Wed 21 Sep 2022 08:03:47 AM EDT
Natural Numbers And A New Continuum Hypothesis
From the meandering on https://groups.google.com/g/sci.math/c/02xv2Bv1ZxE "Could dS be the polysign differential? Integral( dS over S ) = 0":
where the seed got planted.

An infinite value such as 333...34 has been posited. To make these values cohere an aleph mark | is needed so that 10x and x can commense. This aleph is done at the head of the value:
x = |333...32
10x = 3|333...320
and now any confusion over the relative position of this (infinite) head digit is defused. Clearly 10x is larger than x thanks only to this augmentation. Granting these values some fundamental existence we can thus claim a continuum hypothesis. In effect, let's replace the aleph mark with a decimal point:
x = 0.333...32
10x = 3.333...320
This is satisfactory to some people's sensibility of the continuous value.
In effect working at aleph one is natural. The spirit of the natural number as going on and on is present within the value. This is merely an act of interpretation, and I prefer the finite forms myself, but as has been described elsewhere we are on a branch of mathematics here which is treating the ellipsis quite seriously. In physical terms the act of counting a unital is nonexistent, along with the successor, and here we see that roughly the same is true. Instead we've arrived at a macro scale at aleph one where physical work takes place. As we select a unit, whether in geometry on paper with some dividers, or via the King's thumb, or whatever glass ensconced reference you'd like to copy, or some elemental wavelength of light whose interferometric waves you can count, well, this latter possibly not; these are interpretations of unity. This arbitrary nature of unity at the macro level; at the physical level; is obvious to most who have spent time here.

Strangely enough, it is as if we've left the natural value in the dust, so to speak, as we've built out of it such gargantuan values. That this route forgoes the rational value and instead relies upon a sense of infinity that many minds refuse to leave behind: here lays the tantalizing crux: this model is in the minds of men already. There is another way, but this way does seem to hold upon granting the inductive digital infinity such as
333...34
which thence requires a mark to negotiate computations which is very nearly consistent with the decimal point of old.
As I entertain this route, I should make clear that it is taken via the incessant ramblings and ranting on usenet of infinite concerns. Another route exists down lower by denying the rational value and its lack of closure; as if two integers could yield a continuum through the non-fundamental operator known as division. Did products behave this way? Well then, there is another attack on the rational value. Structural analysis exposes that secondary forms of unity serve far more simply as augmentations to the natural value. So a non-infinite route that is not so far away from the natural value exists as well.

onsdag 21 september 2022 kl. 14:28:11 UTC+2 skrev timba...@gmail.com:
> Wed 21 Sep 2022 08:03:47 AM EDT
> Natural Numbers And A New Continuum Hypothesis
> From the meandering on https://groups.google.com/g/sci.math/c/02xv2Bv1ZxE "Could dS be the polysign differential? Integral( dS over S ) = 0":
> where the seed got planted.
>
> An infinite value such as 333...34 has been posited. To make these values cohere an aleph mark | is needed so that 10x and x can commense. This aleph is done at the head of the value:
> x = |333...32
> 10x = 3|333...320
> and now any confusion over the relative position of this (infinite) head digit is defused. Clearly 10x is larger than x thanks only to this augmentation. Granting these values some fundamental existence we can thus claim a continuum hypothesis. In effect, let's replace the aleph mark with a decimal point:
> x = 0.333...32
> 10x = 3.333...320
> This is satisfactory to some people's sensibility of the continuous value..
> In effect working at aleph one is natural. The spirit of the natural number as going on and on is present within the value. This is merely an act of interpretation, and I prefer the finite forms myself, but as has been described elsewhere we are on a branch of mathematics here which is treating the ellipsis quite seriously. In physical terms the act of counting a unital is nonexistent, along with the successor, and here we see that roughly the same is true. Instead we've arrived at a macro scale at aleph one where physical work takes place. As we select a unit, whether in geometry on paper with some dividers, or via the King's thumb, or whatever glass ensconced reference you'd like to copy, or some elemental wavelength of light whose interferometric waves you can count, well, this latter possibly not; these are interpretations of unity. This arbitrary nature of unity at the macro level; at the physical level; is obvious to most who have spent time here.
>
> Strangely enough, it is as if we've left the natural value in the dust, so to speak, as we've built out of it such gargantuan values. That this route forgoes the rational value and instead relies upon a sense of infinity that many minds refuse to leave behind: here lays the tantalizing crux: this model is in the minds of men already. There is another way, but this way does seem to hold upon granting the inductive digital infinity such as
> 333...34
> which thence requires a mark to negotiate computations which is very nearly consistent with the decimal point of old.
>
> As I entertain this route, I should make clear that it is taken via the incessant ramblings and ranting on usenet of infinite concerns. Another route exists down lower by denying the rational value and its lack of closure; as if two integers could yield a continuum through the non-fundamental operator known as division. Did products behave this way? Well then, there is another attack on the rational value. Structural analysis exposes that secondary forms of unity serve far more simply as augmentations to the natural value. So a non-infinite route that is not so far away from the natural value exists as well.
your "polysign" does not work.

Hey. Would You like to talk a little bit ?

Call >>>>>>>>>>>>> 3 7 2 [[[[[[[country]]]]]]]]]]]] and then 56330687

timba...@gmail.com kirjutas Kolmapäev, 21. september 2022 kl 15:28:11 UTC+3:
> Wed 21 Sep 2022 08:03:47 AM EDT
> Natural Numbers And A New Continuum Hypothesis
> From the meandering on https://groups.google.com/g/sci.math/c/02xv2Bv1ZxE "Could dS be the polysign differential? Integral( dS over S ) = 0":
> where the seed got planted.
>
> An infinite value such as 333...34 has been posited. To make these values cohere an aleph mark | is needed so that 10x and x can commense. This aleph is done at the head of the value:
> x = |333...32
> 10x = 3|333...320
> and now any confusion over the relative position of this (infinite) head digit is defused. Clearly 10x is larger than x thanks only to this augmentation. Granting these values some fundamental existence we can thus claim a continuum hypothesis. In effect, let's replace the aleph mark with a decimal point:
> x = 0.333...32
> 10x = 3.333...320
> This is satisfactory to some people's sensibility of the continuous value..
> In effect working at aleph one is natural. The spirit of the natural number as going on and on is present within the value. This is merely an act of interpretation, and I prefer the finite forms myself, but as has been described elsewhere we are on a branch of mathematics here which is treating the ellipsis quite seriously. In physical terms the act of counting a unital is nonexistent, along with the successor, and here we see that roughly the same is true. Instead we've arrived at a macro scale at aleph one where physical work takes place. As we select a unit, whether in geometry on paper with some dividers, or via the King's thumb, or whatever glass ensconced reference you'd like to copy, or some elemental wavelength of light whose interferometric waves you can count, well, this latter possibly not; these are interpretations of unity. This arbitrary nature of unity at the macro level; at the physical level; is obvious to most who have spent time here.
>
> Strangely enough, it is as if we've left the natural value in the dust, so to speak, as we've built out of it such gargantuan values. That this route forgoes the rational value and instead relies upon a sense of infinity that many minds refuse to leave behind: here lays the tantalizing crux: this model is in the minds of men already. There is another way, but this way does seem to hold upon granting the inductive digital infinity such as
> 333...34
> which thence requires a mark to negotiate computations which is very nearly consistent with the decimal point of old.
>
> As I entertain this route, I should make clear that it is taken via the incessant ramblings and ranting on usenet of infinite concerns. Another route exists down lower by denying the rational value and its lack of closure; as if two integers could yield a continuum through the non-fundamental operator known as division. Did products behave this way? Well then, there is another attack on the rational value. Structural analysis exposes that secondary forms of unity serve far more simply as augmentations to the natural value. So a non-infinite route that is not so far away from the natural value exists as well.

On Wednesday, September 21, 2022 at 6:00:35 AM UTC-7, zelos...@gmail.com wrote:
> onsdag 21 september 2022 kl. 14:28:11 UTC+2 skrev timba...@gmail.com:
> > Wed 21 Sep 2022 08:03:47 AM EDT
> > Natural Numbers And A New Continuum Hypothesis
> > From the meandering on https://groups.google.com/g/sci.math/c/02xv2Bv1ZxE "Could dS be the polysign differential? Integral( dS over S ) = 0":
> > where the seed got planted.
> >
> > An infinite value such as 333...34 has been posited. To make these values cohere an aleph mark | is needed so that 10x and x can commense. This aleph is done at the head of the value:
> > x = |333...32
> > 10x = 3|333...320
> > and now any confusion over the relative position of this (infinite) head digit is defused. Clearly 10x is larger than x thanks only to this augmentation. Granting these values some fundamental existence we can thus claim a continuum hypothesis. In effect, let's replace the aleph mark with a decimal point:
> > x = 0.333...32
> > 10x = 3.333...320
> > This is satisfactory to some people's sensibility of the continuous value.
> > In effect working at aleph one is natural. The spirit of the natural number as going on and on is present within the value. This is merely an act of interpretation, and I prefer the finite forms myself, but as has been described elsewhere we are on a branch of mathematics here which is treating the ellipsis quite seriously. In physical terms the act of counting a unital is nonexistent, along with the successor, and here we see that roughly the same is true. Instead we've arrived at a macro scale at aleph one where physical work takes place. As we select a unit, whether in geometry on paper with some dividers, or via the King's thumb, or whatever glass ensconced reference you'd like to copy, or some elemental wavelength of light whose interferometric waves you can count, well, this latter possibly not; these are interpretations of unity. This arbitrary nature of unity at the macro level; at the physical level; is obvious to most who have spent time here.
> >
> > Strangely enough, it is as if we've left the natural value in the dust, so to speak, as we've built out of it such gargantuan values. That this route forgoes the rational value and instead relies upon a sense of infinity that many minds refuse to leave behind: here lays the tantalizing crux: this model is in the minds of men already. There is another way, but this way does seem to hold upon granting the inductive digital infinity such as
> > 333...34
> > which thence requires a mark to negotiate computations which is very nearly consistent with the decimal point of old.
> >
> > As I entertain this route, I should make clear that it is taken via the incessant ramblings and ranting on usenet of infinite concerns. Another route exists down lower by denying the rational value and its lack of closure; as if two integers could yield a continuum through the non-fundamental operator known as division. Did products behave this way? Well then, there is another attack on the rational value. Structural analysis exposes that secondary forms of unity serve far more simply as augmentations to the natural value. So a non-infinite route that is not so far away from the natural value exists as well.
> your "polysign" does not work.

It's like a Brouwerian continuum you should look it up.

On Wednesday, September 21, 2022 at 11:46:06 AM UTC-4, Ross A. Finlayson wrote:
> On Wednesday, September 21, 2022 at 6:00:35 AM UTC-7, zelos...@gmail.com wrote:
> > onsdag 21 september 2022 kl. 14:28:11 UTC+2 skrev timba...@gmail.com:
> > > Wed 21 Sep 2022 08:03:47 AM EDT
> > > Natural Numbers And A New Continuum Hypothesis
> > > From the meandering on https://groups.google.com/g/sci.math/c/02xv2Bv1ZxE "Could dS be the polysign differential? Integral( dS over S ) = 0":
> > > where the seed got planted.
> > >
> > > An infinite value such as 333...34 has been posited. To make these values cohere an aleph mark | is needed so that 10x and x can commense. This aleph is done at the head of the value:
> > > x = |333...32
> > > 10x = 3|333...320
> > > and now any confusion over the relative position of this (infinite) head digit is defused. Clearly 10x is larger than x thanks only to this augmentation. Granting these values some fundamental existence we can thus claim a continuum hypothesis. In effect, let's replace the aleph mark with a decimal point:
> > > x = 0.333...32
> > > 10x = 3.333...320
> > > This is satisfactory to some people's sensibility of the continuous value.
> > > In effect working at aleph one is natural. The spirit of the natural number as going on and on is present within the value. This is merely an act of interpretation, and I prefer the finite forms myself, but as has been described elsewhere we are on a branch of mathematics here which is treating the ellipsis quite seriously. In physical terms the act of counting a unital is nonexistent, along with the successor, and here we see that roughly the same is true. Instead we've arrived at a macro scale at aleph one where physical work takes place. As we select a unit, whether in geometry on paper with some dividers, or via the King's thumb, or whatever glass ensconced reference you'd like to copy, or some elemental wavelength of light whose interferometric waves you can count, well, this latter possibly not; these are interpretations of unity. This arbitrary nature of unity at the macro level; at the physical level; is obvious to most who have spent time here.
> > >
> > > Strangely enough, it is as if we've left the natural value in the dust, so to speak, as we've built out of it such gargantuan values. That this route forgoes the rational value and instead relies upon a sense of infinity that many minds refuse to leave behind: here lays the tantalizing crux: this model is in the minds of men already. There is another way, but this way does seem to hold upon granting the inductive digital infinity such as
> > > 333...34
> > > which thence requires a mark to negotiate computations which is very nearly consistent with the decimal point of old.
> > >
> > > As I entertain this route, I should make clear that it is taken via the incessant ramblings and ranting on usenet of infinite concerns. Another route exists down lower by denying the rational value and its lack of closure; as if two integers could yield a continuum through the non-fundamental operator known as division. Did products behave this way? Well then, there is another attack on the rational value. Structural analysis exposes that secondary forms of unity serve far more simply as augmentations to the natural value. So a non-infinite route that is not so far away from the natural value exists as well.
> > your "polysign" does not work.
> It's like a Brouwerian continuum you should look it up.

https://en.wikipedia.org/wiki/L._E._J._Brouwer

Thanks, Ross. You are well read.
I didn't really find anything on his continuum concept though. Maybe I am misusing the term 'continuum hypotheses' since it is already in use like the 'axiom of choice' ; sounds so broad but the term takes a narrowness via its past contributors. In that axioms are choices then constructive freedom is available to all who are willing to tamper with them. Consistency and correspondence are really quite important as metrics down at such a fundamental level. Consequences too. Nobody seems to want to touch the ellipsis as ambiguous. There are problems with these untouchables whose meanings are implied elsewhere so that a form of double talk is taking place. Perhaps this is cause to disrespect the natural number and head back to the drawing board to recover the continuum. That modulo principles are at work beneath the modulo neutered natural value is a cause to me. Of course the same is true of real analysis as well, so it's not as if they are any better. Indeed they start by supersetting the natural value. So far I've stayed with that course and found that the real can be had with the augmentation of the decimal point reassigning unity; a secondary form of unity. This idea that numbers do have structure within them has not really taken the digit as a serious element yet. I've still not decompiled down to that level. Perhaps there is something there that will marry into sign as well and yield some benefit.

On Thursday, September 22, 2022 at 8:28:20 AM UTC-7, timba...@gmail.com wrote:
> On Wednesday, September 21, 2022 at 11:46:06 AM UTC-4, Ross A. Finlayson wrote:
> > On Wednesday, September 21, 2022 at 6:00:35 AM UTC-7, zelos...@gmail.com wrote:
> > > onsdag 21 september 2022 kl. 14:28:11 UTC+2 skrev timba...@gmail.com:
> > > > Wed 21 Sep 2022 08:03:47 AM EDT
> > > > Natural Numbers And A New Continuum Hypothesis
> > > > From the meandering on https://groups.google.com/g/sci.math/c/02xv2Bv1ZxE "Could dS be the polysign differential? Integral( dS over S ) = 0":
> > > > where the seed got planted.
> > > >
> > > > An infinite value such as 333...34 has been posited. To make these values cohere an aleph mark | is needed so that 10x and x can commense. This aleph is done at the head of the value:
> > > > x = |333...32
> > > > 10x = 3|333...320
> > > > and now any confusion over the relative position of this (infinite) head digit is defused. Clearly 10x is larger than x thanks only to this augmentation. Granting these values some fundamental existence we can thus claim a continuum hypothesis. In effect, let's replace the aleph mark with a decimal point:
> > > > x = 0.333...32
> > > > 10x = 3.333...320
> > > > This is satisfactory to some people's sensibility of the continuous value.
> > > > In effect working at aleph one is natural. The spirit of the natural number as going on and on is present within the value. This is merely an act of interpretation, and I prefer the finite forms myself, but as has been described elsewhere we are on a branch of mathematics here which is treating the ellipsis quite seriously. In physical terms the act of counting a unital is nonexistent, along with the successor, and here we see that roughly the same is true. Instead we've arrived at a macro scale at aleph one where physical work takes place. As we select a unit, whether in geometry on paper with some dividers, or via the King's thumb, or whatever glass ensconced reference you'd like to copy, or some elemental wavelength of light whose interferometric waves you can count, well, this latter possibly not; these are interpretations of unity. This arbitrary nature of unity at the macro level; at the physical level; is obvious to most who have spent time here.
> > > >
> > > > Strangely enough, it is as if we've left the natural value in the dust, so to speak, as we've built out of it such gargantuan values. That this route forgoes the rational value and instead relies upon a sense of infinity that many minds refuse to leave behind: here lays the tantalizing crux: this model is in the minds of men already. There is another way, but this way does seem to hold upon granting the inductive digital infinity such as
> > > > 333...34
> > > > which thence requires a mark to negotiate computations which is very nearly consistent with the decimal point of old.
> > > >
> > > > As I entertain this route, I should make clear that it is taken via the incessant ramblings and ranting on usenet of infinite concerns. Another route exists down lower by denying the rational value and its lack of closure; as if two integers could yield a continuum through the non-fundamental operator known as division. Did products behave this way? Well then, there is another attack on the rational value. Structural analysis exposes that secondary forms of unity serve far more simply as augmentations to the natural value. So a non-infinite route that is not so far away from the natural value exists as well.
> > > your "polysign" does not work.
> > It's like a Brouwerian continuum you should look it up.
> https://en.wikipedia.org/wiki/L._E._J._Brouwer
>
> Thanks, Ross. You are well read.
> I didn't really find anything on his continuum concept though. Maybe I am misusing the term 'continuum hypotheses' since it is already in use like the 'axiom of choice' ; sounds so broad but the term takes a narrowness via its past contributors. In that axioms are choices then constructive freedom is available to all who are willing to tamper with them. Consistency and correspondence are really quite important as metrics down at such a fundamental level. Consequences too. Nobody seems to want to touch the ellipsis as ambiguous. There are problems with these untouchables whose meanings are implied elsewhere so that a form of double talk is taking place. Perhaps this is cause to disrespect the natural number and head back to the drawing board to recover the continuum. That modulo principles are at work beneath the modulo neutered natural value is a cause to me. Of course the same is true of real analysis as well, so it's not as if they are any better. Indeed they start by supersetting the natural value. So far I've stayed with that course and found that the real can be had with the augmentation of the decimal point reassigning unity; a secondary form of unity. This idea that numbers do have structure within them has not really taken the digit as a serious element yet. I've still not decompiled down to that level. Perhaps there is something there that will marry into sign as well and yield some benefit.

About continuum analysis there's like Veronese and Stolz and Peano and Poincare and Brouwer
later after Cantor with Riemann and Lebesque and Stieltjes also these days, there is even
continuum analysis from the ancients, Newton and Leibniz have their great thing, there's
Bell and Robinson and Conway, Nelson and for example Woodin, Dodgson, Democritus,
Eudoxus/Dedekind/Cauchy, there's mensuration of area and measurement of time,
there's what's called a continuous manifold in a field theory that's a gauge theory,
there's the quasi-invariant in emasure theory, there's line reals, field reals, signal reals, ....

There's Vitali and for measure theory, while, also singularity theory becomes a pivotal
feature, convergence and tangency while preservation of angle, the continuum of [0,1]
the line segment with line drawing is about Zeno and constant motion and Democritus or atomism,
while the spiral-space-filling curve founding geometry is not just the Archimedean but the ancient.

So, I wrote a sigma algebra for real character for line reals, it's really quite simple.
I wrote a variety of sigma algebras and also noted that in this modern model of
continuum infinitesimal analysis, I wrote field operations for [-1, 1], and then
that as a special function this generalized-distribution/limit-of-functions/model-of-
real-function also defines a probability distribution of integers, and surprisingly
not a unique one, and also is special about the exponential or exponential operators,
being its own antiderivative.

So, if I wasn't well-read then getting into all these features of numbers, then I
wouldn't know operator theory for example, or probability theory.
Here that "the natural/unit equivalency function is integrable and equals one",
it's a very cool, special function, and in at least one theory, the prototype and
the simplest model for a continuous domain. It's about the best and one of
the most special functions in mathematics.

Then, I think a lot of what you got going there is arithmetic coding, which basically
reflects two forms at a distance maintain their proportion. Then, properties
of continuity balance "parallax" and "perspective".

Du Bois-Reymond, Schmieden and Laugwitz, Cohen and so on and for
along the lines of "anything about infinity is also about continuity",
DesCartes and Duns Scotus and Spinoza and friends, a "continuum"
and its "individua" follows that there are as above here eight or ten
named approaches to "what is a continuum", that yours is akin to Brouwer's,
though, his branch and grow, yours branch and grow with a straight line in the middle.

Most of the successful analytical approaches take an area and divide it,
here also there are synthetic approaches (to analysis).

So, that condenses a lot.

On Thursday, September 22, 2022 at 9:28:23 AM UTC-7, Ross A. Finlayson wrote:
> On Thursday, September 22, 2022 at 8:28:20 AM UTC-7, timba...@gmail.com wrote:
> > On Wednesday, September 21, 2022 at 11:46:06 AM UTC-4, Ross A. Finlayson wrote:
> > > On Wednesday, September 21, 2022 at 6:00:35 AM UTC-7, zelos...@gmail.com wrote:
> > > > onsdag 21 september 2022 kl. 14:28:11 UTC+2 skrev timba...@gmail.com:
> > > > > Wed 21 Sep 2022 08:03:47 AM EDT
> > > > > Natural Numbers And A New Continuum Hypothesis
> > > > > From the meandering on https://groups.google.com/g/sci.math/c/02xv2Bv1ZxE "Could dS be the polysign differential? Integral( dS over S ) = 0":
> > > > > where the seed got planted.
> > > > >
> > > > > An infinite value such as 333...34 has been posited. To make these values cohere an aleph mark | is needed so that 10x and x can commense. This aleph is done at the head of the value:
> > > > > x = |333...32
> > > > > 10x = 3|333...320
> > > > > and now any confusion over the relative position of this (infinite) head digit is defused. Clearly 10x is larger than x thanks only to this augmentation. Granting these values some fundamental existence we can thus claim a continuum hypothesis. In effect, let's replace the aleph mark with a decimal point:
> > > > > x = 0.333...32
> > > > > 10x = 3.333...320
> > > > > This is satisfactory to some people's sensibility of the continuous value.
> > > > > In effect working at aleph one is natural. The spirit of the natural number as going on and on is present within the value. This is merely an act of interpretation, and I prefer the finite forms myself, but as has been described elsewhere we are on a branch of mathematics here which is treating the ellipsis quite seriously. In physical terms the act of counting a unital is nonexistent, along with the successor, and here we see that roughly the same is true. Instead we've arrived at a macro scale at aleph one where physical work takes place. As we select a unit, whether in geometry on paper with some dividers, or via the King's thumb, or whatever glass ensconced reference you'd like to copy, or some elemental wavelength of light whose interferometric waves you can count, well, this latter possibly not; these are interpretations of unity. This arbitrary nature of unity at the macro level; at the physical level; is obvious to most who have spent time here.
> > > > >
> > > > > Strangely enough, it is as if we've left the natural value in the dust, so to speak, as we've built out of it such gargantuan values. That this route forgoes the rational value and instead relies upon a sense of infinity that many minds refuse to leave behind: here lays the tantalizing crux: this model is in the minds of men already. There is another way, but this way does seem to hold upon granting the inductive digital infinity such as
> > > > > 333...34
> > > > > which thence requires a mark to negotiate computations which is very nearly consistent with the decimal point of old.
> > > > >
> > > > > As I entertain this route, I should make clear that it is taken via the incessant ramblings and ranting on usenet of infinite concerns. Another route exists down lower by denying the rational value and its lack of closure; as if two integers could yield a continuum through the non-fundamental operator known as division. Did products behave this way? Well then, there is another attack on the rational value. Structural analysis exposes that secondary forms of unity serve far more simply as augmentations to the natural value. So a non-infinite route that is not so far away from the natural value exists as well.
> > > > your "polysign" does not work.
> > > It's like a Brouwerian continuum you should look it up.
> > https://en.wikipedia.org/wiki/L._E._J._Brouwer
> >
> > Thanks, Ross. You are well read.
> > I didn't really find anything on his continuum concept though. Maybe I am misusing the term 'continuum hypotheses' since it is already in use like the 'axiom of choice' ; sounds so broad but the term takes a narrowness via its past contributors. In that axioms are choices then constructive freedom is available to all who are willing to tamper with them. Consistency and correspondence are really quite important as metrics down at such a fundamental level. Consequences too. Nobody seems to want to touch the ellipsis as ambiguous. There are problems with these untouchables whose meanings are implied elsewhere so that a form of double talk is taking place. Perhaps this is cause to disrespect the natural number and head back to the drawing board to recover the continuum. That modulo principles are at work beneath the modulo neutered natural value is a cause to me. Of course the same is true of real analysis as well, so it's not as if they are any better. Indeed they start by supersetting the natural value. So far I've stayed with that course and found that the real can be had with the augmentation of the decimal point reassigning unity; a secondary form of unity. This idea that numbers do have structure within them has not really taken the digit as a serious element yet. I've still not decompiled down to that level. Perhaps there is something there that will marry into sign as well and yield some benefit.
> About continuum analysis there's like Veronese and Stolz and Peano and Poincare and Brouwer
> later after Cantor with Riemann and Lebesque and Stieltjes also these days, there is even
> continuum analysis from the ancients, Newton and Leibniz have their great thing, there's
> Bell and Robinson and Conway, Nelson and for example Woodin, Dodgson, Democritus,
> Eudoxus/Dedekind/Cauchy, there's mensuration of area and measurement of time,
> there's what's called a continuous manifold in a field theory that's a gauge theory,
> there's the quasi-invariant in emasure theory, there's line reals, field reals, signal reals, ....
>
> There's Vitali and for measure theory, while, also singularity theory becomes a pivotal
> feature, convergence and tangency while preservation of angle, the continuum of [0,1]
> the line segment with line drawing is about Zeno and constant motion and Democritus or atomism,
> while the spiral-space-filling curve founding geometry is not just the Archimedean but the ancient.
>
> So, I wrote a sigma algebra for real character for line reals, it's really quite simple.
> I wrote a variety of sigma algebras and also noted that in this modern model of
> continuum infinitesimal analysis, I wrote field operations for [-1, 1], and then
> that as a special function this generalized-distribution/limit-of-functions/model-of-
> real-function also defines a probability distribution of integers, and surprisingly
> not a unique one, and also is special about the exponential or exponential operators,
> being its own antiderivative.
>
> So, if I wasn't well-read then getting into all these features of numbers, then I
> wouldn't know operator theory for example, or probability theory.
> Here that "the natural/unit equivalency function is integrable and equals one",
> it's a very cool, special function, and in at least one theory, the prototype and
> the simplest model for a continuous domain. It's about the best and one of
> the most special functions in mathematics.
>
>
> Then, I think a lot of what you got going there is arithmetic coding, which basically
> reflects two forms at a distance maintain their proportion. Then, properties
> of continuity balance "parallax" and "perspective".
>
> Du Bois-Reymond, Schmieden and Laugwitz, Cohen and so on and for
> along the lines of "anything about infinity is also about continuity",
> DesCartes and Duns Scotus and Spinoza and friends, a "continuum"
> and its "individua" follows that there are as above here eight or ten
> named approaches to "what is a continuum", that yours is akin to Brouwer's,
> though, his branch and grow, yours branch and grow with a straight line in the middle.
>
>
> Most of the successful analytical approaches take an area and divide it,
> here also there are synthetic approaches (to analysis).
>
> So, that condenses a lot.

The original continuity of quantity is real. As an infinite sequence
of calculus infinitesimal concept. An infinity of the infinitely small
is One.

On Thursday, September 22, 2022 at 12:28:23 PM UTC-4, Ross A. Finlayson wrote:
> On Thursday, September 22, 2022 at 8:28:20 AM UTC-7, timba...@gmail.com wrote:
> > On Wednesday, September 21, 2022 at 11:46:06 AM UTC-4, Ross A. Finlayson wrote:
> > > On Wednesday, September 21, 2022 at 6:00:35 AM UTC-7, zelos...@gmail.com wrote:
> > > > onsdag 21 september 2022 kl. 14:28:11 UTC+2 skrev timba...@gmail.com:
> > > > > Wed 21 Sep 2022 08:03:47 AM EDT
> > > > > Natural Numbers And A New Continuum Hypothesis
> > > > > From the meandering on https://groups.google.com/g/sci.math/c/02xv2Bv1ZxE "Could dS be the polysign differential? Integral( dS over S ) = 0":
> > > > > where the seed got planted.
> > > > >
> > > > > An infinite value such as 333...34 has been posited. To make these values cohere an aleph mark | is needed so that 10x and x can commense. This aleph is done at the head of the value:
> > > > > x = |333...32
> > > > > 10x = 3|333...320
> > > > > and now any confusion over the relative position of this (infinite) head digit is defused. Clearly 10x is larger than x thanks only to this augmentation. Granting these values some fundamental existence we can thus claim a continuum hypothesis. In effect, let's replace the aleph mark with a decimal point:
> > > > > x = 0.333...32
> > > > > 10x = 3.333...320
> > > > > This is satisfactory to some people's sensibility of the continuous value.
> > > > > In effect working at aleph one is natural. The spirit of the natural number as going on and on is present within the value. This is merely an act of interpretation, and I prefer the finite forms myself, but as has been described elsewhere we are on a branch of mathematics here which is treating the ellipsis quite seriously. In physical terms the act of counting a unital is nonexistent, along with the successor, and here we see that roughly the same is true. Instead we've arrived at a macro scale at aleph one where physical work takes place. As we select a unit, whether in geometry on paper with some dividers, or via the King's thumb, or whatever glass ensconced reference you'd like to copy, or some elemental wavelength of light whose interferometric waves you can count, well, this latter possibly not; these are interpretations of unity. This arbitrary nature of unity at the macro level; at the physical level; is obvious to most who have spent time here.
> > > > >
> > > > > Strangely enough, it is as if we've left the natural value in the dust, so to speak, as we've built out of it such gargantuan values. That this route forgoes the rational value and instead relies upon a sense of infinity that many minds refuse to leave behind: here lays the tantalizing crux: this model is in the minds of men already. There is another way, but this way does seem to hold upon granting the inductive digital infinity such as
> > > > > 333...34
> > > > > which thence requires a mark to negotiate computations which is very nearly consistent with the decimal point of old.
> > > > >
> > > > > As I entertain this route, I should make clear that it is taken via the incessant ramblings and ranting on usenet of infinite concerns. Another route exists down lower by denying the rational value and its lack of closure; as if two integers could yield a continuum through the non-fundamental operator known as division. Did products behave this way? Well then, there is another attack on the rational value. Structural analysis exposes that secondary forms of unity serve far more simply as augmentations to the natural value. So a non-infinite route that is not so far away from the natural value exists as well.
> > > > your "polysign" does not work.
> > > It's like a Brouwerian continuum you should look it up.
> > https://en.wikipedia.org/wiki/L._E._J._Brouwer
> >
> > Thanks, Ross. You are well read.
> > I didn't really find anything on his continuum concept though. Maybe I am misusing the term 'continuum hypotheses' since it is already in use like the 'axiom of choice' ; sounds so broad but the term takes a narrowness via its past contributors. In that axioms are choices then constructive freedom is available to all who are willing to tamper with them. Consistency and correspondence are really quite important as metrics down at such a fundamental level. Consequences too. Nobody seems to want to touch the ellipsis as ambiguous. There are problems with these untouchables whose meanings are implied elsewhere so that a form of double talk is taking place. Perhaps this is cause to disrespect the natural number and head back to the drawing board to recover the continuum. That modulo principles are at work beneath the modulo neutered natural value is a cause to me. Of course the same is true of real analysis as well, so it's not as if they are any better. Indeed they start by supersetting the natural value. So far I've stayed with that course and found that the real can be had with the augmentation of the decimal point reassigning unity; a secondary form of unity. This idea that numbers do have structure within them has not really taken the digit as a serious element yet. I've still not decompiled down to that level. Perhaps there is something there that will marry into sign as well and yield some benefit.
> About continuum analysis there's like Veronese and Stolz and Peano and Poincare and Brouwer
> later after Cantor with Riemann and Lebesque and Stieltjes also these days, there is even
> continuum analysis from the ancients, Newton and Leibniz have their great thing, there's
> Bell and Robinson and Conway, Nelson and for example Woodin, Dodgson, Democritus,
> Eudoxus/Dedekind/Cauchy, there's mensuration of area and measurement of time,
> there's what's called a continuous manifold in a field theory that's a gauge theory,
> there's the quasi-invariant in emasure theory, there's line reals, field reals, signal reals, ....
>
> There's Vitali and for measure theory, while, also singularity theory becomes a pivotal
> feature, convergence and tangency while preservation of angle, the continuum of [0,1]
> the line segment with line drawing is about Zeno and constant motion and Democritus or atomism,
> while the spiral-space-filling curve founding geometry is not just the Archimedean but the ancient.
>
> So, I wrote a sigma algebra for real character for line reals, it's really quite simple.
> I wrote a variety of sigma algebras and also noted that in this modern model of
> continuum infinitesimal analysis, I wrote field operations for [-1, 1], and then
> that as a special function this generalized-distribution/limit-of-functions/model-of-
> real-function also defines a probability distribution of integers, and surprisingly
> not a unique one, and also is special about the exponential or exponential operators,
> being its own antiderivative.
>
> So, if I wasn't well-read then getting into all these features of numbers, then I
> wouldn't know operator theory for example, or probability theory.
> Here that "the natural/unit equivalency function is integrable and equals one",
> it's a very cool, special function, and in at least one theory, the prototype and
> the simplest model for a continuous domain. It's about the best and one of
> the most special functions in mathematics.
>
>
> Then, I think a lot of what you got going there is arithmetic coding, which basically
> reflects two forms at a distance maintain their proportion. Then, properties
> of continuity balance "parallax" and "perspective".
>
> Du Bois-Reymond, Schmieden and Laugwitz, Cohen and so on and for
> along the lines of "anything about infinity is also about continuity",
> DesCartes and Duns Scotus and Spinoza and friends, a "continuum"
> and its "individua" follows that there are as above here eight or ten
> named approaches to "what is a continuum", that yours is akin to Brouwer's,
> though, his branch and grow, yours branch and grow with a straight line in the middle.
>
>
> Most of the successful analytical approaches take an area and divide it,
> here also there are synthetic approaches (to analysis).
>
> So, that condenses a lot.

Wow. Thanks for your extended thinking. I get that you get it here.
To what degree numbers branch is a nifty context.
Especially for mathematicians who dabble with an unknown x that such happenings are refining their value; then too to ponder whether fourths are finer than thirds, and for the halves that halve not: they won't ever get a say in the thing. So it is with the rational analysis. Amiss is putting it lightly.
Dodgulo numbers, so to speak. All modulo principals aside, sir: please, step aside; we have no room for that sort of language here.

I honestly have refuted the infinite branch already and find that position likable. It is so strange to then go on and validate these infinite forms that seem so strange to begin with. I honestly have seen infinity as I understood the forefathers' positions pretty close to Newton's form: something that can be approached but cannot be exceeded. Now these instances do challenge that notion. Instead the disgusting proliferation of these values and their ability to do operations does hold up. I guarantee you that the aleph mark is no contrivance, either. That it can wrap around, graphically hold it's place, and take the unitary stand is to say quite a bit about the mark. Nextly the double form: to what degree have we discovered that the product is indeed two dimensional? that we have the freedom again to state a unique value here constructively is a move I would not take without the cause of the product, but indeed since the product requires it; and the aleph marks too; Informationally we are at a two dimensional value out of one dimension, and we can't just get rid of the thing, either. I'm mean, I guess you could just take the gross part and ditch the miniscule part... you know stuff like that happens in complex analysis all the time. You just keep chucking half the thing out the window. Magnitude and phase wind up deadlocked, not identically but to the point where one dictates the other and you wind up with cardiod patterns; smith charts, and cancelling complex impedances when things are properly matched. This is something that the decimal point does not do.

On Thursday, September 22, 2022 at 3:20:58 PM UTC-4, mitchr...@gmail.com wrote:
> On Thursday, September 22, 2022 at 9:28:23 AM UTC-7, Ross A. Finlayson wrote:
> > On Thursday, September 22, 2022 at 8:28:20 AM UTC-7, timba...@gmail.com wrote:
> > > On Wednesday, September 21, 2022 at 11:46:06 AM UTC-4, Ross A. Finlayson wrote:
> > > > On Wednesday, September 21, 2022 at 6:00:35 AM UTC-7, zelos...@gmail.com wrote:
> > > > > onsdag 21 september 2022 kl. 14:28:11 UTC+2 skrev timba...@gmail.com:
> > > > > > Wed 21 Sep 2022 08:03:47 AM EDT
> > > > > > Natural Numbers And A New Continuum Hypothesis
> > > > > > From the meandering on https://groups.google.com/g/sci.math/c/02xv2Bv1ZxE "Could dS be the polysign differential? Integral( dS over S ) = 0":
> > > > > > where the seed got planted.
> > > > > >
> > > > > > An infinite value such as 333...34 has been posited. To make these values cohere an aleph mark | is needed so that 10x and x can commense. This aleph is done at the head of the value:
> > > > > > x = |333...32
> > > > > > 10x = 3|333...320
> > > > > > and now any confusion over the relative position of this (infinite) head digit is defused. Clearly 10x is larger than x thanks only to this augmentation. Granting these values some fundamental existence we can thus claim a continuum hypothesis. In effect, let's replace the aleph mark with a decimal point:
> > > > > > x = 0.333...32
> > > > > > 10x = 3.333...320
> > > > > > This is satisfactory to some people's sensibility of the continuous value.
> > > > > > In effect working at aleph one is natural. The spirit of the natural number as going on and on is present within the value. This is merely an act of interpretation, and I prefer the finite forms myself, but as has been described elsewhere we are on a branch of mathematics here which is treating the ellipsis quite seriously. In physical terms the act of counting a unital is nonexistent, along with the successor, and here we see that roughly the same is true. Instead we've arrived at a macro scale at aleph one where physical work takes place. As we select a unit, whether in geometry on paper with some dividers, or via the King's thumb, or whatever glass ensconced reference you'd like to copy, or some elemental wavelength of light whose interferometric waves you can count, well, this latter possibly not; these are interpretations of unity. This arbitrary nature of unity at the macro level; at the physical level; is obvious to most who have spent time here.
> > > > > >
> > > > > > Strangely enough, it is as if we've left the natural value in the dust, so to speak, as we've built out of it such gargantuan values. That this route forgoes the rational value and instead relies upon a sense of infinity that many minds refuse to leave behind: here lays the tantalizing crux: this model is in the minds of men already. There is another way, but this way does seem to hold upon granting the inductive digital infinity such as
> > > > > > 333...34
> > > > > > which thence requires a mark to negotiate computations which is very nearly consistent with the decimal point of old.
> > > > > >
> > > > > > As I entertain this route, I should make clear that it is taken via the incessant ramblings and ranting on usenet of infinite concerns. Another route exists down lower by denying the rational value and its lack of closure; as if two integers could yield a continuum through the non-fundamental operator known as division. Did products behave this way? Well then, there is another attack on the rational value. Structural analysis exposes that secondary forms of unity serve far more simply as augmentations to the natural value. So a non-infinite route that is not so far away from the natural value exists as well.
> > > > > your "polysign" does not work.
> > > > It's like a Brouwerian continuum you should look it up.
> > > https://en.wikipedia.org/wiki/L._E._J._Brouwer
> > >
> > > Thanks, Ross. You are well read.
> > > I didn't really find anything on his continuum concept though. Maybe I am misusing the term 'continuum hypotheses' since it is already in use like the 'axiom of choice' ; sounds so broad but the term takes a narrowness via its past contributors. In that axioms are choices then constructive freedom is available to all who are willing to tamper with them. Consistency and correspondence are really quite important as metrics down at such a fundamental level. Consequences too. Nobody seems to want to touch the ellipsis as ambiguous. There are problems with these untouchables whose meanings are implied elsewhere so that a form of double talk is taking place. Perhaps this is cause to disrespect the natural number and head back to the drawing board to recover the continuum. That modulo principles are at work beneath the modulo neutered natural value is a cause to me. Of course the same is true of real analysis as well, so it's not as if they are any better. Indeed they start by supersetting the natural value. So far I've stayed with that course and found that the real can be had with the augmentation of the decimal point reassigning unity; a secondary form of unity. This idea that numbers do have structure within them has not really taken the digit as a serious element yet. I've still not decompiled down to that level. Perhaps there is something there that will marry into sign as well and yield some benefit.
> > About continuum analysis there's like Veronese and Stolz and Peano and Poincare and Brouwer
> > later after Cantor with Riemann and Lebesque and Stieltjes also these days, there is even
> > continuum analysis from the ancients, Newton and Leibniz have their great thing, there's
> > Bell and Robinson and Conway, Nelson and for example Woodin, Dodgson, Democritus,
> > Eudoxus/Dedekind/Cauchy, there's mensuration of area and measurement of time,
> > there's what's called a continuous manifold in a field theory that's a gauge theory,
> > there's the quasi-invariant in emasure theory, there's line reals, field reals, signal reals, ....
> >
> > There's Vitali and for measure theory, while, also singularity theory becomes a pivotal
> > feature, convergence and tangency while preservation of angle, the continuum of [0,1]
> > the line segment with line drawing is about Zeno and constant motion and Democritus or atomism,
> > while the spiral-space-filling curve founding geometry is not just the Archimedean but the ancient.
> >
> > So, I wrote a sigma algebra for real character for line reals, it's really quite simple.
> > I wrote a variety of sigma algebras and also noted that in this modern model of
> > continuum infinitesimal analysis, I wrote field operations for [-1, 1], and then
> > that as a special function this generalized-distribution/limit-of-functions/model-of-
> > real-function also defines a probability distribution of integers, and surprisingly
> > not a unique one, and also is special about the exponential or exponential operators,
> > being its own antiderivative.
> >
> > So, if I wasn't well-read then getting into all these features of numbers, then I
> > wouldn't know operator theory for example, or probability theory.
> > Here that "the natural/unit equivalency function is integrable and equals one",
> > it's a very cool, special function, and in at least one theory, the prototype and
> > the simplest model for a continuous domain. It's about the best and one of
> > the most special functions in mathematics.
> >
> >
> > Then, I think a lot of what you got going there is arithmetic coding, which basically
> > reflects two forms at a distance maintain their proportion. Then, properties
> > of continuity balance "parallax" and "perspective".
> >
> > Du Bois-Reymond, Schmieden and Laugwitz, Cohen and so on and for
> > along the lines of "anything about infinity is also about continuity",
> > DesCartes and Duns Scotus and Spinoza and friends, a "continuum"
> > and its "individua" follows that there are as above here eight or ten
> > named approaches to "what is a continuum", that yours is akin to Brouwer's,
> > though, his branch and grow, yours branch and grow with a straight line in the middle.
> >
> >
> > Most of the successful analytical approaches take an area and divide it,
> > here also there are synthetic approaches (to analysis).
> >
> > So, that condenses a lot.
> The original continuity of quantity is real. As an infinite sequence
> of calculus infinitesimal concept. An infinity of the infinitely small
> is One.

Well Mitch, this is sort of a way to do what you are describing. Taking unity at Aleph one within this digital system:
1|00..
is compact notation, and the aleph bar simply takes the interpretation of the decimal point, and now your real has infinite length precision. This is the usual mathematician's sense of perfect real values. I argue against this style, as the decimal point can already reassign the unital position of the natural value without any infinite treatment. This style though yields a continuum that is gray. The digits are not implied at all beyond the last digit. I guess that will make this a confusing statement since I'm comparing two interpretations inline. They are not equivalent. Raising the confusion further we can enter the rationals like 1/3 that yield the repeating form, but I've already refuted this last option.