1×1 order of a sequence.
> > >
> > > In a sequence,
> > > there is 'successor' and 'predecessor'.
> > > Nothing else from the sequence is between
> > > successor and predecessor.
> > >
> > > However,
> > > between each two x,y ∈ {x:ℕ→{0,1}}:
> > > there are z ∈ {x:ℕ→{0,1}}: x < z < y
> > >
> > > -- mostly there are z between.
> > > Suppose
> > > ∀i<j: x(i) = y(i)
> > > x(j) = 0 ∧ y(j) = 1
> > > ∀i>j: x(i) = 1 ∧ y(i) = 0
> > >
> > > Still, {x:ℕ→{0,1}} is mostly not 1×1, not a sequence.
> > >
> > > > [...] while "Square Cantor space" is "square",
> > > > countable down and across,
> > >
> > > {x:ℕ→{0,1}} is countable across,
> > > uncountable down in any order.
> > > > This is where "diagonalization" as it were
> > > > is usually opposite the intent,
> > > > "anti-diagonalization",
> > > > when it really means to satisfy what's there
> > > > (that countability naturally leaves out).
> > > The anti-diagonal is not an entry.
> > > There is always an anti-diagonal,
> > > so there is no order in which
> > > the list does not miss some.
> > > > It's the first nonstandard function
> > > > "what is it?
> > > > It's drawing a line and saying
> > > > it was points in order,
> > > Yes.
> > > Trichotomous and transitive.
> > > > as by integers in order,
> > > No.
> > > Mostly not 1×1
> > >
> > > > must be infinite".
> > >
> > > must not be listable.
> > > > I came looking for mathematics and
> > > > when I found out that there were missing
> > > > formalisms for what I knew, I was like
> > > > "where's the entire world of naming
> > > > how this all fits together
> > > > these modes of continuity", "modes",
> > > > and it was like "make one yourself"
> > > > and thusly it would be unconscionable
> > > > to not say
> > > > "here let me give this an opinion and
> > > > an argument for itself",
> > > > "must be so",
> > > > and I was like
> > > > "really? I get to do all this myself?"
> > > > and it's like
> > > > "I guess it depends if they want to go along".
> > > Mathematicians are notoriously unwilling
> > > to just go along with the crowd.
> > >
> > > When they agree, it is because
> > > there is no alternative.
> > >
> > > There is certainly a place for
> > > thinking outside the box in mathematics.
> > >
> > > But being outside the box in itself
> > > does not confer honor or legitimacy.
> > > Remember what they are notorious for.
> > > That would be like expecting applause
> > > for getting up in the morning.
> > >
> > > Showing that _there is no alternative_
> > > to leaving the box, ah, now we're talking
> > > big legitimacy, big honor.
> > >
> > > If All The Mathematicians
> > > go along with the crowd
> > > by disagreeing with you,
> > > it is very likely that
> > > there is no alternative.
> > Wonderful, I'll write a fuller reply later, about the
> > "modern notions of a continuous domain", then
> > as specifically about the theory of functions, and,
> > a non-standard function, though with a particular character,
> > of its in-decomposability, and, the establishment from its
> > countable and integer domain, to a subset of [0,1] as
> > magnitudes in real values, the extent, density, completeness,
> > and measure, properties of a continuous domain in real-valued
> > variables, in real-valued quantites, for the purposes of mathematics.
> >
> > Mathematics is discovered not invented /
> > can't be ignored.
> Well, a more fuller reply, here is not so much. Everyone here has had to sit through
> my pointed writings which basically result an extended expression.
>
> This is to include that mathematics is a common language, there's that "mathematics
> is the common language". Symbols are the common language, often words.
>
> So, first, I axiomatize the entire notion of "0 to 1 in the real numbers is zero to infinity
> in the integers", "must be infinite", "includes infinity".
>
> Then, any unit line segment is that, one of those, each one a different copy of an integers,
> an infinitude of integers.
>
> By what end it is, either end, it's the integers up and down, or up and down.
>
> I.e., the "Equivalency Function", "natural/unit", of course is a "discrete/continuous" function,
> a function, one side actually discrete, if infinite, and the other side continuous "[0,1] real numbers".
>
> Then, "sweep", is the principle, for example axiomatized above.
>
> It's like I'm reading Kepler and Courant, and Courant says "of course we must implicitly axiomatize
> _away_ any conflicts existence _not_ contradicting, sweep doesn't exist".
>
> Then of course it sits perfectly satisfied in defining accumulation and limit and critical points,
> continuous and uniform continuous, Courant's.
>
> Then, where it does, for example is "though it uses otherwise the singleton copy of the integers,
> in any space its its own book-keeping for geometry in the same sense as an integer axis in an integer
> lattice, also is".
>
> So, not only is that standard, leaving that out because mathematics is satisfied having "a" copy
> of the integers if not "the copies", in terms of a value from integers having a given value.
>
> Reading Kepler is pretty great, he refers to at least a few cases of halving formulas and doubling
> formulas, for doubling spaces in continuity, and, in the terms, where for example Courant points
> directly at the Dirichlet function as "rationals 1, irrationals 0": "everywhere discontinuous, no measure".
>
> As a function thusly it's in a separate category of function, from the usual definition of function in
> axiomatic theory, or vice-versa, whether rationals and the field, usually, are more than "functions",
> their book-keeping, with that basically the line-reals "integers" are infinite, that inverse f 1/2: is "infinity/2".
>
> Then, there's that "infinite" is "any two-ended sequence, infinite both ways". Or, it's just as simply any greater,
> just that finite is sometimes great enough to have no greater.
>
> Basically the definition of limit shows that for f(n) = n/d, n -> d, d ->oo, there's
>
> extent
> density
> completeness
> measure
>
> I wrote these particularly and
> "for any x [0,1], f^-1 x is in dom(f)",
> also, for bit-strings, elements of [0,1] as necessarily "the representations
> in Square Cantor space", that
> "for any x [0,1], ~x e ran(f)".
>
> The extent is [0,1], accumulates, is complete somewhat trivially as any greater upper bound
> would at most be, "next", then a variety of sigma algebras for measure were written, as sets
> in set theory, with a particular attachment of "length assignment" geometrically, to reflect
> any association with "a measurable quantity" and "measure 1.0".
>
> Then for the rest apologetics is the slates, basically "sweep" in principle and for "intuition"
> and "axiomless natural deduction".
>
> I.e. "sweep the function" is defined briefly enough like so.
>
>
>
>
> If you ask most people in mathematics they could quite agree "yes that's the usual way to
> think about continuity from numbers, it makes a measure, and with that I can count with my hands".
>
> Also, specifically, "we were all told in school that mathematics does it the opposite way,
> that numbers are always divisible and never at once divided, except zero not but annihilating,
> which is fair that way, everybody knows that and we really identify it with the differential
> and it's called constant infinitesimals or iota-values, standard infinitesimals".
>
> That's just saying there's nothing but denying that I took only the _absolute_ _easiest_ ball,
> and ran with it, "pre-calculus" and "differentials and iota", this is nothing but paste stock in trade,
> this way it was at least easy to carry, wading the field.
>
> The rest of the field then I just put down on my slates, "justifications"..
>
> Then applications are coming down from deductive properties of the space, and
> thusly for example statistical properties, what seem false and even fail convergence
> tests, thus implying that as they are accurate, under the manipulation of their quantities
> mathematically, set down that other identities in constants follow.
>
> I.e. I put down some nonstandard applications, theoretically, because of course usual sorts
> derivations like having gone all the way to school. (Which I figure you can do yourself.)
>
> Having the easiest carry, then also let plenty room to setup the side, but I really made it
> easy on myself, so far as always agreeing with myself, I'm most technically flawless.
>
>
> As far as I know I always feel to have gotten my "last word", technically impervious forever.
>
> That way you can blame me for anything left unsaid.
>
>
> Courant's "sweep" that doesn't exist: is the same one here that does.
>
> It does for him, too: where it does.
>
>
> Again, just to re-iterate, this is only the most usual and obvious conception of "points in a line",
> "an infinitude from beginning to end", "that just so happens to equal 1.0", and here,
> "that just so happens to only always equal 1.0".
>
>
> So, I built all these apologetics so it's not contradictory with set theory at all.
>
> It's in a bigger one, ..., a regular set theory (if outside on the extra-ordinary itself).
>
>
> So, it's mostly not just "not much to see here" but also "what you already know",
> this "line-continuity" is _exactly_ "this is the perfect line-continuity that we _don't_
> talk about", basically for giving it its own piece on the board but never letting it cross.
> There it's "these perfect delta-epsilonics, divide these numbers forever and add them up".
>
> This way, it just "fixes set theory" for continuity, like I said, simple.

On Thursday, October 19, 2023 at 8:21:35 PM UTC-7, Ross Finlayson wrote:
> On Monday, October 16, 2023 at 10:24:05 PM UTC-7, Ross Finlayson wrote:
> > On Monday, October 16, 2023 at 12:04:19 PM UTC-7, Ross Finlayson wrote:
> > > On Sunday, October 15, 2023 at 3:35:07 PM UTC-7, Jim Burns wrote:
> > > > On 10/13/2023 10:00 PM, Ross Finlayson wrote:
> > > > > On Friday, October 13, 2023
> > > > > at 11:44:33 AM UTC-7, Jim Burns wrote:
> > > > > [...]
> > > > > It is commendable, that
> > > > > continuous domains, like time, make for
> > > > > continuous functions
> > > > > the properties of continuous functions.
> > > > It's a good thing to remember.
> > > >
> > > > I think that that is why
> > > > the real numbers are so beloved by physicists,
> > > > even though they use almost none of them.
> > > > It is the continuous functions which are
> > > > their true love.
> > > > Real numbers are a means to that end.
> > > > > Identity's a continuous function on
> > > > > a continuous domain.
> > > > Yes,
> > > > identity is not one of the functions
> > > > which we intend to rule out for jumping.
> > > >
> > > > Identity is exceptionally well-behaved.
> > > >
> > > > There are other functions less well-behaved,
> > > > which are only piecewise well-behaved.
> > > > A function may be continuous on
> > > > each of the internally-connected components of
> > > > of its domain, but it may jump
> > > > from one disconnected component to another.
> > > >
> > > > Consider
> > > > H(x) = [0≤x]
> > > > H(x) = 1 if 0≤x
> > > > H(x) = 0 if x<0
> > > >
> > > > H(x) is continuous on
> > > > the disconnected domain ℝ\{0}
> > > > H(x) is discontinuous at 0
> > > > on the connected domain ℝ
> > > >
> > > > We can't rule out H: ℝ\{0} -> {0,1}
> > > > by requiring that
> > > > there is no point at which it jumps.
> > > > There isn't such a point.
> > > > 0 is not in its domain.
> > > >
> > > > We can rule out the domain ℝ\{0} of H(x)
> > > > by requiring that
> > > > there are no two disjoint open sets
> > > > into which its domain can be split.
> > > > There are.
> > > >
> > > > No jumps in a connected domain
> > > > gives us no jumps, period,
> > > > our Holy Grail.
> > > > > The Cantor space, is the set of,
> > > > > all the sequences, infinite, of the 0's and 1's.
> > > > {x:ℕ→{0,1}}
> > > > > The square Cantor space:
> > > > > is those in order, their natural order,
> > > > > as that the series are expansions.
> > > > ∀x,y ∈ {x:ℕ→{0,1}}:
> > > > x < y :⇔
> > > > ∃j ∈ ℕ: x(j) < y(j) ∧ ∀i<j: x(i) = y(i)
> > > > > The other use of powerset representation of
> > > > > the Cantor space is to represent each of
> > > > > a countable domain's, members, presence, in
> > > > > any of the powersets or "sets of all subsets",
> > > > > the set.
> > > > >
> > > > > So, Cantor space is uncountable,
> > > > > but square Cantor space is "countable",
> > > > The square Cantor space is uncountable.
> > > > Order doesn't change cardinality.
> > > > > after a geometrization and length assignment
> > > > > the space, line-continuity's,
> > > > > "including the diagonal".
> > > > {x:ℕ→{0,1}} has a natural order.
> > > > That order is not the 1×1 order of a sequence.
> > > >
> > > > In a sequence,
> > > > there is 'successor' and 'predecessor'.
> > > > Nothing else from the sequence is between
> > > > successor and predecessor.
> > > >
> > > > However,
> > > > between each two x,y ∈ {x:ℕ→{0,1}}:
> > > > there are z ∈ {x:ℕ→{0,1}}: x < z < y
> > > >
> > > > -- mostly there are z between.
> > > > Suppose
> > > > ∀i<j: x(i) = y(i)
> > > > x(j) = 0 ∧ y(j) = 1
> > > > ∀i>j: x(i) = 1 ∧ y(i) = 0
> > > >
> > > > Still, {x:ℕ→{0,1}} is mostly not 1×1, not a sequence.
> > > >
> > > > > [...] while "Square Cantor space" is "square",
> > > > > countable down and across,
> > > >
> > > > {x:ℕ→{0,1}} is countable across,
> > > > uncountable down in any order.
> > > > > This is where "diagonalization" as it were
> > > > > is usually opposite the intent,
> > > > > "anti-diagonalization",
> > > > > when it really means to satisfy what's there
> > > > > (that countability naturally leaves out).
> > > > The anti-diagonal is not an entry.
> > > > There is always an anti-diagonal,
> > > > so there is no order in which
> > > > the list does not miss some.
> > > > > It's the first nonstandard function
> > > > > "what is it?
> > > > > It's drawing a line and saying
> > > > > it was points in order,
> > > > Yes.
> > > > Trichotomous and transitive.
> > > > > as by integers in order,
> > > > No.
> > > > Mostly not 1×1
> > > >
> > > > > must be infinite".
> > > >
> > > > must not be listable.
> > > > > I came looking for mathematics and
> > > > > when I found out that there were missing
> > > > > formalisms for what I knew, I was like
> > > > > "where's the entire world of naming
> > > > > how this all fits together
> > > > > these modes of continuity", "modes",
> > > > > and it was like "make one yourself"
> > > > > and thusly it would be unconscionable
> > > > > to not say
> > > > > "here let me give this an opinion and
> > > > > an argument for itself",
> > > > > "must be so",
> > > > > and I was like
> > > > > "really? I get to do all this myself?"
> > > > > and it's like
> > > > > "I guess it depends if they want to go along".
> > > > Mathematicians are notoriously unwilling
> > > > to just go along with the crowd.
> > > >
> > > > When they agree, it is because
> > > > there is no alternative.
> > > >
> > > > There is certainly a place for
> > > > thinking outside the box in mathematics.
> > > >
> > > > But being outside the box in itself
> > > > does not confer honor or legitimacy.
> > > > Remember what they are notorious for.
> > > > That would be like expecting applause
> > > > for getting up in the morning.
> > > >
> > > > Showing that _there is no alternative_
> > > > to leaving the box, ah, now we're talking
> > > > big legitimacy, big honor.
> > > >
> > > > If All The Mathematicians
> > > > go along with the crowd
> > > > by disagreeing with you,
> > > > it is very likely that
> > > > there is no alternative.
> > > Wonderful, I'll write a fuller reply later, about the
> > > "modern notions of a continuous domain", then
> > > as specifically about the theory of functions, and,
> > > a non-standard function, though with a particular character,
> > > of its in-decomposability, and, the establishment from its
> > > countable and integer domain, to a subset of [0,1] as
> > > magnitudes in real values, the extent, density, completeness,
> > > and measure, properties of a continuous domain in real-valued
> > > variables, in real-valued quantites, for the purposes of mathematics.
> > >
> > > Mathematics is discovered not invented /
> > > can't be ignored.
> > Well, a more fuller reply, here is not so much. Everyone here has had to sit through
> > my pointed writings which basically result an extended expression.
> >
> > This is to include that mathematics is a common language, there's that "mathematics
> > is the common language". Symbols are the common language, often words.
> >
> > So, first, I axiomatize the entire notion of "0 to 1 in the real numbers is zero to infinity
> > in the integers", "must be infinite", "includes infinity".
> >
> > Then, any unit line segment is that, one of those, each one a different copy of an integers,
> > an infinitude of integers.
> >
> > By what end it is, either end, it's the integers up and down, or up and down.
> >
> > I.e., the "Equivalency Function", "natural/unit", of course is a "discrete/continuous" function,
> > a function, one side actually discrete, if infinite, and the other side continuous "[0,1] real numbers".
> >
> > Then, "sweep", is the principle, for example axiomatized above.
> >
> > It's like I'm reading Kepler and Courant, and Courant says "of course we must implicitly axiomatize
> > _away_ any conflicts existence _not_ contradicting, sweep doesn't exist".
> >
> > Then of course it sits perfectly satisfied in defining accumulation and limit and critical points,
> > continuous and uniform continuous, Courant's.
> >
> > Then, where it does, for example is "though it uses otherwise the singleton copy of the integers,
> > in any space its its own book-keeping for geometry in the same sense as an integer axis in an integer
> > lattice, also is".
> >
> > So, not only is that standard, leaving that out because mathematics is satisfied having "a" copy
> > of the integers if not "the copies", in terms of a value from integers having a given value.
> >
> > Reading Kepler is pretty great, he refers to at least a few cases of halving formulas and doubling
> > formulas, for doubling spaces in continuity, and, in the terms, where for example Courant points
> > directly at the Dirichlet function as "rationals 1, irrationals 0": "everywhere discontinuous, no measure".
> >
> > As a function thusly it's in a separate category of function, from the usual definition of function in
> > axiomatic theory, or vice-versa, whether rationals and the field, usually, are more than "functions",
> > their book-keeping, with that basically the line-reals "integers" are infinite, that inverse f 1/2: is "infinity/2".
> >
> > Then, there's that "infinite" is "any two-ended sequence, infinite both ways". Or, it's just as simply any greater,
> > just that finite is sometimes great enough to have no greater.
> >
> > Basically the definition of limit shows that for f(n) = n/d, n -> d, d ->oo, there's
> >
> > extent
> > density
> > completeness
> > measure
> >
> > I wrote these particularly and
> > "for any x [0,1], f^-1 x is in dom(f)",
> > also, for bit-strings, elements of [0,1] as necessarily "the representations
> > in Square Cantor space", that
> > "for any x [0,1], ~x e ran(f)".
> >
> > The extent is [0,1], accumulates, is complete somewhat trivially as any greater upper bound
> > would at most be, "next", then a variety of sigma algebras for measure were written, as sets
> > in set theory, with a particular attachment of "length assignment" geometrically, to reflect
> > any association with "a measurable quantity" and "measure 1.0".
> >
> > Then for the rest apologetics is the slates, basically "sweep" in principle and for "intuition"
> > and "axiomless natural deduction".
> >
> > I.e. "sweep the function" is defined briefly enough like so.
> >
> >
> >
> >
> > If you ask most people in mathematics they could quite agree "yes that's the usual way to
> > think about continuity from numbers, it makes a measure, and with that I can count with my hands".
> >
> > Also, specifically, "we were all told in school that mathematics does it the opposite way,
> > that numbers are always divisible and never at once divided, except zero not but annihilating,
> > which is fair that way, everybody knows that and we really identify it with the differential
> > and it's called constant infinitesimals or iota-values, standard infinitesimals".
> >
> > That's just saying there's nothing but denying that I took only the _absolute_ _easiest_ ball,
> > and ran with it, "pre-calculus" and "differentials and iota", this is nothing but paste stock in trade,
> > this way it was at least easy to carry, wading the field.
> >
> > The rest of the field then I just put down on my slates, "justifications".
> >
> > Then applications are coming down from deductive properties of the space, and
> > thusly for example statistical properties, what seem false and even fail convergence
> > tests, thus implying that as they are accurate, under the manipulation of their quantities
> > mathematically, set down that other identities in constants follow.
> >
> > I.e. I put down some nonstandard applications, theoretically, because of course usual sorts
> > derivations like having gone all the way to school. (Which I figure you can do yourself.)
> >
> > Having the easiest carry, then also let plenty room to setup the side, but I really made it
> > easy on myself, so far as always agreeing with myself, I'm most technically flawless.
> >
> >
> > As far as I know I always feel to have gotten my "last word", technically impervious forever.
> >
> > That way you can blame me for anything left unsaid.
> >
> >
> > Courant's "sweep" that doesn't exist: is the same one here that does.
> >
> > It does for him, too: where it does.
> >
> >
> > Again, just to re-iterate, this is only the most usual and obvious conception of "points in a line",
> > "an infinitude from beginning to end", "that just so happens to equal 1..0", and here,
> > "that just so happens to only always equal 1.0".
> >
> >
> > So, I built all these apologetics so it's not contradictory with set theory at all.
> >
> > It's in a bigger one, ..., a regular set theory (if outside on the extra-ordinary itself).
> >
> >
> > So, it's mostly not just "not much to see here" but also "what you already know",
> > this "line-continuity" is _exactly_ "this is the perfect line-continuity that we _don't_
> > talk about", basically for giving it its own piece on the board but never letting it cross.
> > There it's "these perfect delta-epsilonics, divide these numbers forever and add them up".
> >
> > This way, it just "fixes set theory" for continuity, like I said, simple.
>
> Really now as we've broached the topic of "continuous domains",
> think of it as a concept that math has been putting off -
> putting the concept of "continuous functions" and having
> this "complete ordered field" continuous domain, that theorems
> and formulas about continuous functions, mostly are as analytically
> formulas, and theorems, as continuous domains,
> of whatever sort, here mostly "for example the complete ordered field".
>
> Thusly, the _language_ of sci.math changed, "continuous domains",
> is now a word, and it's the same throughout mathematics.
>
> Of course, the language didn't "change" so much, as that's what it "means",
> "continuous domains".

On Sunday, November 5, 2023 at 9:43:14 AM UTC-8, Ross Finlayson wrote:
> On Thursday, October 19, 2023 at 8:21:35 PM UTC-7, Ross Finlayson wrote:
> > On Monday, October 16, 2023 at 10:24:05 PM UTC-7, Ross Finlayson wrote:
> > > On Monday, October 16, 2023 at 12:04:19 PM UTC-7, Ross Finlayson wrote:
> > > > On Sunday, October 15, 2023 at 3:35:07 PM UTC-7, Jim Burns wrote:
> > > > > On 10/13/2023 10:00 PM, Ross Finlayson wrote:
> > > > > > On Friday, October 13, 2023
> > > > > > at 11:44:33 AM UTC-7, Jim Burns wrote:
> > > > > > [...]
> > > > > > It is commendable, that
> > > > > > continuous domains, like time, make for
> > > > > > continuous functions
> > > > > > the properties of continuous functions.
> > > > > It's a good thing to remember.
> > > > >
> > > > > I think that that is why
> > > > > the real numbers are so beloved by physicists,
> > > > > even though they use almost none of them.
> > > > > It is the continuous functions which are
> > > > > their true love.
> > > > > Real numbers are a means to that end.
> > > > > > Identity's a continuous function on
> > > > > > a continuous domain.
> > > > > Yes,
> > > > > identity is not one of the functions
> > > > > which we intend to rule out for jumping.
> > > > >
> > > > > Identity is exceptionally well-behaved.
> > > > >
> > > > > There are other functions less well-behaved,
> > > > > which are only piecewise well-behaved.
> > > > > A function may be continuous on
> > > > > each of the internally-connected components of
> > > > > of its domain, but it may jump
> > > > > from one disconnected component to another.
> > > > >
> > > > > Consider
> > > > > H(x) = [0≤x]
> > > > > H(x) = 1 if 0≤x
> > > > > H(x) = 0 if x<0
> > > > >
> > > > > H(x) is continuous on
> > > > > the disconnected domain ℝ\{0}
> > > > > H(x) is discontinuous at 0
> > > > > on the connected domain ℝ
> > > > >
> > > > > We can't rule out H: ℝ\{0} -> {0,1}
> > > > > by requiring that
> > > > > there is no point at which it jumps.
> > > > > There isn't such a point.
> > > > > 0 is not in its domain.
> > > > >
> > > > > We can rule out the domain ℝ\{0} of H(x)
> > > > > by requiring that
> > > > > there are no two disjoint open sets
> > > > > into which its domain can be split.
> > > > > There are.
> > > > >
> > > > > No jumps in a connected domain
> > > > > gives us no jumps, period,
> > > > > our Holy Grail.
> > > > > > The Cantor space, is the set of,
> > > > > > all the sequences, infinite, of the 0's and 1's.
> > > > > {x:ℕ→{0,1}}
> > > > > > The square Cantor space:
> > > > > > is those in order, their natural order,
> > > > > > as that the series are expansions.
> > > > > ∀x,y ∈ {x:ℕ→{0,1}}:
> > > > > x < y :⇔
> > > > > ∃j ∈ ℕ: x(j) < y(j) ∧ ∀i<j: x(i) = y(i)
> > > > > > The other use of powerset representation of
> > > > > > the Cantor space is to represent each of
> > > > > > a countable domain's, members, presence, in
> > > > > > any of the powersets or "sets of all subsets",
> > > > > > the set.
> > > > > >
> > > > > > So, Cantor space is uncountable,
> > > > > > but square Cantor space is "countable",
> > > > > The square Cantor space is uncountable.
> > > > > Order doesn't change cardinality.
> > > > > > after a geometrization and length assignment
> > > > > > the space, line-continuity's,
> > > > > > "including the diagonal".
> > > > > {x:ℕ→{0,1}} has a natural order.
> > > > > That order is not the 1×1 order of a sequence.
> > > > >
> > > > > In a sequence,
> > > > > there is 'successor' and 'predecessor'.
> > > > > Nothing else from the sequence is between
> > > > > successor and predecessor.
> > > > >
> > > > > However,
> > > > > between each two x,y ∈ {x:ℕ→{0,1}}:
> > > > > there are z ∈ {x:ℕ→{0,1}}: x < z < y
> > > > >
> > > > > -- mostly there are z between.
> > > > > Suppose
> > > > > ∀i<j: x(i) = y(i)
> > > > > x(j) = 0 ∧ y(j) = 1
> > > > > ∀i>j: x(i) = 1 ∧ y(i) = 0
> > > > >
> > > > > Still, {x:ℕ→{0,1}} is mostly not 1×1, not a sequence.
> > > > >
> > > > > > [...] while "Square Cantor space" is "square",
> > > > > > countable down and across,
> > > > >
> > > > > {x:ℕ→{0,1}} is countable across,
> > > > > uncountable down in any order.
> > > > > > This is where "diagonalization" as it were
> > > > > > is usually opposite the intent,
> > > > > > "anti-diagonalization",
> > > > > > when it really means to satisfy what's there
> > > > > > (that countability naturally leaves out).
> > > > > The anti-diagonal is not an entry.
> > > > > There is always an anti-diagonal,
> > > > > so there is no order in which
> > > > > the list does not miss some.
> > > > > > It's the first nonstandard function
> > > > > > "what is it?
> > > > > > It's drawing a line and saying
> > > > > > it was points in order,
> > > > > Yes.
> > > > > Trichotomous and transitive.
> > > > > > as by integers in order,
> > > > > No.
> > > > > Mostly not 1×1
> > > > >
> > > > > > must be infinite".
> > > > >
> > > > > must not be listable.
> > > > > > I came looking for mathematics and
> > > > > > when I found out that there were missing
> > > > > > formalisms for what I knew, I was like
> > > > > > "where's the entire world of naming
> > > > > > how this all fits together
> > > > > > these modes of continuity", "modes",
> > > > > > and it was like "make one yourself"
> > > > > > and thusly it would be unconscionable
> > > > > > to not say
> > > > > > "here let me give this an opinion and
> > > > > > an argument for itself",
> > > > > > "must be so",
> > > > > > and I was like
> > > > > > "really? I get to do all this myself?"
> > > > > > and it's like
> > > > > > "I guess it depends if they want to go along".
> > > > > Mathematicians are notoriously unwilling
> > > > > to just go along with the crowd.
> > > > >
> > > > > When they agree, it is because
> > > > > there is no alternative.
> > > > >
> > > > > There is certainly a place for
> > > > > thinking outside the box in mathematics.
> > > > >
> > > > > But being outside the box in itself
> > > > > does not confer honor or legitimacy.
> > > > > Remember what they are notorious for.
> > > > > That would be like expecting applause
> > > > > for getting up in the morning.
> > > > >
> > > > > Showing that _there is no alternative_
> > > > > to leaving the box, ah, now we're talking
> > > > > big legitimacy, big honor.
> > > > >
> > > > > If All The Mathematicians
> > > > > go along with the crowd
> > > > > by disagreeing with you,
> > > > > it is very likely that
> > > > > there is no alternative.
> > > > Wonderful, I'll write a fuller reply later, about the
> > > > "modern notions of a continuous domain", then
> > > > as specifically about the theory of functions, and,
> > > > a non-standard function, though with a particular character,
> > > > of its in-decomposability, and, the establishment from its
> > > > countable and integer domain, to a subset of [0,1] as
> > > > magnitudes in real values, the extent, density, completeness,
> > > > and measure, properties of a continuous domain in real-valued
> > > > variables, in real-valued quantites, for the purposes of mathematics.
> > > >
> > > > Mathematics is discovered not invented /
> > > > can't be ignored.
> > > Well, a more fuller reply, here is not so much. Everyone here has had to sit through
> > > my pointed writings which basically result an extended expression.
> > >
> > > This is to include that mathematics is a common language, there's that "mathematics
> > > is the common language". Symbols are the common language, often words..
> > >
> > > So, first, I axiomatize the entire notion of "0 to 1 in the real numbers is zero to infinity
> > > in the integers", "must be infinite", "includes infinity".
> > >
> > > Then, any unit line segment is that, one of those, each one a different copy of an integers,
> > > an infinitude of integers.
> > >
> > > By what end it is, either end, it's the integers up and down, or up and down.
> > >
> > > I.e., the "Equivalency Function", "natural/unit", of course is a "discrete/continuous" function,
> > > a function, one side actually discrete, if infinite, and the other side continuous "[0,1] real numbers".
> > >
> > > Then, "sweep", is the principle, for example axiomatized above.
> > >
> > > It's like I'm reading Kepler and Courant, and Courant says "of course we must implicitly axiomatize
> > > _away_ any conflicts existence _not_ contradicting, sweep doesn't exist".
> > >
> > > Then of course it sits perfectly satisfied in defining accumulation and limit and critical points,
> > > continuous and uniform continuous, Courant's.
> > >
> > > Then, where it does, for example is "though it uses otherwise the singleton copy of the integers,
> > > in any space its its own book-keeping for geometry in the same sense as an integer axis in an integer
> > > lattice, also is".
> > >
> > > So, not only is that standard, leaving that out because mathematics is satisfied having "a" copy
> > > of the integers if not "the copies", in terms of a value from integers having a given value.
> > >
> > > Reading Kepler is pretty great, he refers to at least a few cases of halving formulas and doubling
> > > formulas, for doubling spaces in continuity, and, in the terms, where for example Courant points
> > > directly at the Dirichlet function as "rationals 1, irrationals 0": "everywhere discontinuous, no measure".
> > >
> > > As a function thusly it's in a separate category of function, from the usual definition of function in
> > > axiomatic theory, or vice-versa, whether rationals and the field, usually, are more than "functions",
> > > their book-keeping, with that basically the line-reals "integers" are infinite, that inverse f 1/2: is "infinity/2".
> > >
> > > Then, there's that "infinite" is "any two-ended sequence, infinite both ways". Or, it's just as simply any greater,
> > > just that finite is sometimes great enough to have no greater.
> > >
> > > Basically the definition of limit shows that for f(n) = n/d, n -> d, d ->oo, there's
> > >
> > > extent
> > > density
> > > completeness
> > > measure
> > >
> > > I wrote these particularly and
> > > "for any x [0,1], f^-1 x is in dom(f)",
> > > also, for bit-strings, elements of [0,1] as necessarily "the representations
> > > in Square Cantor space", that
> > > "for any x [0,1], ~x e ran(f)".
> > >
> > > The extent is [0,1], accumulates, is complete somewhat trivially as any greater upper bound
> > > would at most be, "next", then a variety of sigma algebras for measure were written, as sets
> > > in set theory, with a particular attachment of "length assignment" geometrically, to reflect
> > > any association with "a measurable quantity" and "measure 1.0".
> > >
> > > Then for the rest apologetics is the slates, basically "sweep" in principle and for "intuition"
> > > and "axiomless natural deduction".
> > >
> > > I.e. "sweep the function" is defined briefly enough like so.
> > >
> > >
> > >
> > >
> > > If you ask most people in mathematics they could quite agree "yes that's the usual way to
> > > think about continuity from numbers, it makes a measure, and with that I can count with my hands".
> > >
> > > Also, specifically, "we were all told in school that mathematics does it the opposite way,
> > > that numbers are always divisible and never at once divided, except zero not but annihilating,
> > > which is fair that way, everybody knows that and we really identify it with the differential
> > > and it's called constant infinitesimals or iota-values, standard infinitesimals".
> > >
> > > That's just saying there's nothing but denying that I took only the _absolute_ _easiest_ ball,
> > > and ran with it, "pre-calculus" and "differentials and iota", this is nothing but paste stock in trade,
> > > this way it was at least easy to carry, wading the field.
> > >
> > > The rest of the field then I just put down on my slates, "justifications".
> > >
> > > Then applications are coming down from deductive properties of the space, and
> > > thusly for example statistical properties, what seem false and even fail convergence
> > > tests, thus implying that as they are accurate, under the manipulation of their quantities
> > > mathematically, set down that other identities in constants follow.
> > >
> > > I.e. I put down some nonstandard applications, theoretically, because of course usual sorts
> > > derivations like having gone all the way to school. (Which I figure you can do yourself.)
> > >
> > > Having the easiest carry, then also let plenty room to setup the side, but I really made it
> > > easy on myself, so far as always agreeing with myself, I'm most technically flawless.
> > >
> > >
> > > As far as I know I always feel to have gotten my "last word", technically impervious forever.
> > >
> > > That way you can blame me for anything left unsaid.
> > >
> > >
> > > Courant's "sweep" that doesn't exist: is the same one here that does.
> > >
> > > It does for him, too: where it does.
> > >
> > >
> > > Again, just to re-iterate, this is only the most usual and obvious conception of "points in a line",
> > > "an infinitude from beginning to end", "that just so happens to equal 1.0", and here,
> > > "that just so happens to only always equal 1.0".
> > >
> > >
> > > So, I built all these apologetics so it's not contradictory with set theory at all.
> > >
> > > It's in a bigger one, ..., a regular set theory (if outside on the extra-ordinary itself).
> > >
> > >
> > > So, it's mostly not just "not much to see here" but also "what you already know",
> > > this "line-continuity" is _exactly_ "this is the perfect line-continuity that we _don't_
> > > talk about", basically for giving it its own piece on the board but never letting it cross.
> > > There it's "these perfect delta-epsilonics, divide these numbers forever and add them up".
> > >
> > > This way, it just "fixes set theory" for continuity, like I said, simple.
> >
> > Really now as we've broached the topic of "continuous domains",
> > think of it as a concept that math has been putting off -
> > putting the concept of "continuous functions" and having
> > this "complete ordered field" continuous domain, that theorems
> > and formulas about continuous functions, mostly are as analytically
> > formulas, and theorems, as continuous domains,
> > of whatever sort, here mostly "for example the complete ordered field".
> >
> > Thusly, the _language_ of sci.math changed, "continuous domains",
> > is now a word, and it's the same throughout mathematics.
> >
> > Of course, the language didn't "change" so much, as that's what it "means",
> > "continuous domains".
> Yeah I really, ..., admire Courant, or his refresher on Differential & Integral Calculus,
> it's called Differential & Integral Calculus Volume I, 2'nd ed. 1934, 15th reprinting 1959.

On Thursday, October 19, 2023 at 8:21:35 PM UTC-7, Ross Finlayson wrote:
> On Monday, October 16, 2023 at 10:24:05 PM UTC-7, Ross Finlayson wrote:
> > On Monday, October 16, 2023 at 12:04:19 PM UTC-7, Ross Finlayson wrote:
> > > On Sunday, October 15, 2023 at 3:35:07 PM UTC-7, Jim Burns wrote:
> > > > On 10/13/2023 10:00 PM, Ross Finlayson wrote:
> > > > > On Friday, October 13, 2023
> > > > > at 11:44:33 AM UTC-7, Jim Burns wrote:
> > > > > [...]
> > > > > It is commendable, that
> > > > > continuous domains, like time, make for
> > > > > continuous functions
> > > > > the properties of continuous functions.
> > > > It's a good thing to remember.
> > > >
> > > > I think that that is why
> > > > the real numbers are so beloved by physicists,
> > > > even though they use almost none of them.
> > > > It is the continuous functions which are
> > > > their true love.
> > > > Real numbers are a means to that end.
> > > > > Identity's a continuous function on
> > > > > a continuous domain.
> > > > Yes,
> > > > identity is not one of the functions
> > > > which we intend to rule out for jumping.
> > > >
> > > > Identity is exceptionally well-behaved.
> > > >
> > > > There are other functions less well-behaved,
> > > > which are only piecewise well-behaved.
> > > > A function may be continuous on
> > > > each of the internally-connected components of
> > > > of its domain, but it may jump
> > > > from one disconnected component to another.
> > > >
> > > > Consider
> > > > H(x) = [0≤x]
> > > > H(x) = 1 if 0≤x
> > > > H(x) = 0 if x<0
> > > >
> > > > H(x) is continuous on
> > > > the disconnected domain ℝ\{0}
> > > > H(x) is discontinuous at 0
> > > > on the connected domain ℝ
> > > >
> > > > We can't rule out H: ℝ\{0} -> {0,1}
> > > > by requiring that
> > > > there is no point at which it jumps.
> > > > There isn't such a point.
> > > > 0 is not in its domain.
> > > >
> > > > We can rule out the domain ℝ\{0} of H(x)
> > > > by requiring that
> > > > there are no two disjoint open sets
> > > > into which its domain can be split.
> > > > There are.
> > > >
> > > > No jumps in a connected domain
> > > > gives us no jumps, period,
> > > > our Holy Grail.
> > > > > The Cantor space, is the set of,
> > > > > all the sequences, infinite, of the 0's and 1's.
> > > > {x:ℕ→{0,1}}
> > > > > The square Cantor space:
> > > > > is those in order, their natural order,
> > > > > as that the series are expansions.
> > > > ∀x,y ∈ {x:ℕ→{0,1}}:
> > > > x < y :⇔
> > > > ∃j ∈ ℕ: x(j) < y(j) ∧ ∀i<j: x(i) = y(i)
> > > > > The other use of powerset representation of
> > > > > the Cantor space is to represent each of
> > > > > a countable domain's, members, presence, in
> > > > > any of the powersets or "sets of all subsets",
> > > > > the set.
> > > > >
> > > > > So, Cantor space is uncountable,
> > > > > but square Cantor space is "countable",
> > > > The square Cantor space is uncountable.
> > > > Order doesn't change cardinality.
> > > > > after a geometrization and length assignment
> > > > > the space, line-continuity's,
> > > > > "including the diagonal".
> > > > {x:ℕ→{0,1}} has a natural order.
> > > > That order is not the 1×1 order of a sequence.
> > > >
> > > > In a sequence,
> > > > there is 'successor' and 'predecessor'.
> > > > Nothing else from the sequence is between
> > > > successor and predecessor.
> > > >
> > > > However,
> > > > between each two x,y ∈ {x:ℕ→{0,1}}:
> > > > there are z ∈ {x:ℕ→{0,1}}: x < z < y
> > > >
> > > > -- mostly there are z between.
> > > > Suppose
> > > > ∀i<j: x(i) = y(i)
> > > > x(j) = 0 ∧ y(j) = 1
> > > > ∀i>j: x(i) = 1 ∧ y(i) = 0
> > > >
> > > > Still, {x:ℕ→{0,1}} is mostly not 1×1, not a sequence.
> > > >
> > > > > [...] while "Square Cantor space" is "square",
> > > > > countable down and across,
> > > >
> > > > {x:ℕ→{0,1}} is countable across,
> > > > uncountable down in any order.
> > > > > This is where "diagonalization" as it were
> > > > > is usually opposite the intent,
> > > > > "anti-diagonalization",
> > > > > when it really means to satisfy what's there
> > > > > (that countability naturally leaves out).
> > > > The anti-diagonal is not an entry.
> > > > There is always an anti-diagonal,
> > > > so there is no order in which
> > > > the list does not miss some.
> > > > > It's the first nonstandard function
> > > > > "what is it?
> > > > > It's drawing a line and saying
> > > > > it was points in order,
> > > > Yes.
> > > > Trichotomous and transitive.
> > > > > as by integers in order,
> > > > No.
> > > > Mostly not 1×1
> > > >
> > > > > must be infinite".
> > > >
> > > > must not be listable.
> > > > > I came looking for mathematics and
> > > > > when I found out that there were missing
> > > > > formalisms for what I knew, I was like
> > > > > "where's the entire world of naming
> > > > > how this all fits together
> > > > > these modes of continuity", "modes",
> > > > > and it was like "make one yourself"
> > > > > and thusly it would be unconscionable
> > > > > to not say
> > > > > "here let me give this an opinion and
> > > > > an argument for itself",
> > > > > "must be so",
> > > > > and I was like
> > > > > "really? I get to do all this myself?"
> > > > > and it's like
> > > > > "I guess it depends if they want to go along".
> > > > Mathematicians are notoriously unwilling
> > > > to just go along with the crowd.
> > > >
> > > > When they agree, it is because
> > > > there is no alternative.
> > > >
> > > > There is certainly a place for
> > > > thinking outside the box in mathematics.
> > > >
> > > > But being outside the box in itself
> > > > does not confer honor or legitimacy.
> > > > Remember what they are notorious for.
> > > > That would be like expecting applause
> > > > for getting up in the morning.
> > > >
> > > > Showing that _there is no alternative_
> > > > to leaving the box, ah, now we're talking
> > > > big legitimacy, big honor.
> > > >
> > > > If All The Mathematicians
> > > > go along with the crowd
> > > > by disagreeing with you,
> > > > it is very likely that
> > > > there is no alternative.
> > > Wonderful, I'll write a fuller reply later, about the
> > > "modern notions of a continuous domain", then
> > > as specifically about the theory of functions, and,
> > > a non-standard function, though with a particular character,
> > > of its in-decomposability, and, the establishment from its
> > > countable and integer domain, to a subset of [0,1] as
> > > magnitudes in real values, the extent, density, completeness,
> > > and measure, properties of a continuous domain in real-valued
> > > variables, in real-valued quantites, for the purposes of mathematics.
> > >
> > > Mathematics is discovered not invented /
> > > can't be ignored.
> > Well, a more fuller reply, here is not so much. Everyone here has had to sit through
> > my pointed writings which basically result an extended expression.
> >
> > This is to include that mathematics is a common language, there's that "mathematics
> > is the common language". Symbols are the common language, often words.
> >
> > So, first, I axiomatize the entire notion of "0 to 1 in the real numbers is zero to infinity
> > in the integers", "must be infinite", "includes infinity".
> >
> > Then, any unit line segment is that, one of those, each one a different copy of an integers,
> > an infinitude of integers.
> >
> > By what end it is, either end, it's the integers up and down, or up and down.
> >
> > I.e., the "Equivalency Function", "natural/unit", of course is a "discrete/continuous" function,
> > a function, one side actually discrete, if infinite, and the other side continuous "[0,1] real numbers".
> >
> > Then, "sweep", is the principle, for example axiomatized above.
> >
> > It's like I'm reading Kepler and Courant, and Courant says "of course we must implicitly axiomatize
> > _away_ any conflicts existence _not_ contradicting, sweep doesn't exist".
> >
> > Then of course it sits perfectly satisfied in defining accumulation and limit and critical points,
> > continuous and uniform continuous, Courant's.
> >
> > Then, where it does, for example is "though it uses otherwise the singleton copy of the integers,
> > in any space its its own book-keeping for geometry in the same sense as an integer axis in an integer
> > lattice, also is".
> >
> > So, not only is that standard, leaving that out because mathematics is satisfied having "a" copy
> > of the integers if not "the copies", in terms of a value from integers having a given value.
> >
> > Reading Kepler is pretty great, he refers to at least a few cases of halving formulas and doubling
> > formulas, for doubling spaces in continuity, and, in the terms, where for example Courant points
> > directly at the Dirichlet function as "rationals 1, irrationals 0": "everywhere discontinuous, no measure".
> >
> > As a function thusly it's in a separate category of function, from the usual definition of function in
> > axiomatic theory, or vice-versa, whether rationals and the field, usually, are more than "functions",
> > their book-keeping, with that basically the line-reals "integers" are infinite, that inverse f 1/2: is "infinity/2".
> >
> > Then, there's that "infinite" is "any two-ended sequence, infinite both ways". Or, it's just as simply any greater,
> > just that finite is sometimes great enough to have no greater.
> >
> > Basically the definition of limit shows that for f(n) = n/d, n -> d, d ->oo, there's
> >
> > extent
> > density
> > completeness
> > measure
> >
> > I wrote these particularly and
> > "for any x [0,1], f^-1 x is in dom(f)",
> > also, for bit-strings, elements of [0,1] as necessarily "the representations
> > in Square Cantor space", that
> > "for any x [0,1], ~x e ran(f)".
> >
> > The extent is [0,1], accumulates, is complete somewhat trivially as any greater upper bound
> > would at most be, "next", then a variety of sigma algebras for measure were written, as sets
> > in set theory, with a particular attachment of "length assignment" geometrically, to reflect
> > any association with "a measurable quantity" and "measure 1.0".
> >
> > Then for the rest apologetics is the slates, basically "sweep" in principle and for "intuition"
> > and "axiomless natural deduction".
> >
> > I.e. "sweep the function" is defined briefly enough like so.
> >
> >
> >
> >
> > If you ask most people in mathematics they could quite agree "yes that's the usual way to
> > think about continuity from numbers, it makes a measure, and with that I can count with my hands".
> >
> > Also, specifically, "we were all told in school that mathematics does it the opposite way,
> > that numbers are always divisible and never at once divided, except zero not but annihilating,
> > which is fair that way, everybody knows that and we really identify it with the differential
> > and it's called constant infinitesimals or iota-values, standard infinitesimals".
> >
> > That's just saying there's nothing but denying that I took only the _absolute_ _easiest_ ball,
> > and ran with it, "pre-calculus" and "differentials and iota", this is nothing but paste stock in trade,
> > this way it was at least easy to carry, wading the field.
> >
> > The rest of the field then I just put down on my slates, "justifications".
> >
> > Then applications are coming down from deductive properties of the space, and
> > thusly for example statistical properties, what seem false and even fail convergence
> > tests, thus implying that as they are accurate, under the manipulation of their quantities
> > mathematically, set down that other identities in constants follow.
> >
> > I.e. I put down some nonstandard applications, theoretically, because of course usual sorts
> > derivations like having gone all the way to school. (Which I figure you can do yourself.)
> >
> > Having the easiest carry, then also let plenty room to setup the side, but I really made it
> > easy on myself, so far as always agreeing with myself, I'm most technically flawless.
> >
> >
> > As far as I know I always feel to have gotten my "last word", technically impervious forever.
> >
> > That way you can blame me for anything left unsaid.
> >
> >
> > Courant's "sweep" that doesn't exist: is the same one here that does.
> >
> > It does for him, too: where it does.
> >
> >
> > Again, just to re-iterate, this is only the most usual and obvious conception of "points in a line",
> > "an infinitude from beginning to end", "that just so happens to equal 1..0", and here,
> > "that just so happens to only always equal 1.0".
> >
> >
> > So, I built all these apologetics so it's not contradictory with set theory at all.
> >
> > It's in a bigger one, ..., a regular set theory (if outside on the extra-ordinary itself).
> >
> >
> > So, it's mostly not just "not much to see here" but also "what you already know",
> > this "line-continuity" is _exactly_ "this is the perfect line-continuity that we _don't_
> > talk about", basically for giving it its own piece on the board but never letting it cross.
> > There it's "these perfect delta-epsilonics, divide these numbers forever and add them up".
> >
> > This way, it just "fixes set theory" for continuity, like I said, simple.
>
> Really now as we've broached the topic of "continuous domains",
> think of it as a concept that math has been putting off -
> putting the concept of "continuous functions" and having
> this "complete ordered field" continuous domain, that theorems
> and formulas about continuous functions, mostly are as analytically
> formulas, and theorems, as continuous domains,
> of whatever sort, here mostly "for example the complete ordered field".
>
> Thusly, the _language_ of sci.math changed, "continuous domains",
> is now a word, and it's the same throughout mathematics.
>
> Of course, the language didn't "change" so much, as that's what it "means",
> "continuous domains".

bassam karzeddin schrieb am Samstag, 8. August 2020 um 10:27:08 UTC+2:
> I saw very clearly and since long ago the discontinuity of the
so-called real number in modern mathematics (as simple as it is)
>
> 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

Yes, this is true for all visible real numbers, i.e., real numbers which
can be defined by digits (like integers and ending rational numbers) or
by limits (like periodic rational numbers or irrational numbers).
Between them there is either a sea of dark numbers from which visible
numbers can be created, or nothing from which visible numbers can be
created.

Regards, WM