mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
> zero is not a quantity so it would be absolute
> But above zero is first quantity or infinitesimal
> of the calculus. That is sub finite.

The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point.) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).

Regards, WM

On 1/7/2024 2:34 PM, WM wrote:
> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
>> zero is not a quantity so it would be absolute
>> But above zero is first quantity or infinitesimal
>> of the calculus. That is sub finite.
>
> The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point.) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).
>

There is no _first_ real number above zero.

On 1/7/24 5:34 PM, WM wrote:
> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
>> zero is not a quantity so it would be absolute
>> But above zero is first quantity or infinitesimal
>> of the calculus. That is sub finite.
>
> The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point.) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).
>
> Regards, WM

Nope, your reasoning is "Dark", as it just doesn't understand Unbouded sets.

Can you name a set that only has "Dark" numbers?

You say they can be used collectively, so give a collection that only
has dark numbers!

If not, then are your dark numbers actually in existance?

From your previous comments, the "darkness" isn't actually a property
of the "Number" but of the observes knowledge.

It seems you just don't know how to keep the numbers you are talking
about in the set that they need to be in.

On Sunday, January 7, 2024 at 3:19:25 PM UTC-8, Chris M. Thomasson wrote:
> On 1/7/2024 2:34 PM, WM wrote:
> > mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
> >> zero is not a quantity so it would be absolute
> >> But above zero is first quantity or infinitesimal
> >> of the calculus. That is sub finite.
> >
> > The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point.) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).
> >
> There is no _first_ real number above zero.

Real numbers, sets of, each, are generally considered "continuous domains"
of "elements of a linear continuum". There are at least three models of these,
line reals with infinitesimals [0,1], field reals the complete ordered field, and
signal reals even/odd infinitesimals.

integer continuum -> line reals <-> field reals <-> signal reals <- long-line continuum

So, in line reals there _is- a first number above zero.

This is called, for each "definition of continuity", a "completenes", all, a "repleteness".

That is all, ....

On 1/7/2024 4:24 PM, Ross Finlayson wrote:
> On Sunday, January 7, 2024 at 3:19:25 PM UTC-8, Chris M. Thomasson wrote:
>> On 1/7/2024 2:34 PM, WM wrote:
>>> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
>>>> zero is not a quantity so it would be absolute
>>>> But above zero is first quantity or infinitesimal
>>>> of the calculus. That is sub finite.
>>>
>>> The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point.) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).
>>>
>> There is no _first_ real number above zero.
>
> Real numbers, sets of, each, are generally considered "continuous domains"
> of "elements of a linear continuum". There are at least three models of these,
> line reals with infinitesimals [0,1], field reals the complete ordered field, and
> signal reals even/odd infinitesimals.
>
> integer continuum -> line reals <-> field reals <-> signal reals <- long-line continuum
>
> So, in line reals there _is- a first number above zero.

There is no first real number greater than zero. Which one is first?

..001, or .01?

Oh don't forget .00001. Not to mention .00000000001

These are all reals greater than one... Now, if I define a normal unit,
say 1 / 42, then we can have a first one because I simply defined the
normal unit. Is not infinite anymore. See?

> This is called, for each "definition of continuity", a "completenes", all, a "repleteness".
>
> That is all, ....

We can plot a line in n-ary p0, and p1.

dif = p1 - p0;

Now we have our difference. We can define a normal, say:

n = 2048;
normal_base = 1 / n;

Then we can plot the points:

for (i = 0; i < n; ++i)
{ normal = normal_base * i;
point = p0 + dif * normal;
plot_point(point);
}

Notice that as soon as we define n, its no longer infinite...

On Sunday, January 7, 2024 at 7:15:12 PM UTC-8, Chris M. Thomasson wrote:
> On 1/7/2024 4:24 PM, Ross Finlayson wrote:
> > On Sunday, January 7, 2024 at 3:19:25 PM UTC-8, Chris M. Thomasson wrote:
> >> On 1/7/2024 2:34 PM, WM wrote:
> >>> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
> >>>> zero is not a quantity so it would be absolute
> >>>> But above zero is first quantity or infinitesimal
> >>>> of the calculus. That is sub finite.
> >>>
> >>> The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point.) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).
> >>>
> >> There is no _first_ real number above zero.
> >
> > Real numbers, sets of, each, are generally considered "continuous domains"
> > of "elements of a linear continuum". There are at least three models of these,
> > line reals with infinitesimals [0,1], field reals the complete ordered field, and
> > signal reals even/odd infinitesimals.
> >
> > integer continuum -> line reals <-> field reals <-> signal reals <- long-line continuum
> >
> > So, in line reals there _is- a first number above zero.
> There is no first real number greater than zero. Which one is first?
>
> .001, or .01?
>
> Oh don't forget .00001. Not to mention .00000000001
>
> These are all reals greater than one... Now, if I define a normal unit,
> say 1 / 42, then we can have a first one because I simply defined the
> normal unit. Is not infinite anymore. See?
> > This is called, for each "definition of continuity", a "completenes", all, a "repleteness".
> >
> > That is all, ....
> We can plot a line in n-ary p0, and p1.
>
> dif = p1 - p0;
>
> Now we have our difference. We can define a normal, say:
>
> n = 2048;
> normal_base = 1 / n;
>
> Then we can plot the points:
>
> for (i = 0; i < n; ++i)
> {
> normal = normal_base * i;
> point = p0 + dif * normal;
> plot_point(point);
> }
>
>
> Notice that as soon as we define n, its no longer infinite...

What you do is you get a compass and you get an edge,
and you throw away the compass, and get some chalk,
then you mark down the edge, the chalk, and,
you just wrote infinity points in a row.

It's sort of similar with fixed point, it's just, you know,
1, with however deep a radix in all the storage there is.

I mean, your real numbers have 1/3, right. They have n/d,
for unsigned n, d. Then just let n go from zero to d,
as sizeof(typeof(d)), or in accords with types, goes to infinity.

It's called the equivalency function, its range has a sigma algebra.

See also Ross Finlayson's 10,000's posts to sci.math,
particularly his, "slates".

....

On 1/7/2024 7:21 PM, Ross Finlayson wrote:
> On Sunday, January 7, 2024 at 7:15:12 PM UTC-8, Chris M. Thomasson wrote:
>> On 1/7/2024 4:24 PM, Ross Finlayson wrote:
>>> On Sunday, January 7, 2024 at 3:19:25 PM UTC-8, Chris M. Thomasson wrote:
>>>> On 1/7/2024 2:34 PM, WM wrote:
>>>>> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
>>>>>> zero is not a quantity so it would be absolute
>>>>>> But above zero is first quantity or infinitesimal
>>>>>> of the calculus. That is sub finite.
>>>>>
>>>>> The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point.) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).
>>>>>
>>>> There is no _first_ real number above zero.
>>>
>>> Real numbers, sets of, each, are generally considered "continuous domains"
>>> of "elements of a linear continuum". There are at least three models of these,
>>> line reals with infinitesimals [0,1], field reals the complete ordered field, and
>>> signal reals even/odd infinitesimals.
>>>
>>> integer continuum -> line reals <-> field reals <-> signal reals <- long-line continuum
>>>
>>> So, in line reals there _is- a first number above zero.
>> There is no first real number greater than zero. Which one is first?
>>
>> .001, or .01?
>>
>> Oh don't forget .00001. Not to mention .00000000001
>>
>> These are all reals greater than one... Now, if I define a normal unit,
>> say 1 / 42, then we can have a first one because I simply defined the
>> normal unit. Is not infinite anymore. See?
>>> This is called, for each "definition of continuity", a "completenes", all, a "repleteness".
>>>
>>> That is all, ....
>> We can plot a line in n-ary p0, and p1.
>>
>> dif = p1 - p0;
>>
>> Now we have our difference. We can define a normal, say:
>>
>> n = 2048;
>> normal_base = 1 / n;
>>
>> Then we can plot the points:
>>
>> for (i = 0; i < n; ++i)
>> {
>> normal = normal_base * i;
>> point = p0 + dif * normal;
>> plot_point(point);
>> }
>>
>>
>> Notice that as soon as we define n, its no longer infinite...
>
> What you do is you get a compass and you get an edge,
> and you throw away the compass, and get some chalk,
> then you mark down the edge, the chalk, and,
> you just wrote infinity points in a row.

Yes. Think of drawing a line from p0 to p1, we just drew infinite
points. Fine. However, wrt infinity and real numbers there is no first
one greater than zero:

0---->1/2---->1

Well, we have:

0---->1/4---->1/2

Oh what about:

0---->1/8---->1/4

0---->1/16---->1/8

Humm, oh we also have:

0---->1/32---->1/16

This is basically zooming in on line segments. Those are all covered,
however there is no first real number greater than zero... We can zoom
in forever...

>
>
> It's sort of similar with fixed point, it's just, you know,
> 1, with however deep a radix in all the storage there is.
>
> I mean, your real numbers have 1/3, right. They have n/d,
> for unsigned n, d. Then just let n go from zero to d,
> as sizeof(typeof(d)), or in accords with types, goes to infinity.
>
> It's called the equivalency function, its range has a sigma algebra.
>
> See also Ross Finlayson's 10,000's posts to sci.math,
> particularly his, "slates".
>
> ...
>

On 1/7/2024 7:21 PM, Ross Finlayson wrote:
> On Sunday, January 7, 2024 at 7:15:12 PM UTC-8, Chris M. Thomasson wrote:
>> On 1/7/2024 4:24 PM, Ross Finlayson wrote:
>>> On Sunday, January 7, 2024 at 3:19:25 PM UTC-8, Chris M. Thomasson wrote:
>>>> On 1/7/2024 2:34 PM, WM wrote:
>>>>> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
>>>>>> zero is not a quantity so it would be absolute
>>>>>> But above zero is first quantity or infinitesimal
>>>>>> of the calculus. That is sub finite.
>>>>>
>>>>> The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point.) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).
>>>>>
>>>> There is no _first_ real number above zero.
>>>
>>> Real numbers, sets of, each, are generally considered "continuous domains"
>>> of "elements of a linear continuum". There are at least three models of these,
>>> line reals with infinitesimals [0,1], field reals the complete ordered field, and
>>> signal reals even/odd infinitesimals.
>>>
>>> integer continuum -> line reals <-> field reals <-> signal reals <- long-line continuum
>>>
>>> So, in line reals there _is- a first number above zero.
>> There is no first real number greater than zero. Which one is first?
>>
>> .001, or .01?
>>
>> Oh don't forget .00001. Not to mention .00000000001
>>
>> These are all reals greater than one... Now, if I define a normal unit,
>> say 1 / 42, then we can have a first one because I simply defined the
>> normal unit. Is not infinite anymore. See?
>>> This is called, for each "definition of continuity", a "completenes", all, a "repleteness".
>>>
>>> That is all, ....
>> We can plot a line in n-ary p0, and p1.
>>
>> dif = p1 - p0;
>>
>> Now we have our difference. We can define a normal, say:
>>
>> n = 2048;
>> normal_base = 1 / n;
>>
>> Then we can plot the points:
>>
>> for (i = 0; i < n; ++i)
>> {
>> normal = normal_base * i;
>> point = p0 + dif * normal;
>> plot_point(point);
>> }
>>
>>
>> Notice that as soon as we define n, its no longer infinite...
>
> What you do is you get a compass and you get an edge,
> and you throw away the compass, and get some chalk,
> then you mark down the edge, the chalk, and,
> you just wrote infinity points in a row.
>
>
> It's sort of similar with fixed point, it's just, you know,
> 1, with however deep a radix in all the storage there is.
>
> I mean, your real numbers have 1/3, right. They have n/d,
> for unsigned n, d. Then just let n go from zero to d,
> as sizeof(typeof(d)), or in accords with types, goes to infinity.
>
> It's called the equivalency function, its range has a sigma algebra.
>
> See also Ross Finlayson's 10,000's posts to sci.math,
> particularly his, "slates".

Just keep in mind that the concept of a "first" real number greater than
zero doesn't exist within the framework of real numbers...

On Sunday, January 7, 2024 at 7:32:09 PM UTC-8, Chris M. Thomasson wrote:
> On 1/7/2024 7:21 PM, Ross Finlayson wrote:
> > On Sunday, January 7, 2024 at 7:15:12 PM UTC-8, Chris M. Thomasson wrote:
> >> On 1/7/2024 4:24 PM, Ross Finlayson wrote:
> >>> On Sunday, January 7, 2024 at 3:19:25 PM UTC-8, Chris M. Thomasson wrote:
> >>>> On 1/7/2024 2:34 PM, WM wrote:
> >>>>> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
> >>>>>> zero is not a quantity so it would be absolute
> >>>>>> But above zero is first quantity or infinitesimal
> >>>>>> of the calculus. That is sub finite.
> >>>>>
> >>>>> The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point..) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).
> >>>>>
> >>>> There is no _first_ real number above zero.
> >>>
> >>> Real numbers, sets of, each, are generally considered "continuous domains"
> >>> of "elements of a linear continuum". There are at least three models of these,
> >>> line reals with infinitesimals [0,1], field reals the complete ordered field, and
> >>> signal reals even/odd infinitesimals.
> >>>
> >>> integer continuum -> line reals <-> field reals <-> signal reals <- long-line continuum
> >>>
> >>> So, in line reals there _is- a first number above zero.
> >> There is no first real number greater than zero. Which one is first?
> >>
> >> .001, or .01?
> >>
> >> Oh don't forget .00001. Not to mention .00000000001
> >>
> >> These are all reals greater than one... Now, if I define a normal unit,
> >> say 1 / 42, then we can have a first one because I simply defined the
> >> normal unit. Is not infinite anymore. See?
> >>> This is called, for each "definition of continuity", a "completenes", all, a "repleteness".
> >>>
> >>> That is all, ....
> >> We can plot a line in n-ary p0, and p1.
> >>
> >> dif = p1 - p0;
> >>
> >> Now we have our difference. We can define a normal, say:
> >>
> >> n = 2048;
> >> normal_base = 1 / n;
> >>
> >> Then we can plot the points:
> >>
> >> for (i = 0; i < n; ++i)
> >> {
> >> normal = normal_base * i;
> >> point = p0 + dif * normal;
> >> plot_point(point);
> >> }
> >>
> >>
> >> Notice that as soon as we define n, its no longer infinite...
> >
> > What you do is you get a compass and you get an edge,
> > and you throw away the compass, and get some chalk,
> > then you mark down the edge, the chalk, and,
> > you just wrote infinity points in a row.
> >
> >
> > It's sort of similar with fixed point, it's just, you know,
> > 1, with however deep a radix in all the storage there is.
> >
> > I mean, your real numbers have 1/3, right. They have n/d,
> > for unsigned n, d. Then just let n go from zero to d,
> > as sizeof(typeof(d)), or in accords with types, goes to infinity.
> >
> > It's called the equivalency function, its range has a sigma algebra.
> >
> > See also Ross Finlayson's 10,000's posts to sci.math,
> > particularly his, "slates".
> Just keep in mind that the concept of a "first" real number greater than
> zero doesn't exist within the framework of real numbers...

.... "Standard", "standard" real numbers, indeed, the entire idea of
the "non-standard", real numbers, is these sorts things.

Then it's what's called "continuous domains", and that on the "linear continuum",
which are all, only, and everywhere real numbers, are these standard and non-standard,
real numbers.

For example if you look into Jordan measure it's for a line integral, in the line,
and the Dirichlet function is, half the real numbers, even/odd, or alternating.

Then, it's very simple, real simple, that n/d for n, d natural as n -> d, and d -> oo,
results a model of a "not-a-real-function" a "not-a-standard-real-function",
that for example has least upper bound property, very trivially,
that for example "zooming" never reaches, because it's the ordered field,
not the complete ordered field, which only follows axiomatization of least-upper-bound property.
It has sigma algebras for emasure and LUB for gaplessness so the fundamental
theorems of calculus follow, it's a continuous domain.

Other "not-a-real-function's" of interest include the Dirac delta.
(As modeled by real functions and only true in the limit.)

So, in the wider framework of continuous domains, and real numbers,
are three different sets of real numbers, or frameworks of functions,
that make the equivalency function that is not a Cartesian function,
a continuous domain, and, the usual standard complete ordered field,
which is equivalence classes of sequences that are Cauchy, and for signal reals,
functions in Dirichlet functions that subset the continuous domain while
keeping rational analytical character.

https://www.youtube.com/@rossfinlayson

Then "line-drawing" is called "line-drawing" and it's a fundamental geometrical act.

So, in line reals there is a first number after f(0) = 0, it's f(1), in "iota-values".

It's real valued.

....

Le 08/01/2024 à 00:19, "Chris M. Thomasson" a écrit :
> On 1/7/2024 2:34 PM, WM wrote:
>> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
>>> zero is not a quantity so it would be absolute
>>> But above zero is first quantity or infinitesimal
>>> of the calculus. That is sub finite.
>>
>> The first real numbers above zero are dark in that they can only be used
>> collectively. (But some of them must be there below any defined point.) Moreover,
>> between any two defined points on the real line there are ℵ dark points (if
>> actual infinity exists) or nothing (if infinity is potential only).
>>
>
> There is no _first_ real number above zero.

But there are ℵ real numbers above zero and smaller than every definable
positive real number. Proof: Try to define a positive number x with less
smaller numbers between 0 and x. Impossible.

Regards, WM

Le 08/01/2024 à 00:26, Richard Damon a écrit :
> On 1/7/24 5:34 PM, WM wrote:
>> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
>>> zero is not a quantity so it would be absolute
>>> But above zero is first quantity or infinitesimal
>>> of the calculus. That is sub finite.
>>
>> The first real numbers above zero are dark in that they can only be used
>> collectively. (But some of them must be there below any defined point.) Moreover,
>> between any two defined points on the real line there are ℵ dark points (if
>> actual infinity exists) or nothing (if infinity is potential only).

> Nope, your reasoning is "Dark", as it just doesn't understand Unbouded sets.

I understand that every real x > 0 that you can name has ℵ smaller
positive reals ℵ of which you cannot name.
>
> Can you name a set that only has "Dark" numbers?

If existing {ω, ω+1, ω+2, ...} because there is no finite initial
segment for any of its elements.
>
> You say they can be used collectively, so give a collection that only
> has dark numbers!

That is not a precondition for the existence of dark numbers.
>
> If not, then are your dark numbers actually in existance?

I don't know ehtehre dar numbers are existing. I have proved only that, if
actual infinity exists, then dark numbers exist too.
>
> From your previous comments, the "darkness" isn't actually a property
> of the "Number" but of the observes knowledge.
>
> It seems you just don't know how to keep the numbers you are talking
> about in the set that they need to be in.

Try to understand: Every x > 0 is not the smallest positive number. Either
there are no (not yet) numbers below the smallest defined x > 0, or they
are dark.

Regards, WM

On 1/8/24 5:56 AM, WM wrote:
> Le 08/01/2024 à 00:26, Richard Damon a écrit :
>> On 1/7/24 5:34 PM, WM wrote:
>>> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23
>>> UTC+1:
>>>> zero is not a quantity so it would be absolute
>>>> But above zero is first quantity or infinitesimal
>>>> of the calculus. That is sub finite.
>>>
>>> The first real numbers above zero are dark in that they can only be
>>> used collectively. (But some of them must be there below any defined
>>> point.) Moreover, between any two defined points on the real line
>>> there are ℵ dark points (if actual infinity exists) or nothing (if
>>> infinity is potential only).
>
>> Nope, your reasoning is "Dark", as it just doesn't understand Unbouded
>> sets.
>
> I understand that every real x > 0 that you can name has ℵ smaller
> positive reals ℵ of which you cannot name.

why do you think I can not name any of them?

>>
>> Can you name a set that only has "Dark" numbers?
>
> If existing {ω, ω+1, ω+2, ...} because there is no finite initial
> segment for any of its elements.

But you have explicitly said you are working with the NATURAL numbers,
and ω and such are not Natural Numbers, so apparently your dark numbers
are just Natural Numbers that are not Natural Numbers.

>>
>> You say they can be used collectively, so give a collection that only
>> has dark numbers!
>
> That is not a precondition for the existence of dark numbers.

WHy

>>
>> If not, then are your dark numbers actually in existance?
>
> I don't know ehtehre dar numbers are existing. I have proved only that,
> if actual infinity exists, then dark numbers exist too.

And what do you mean by "actual infinity"?

>>
>>  From your previous comments, the "darkness" isn't actually a property
>> of the "Number" but of the observes knowledge.
>>
>> It seems you just don't know how to keep the numbers you are talking
>> about in the set that they need to be in.
>
> Try to understand: Every x > 0 is not the smallest positive number.
> Either there are no (not yet) numbers below the smallest defined x > 0,
> or they are dark.
>
> Regards, WM
>
>

Your problem is you assume the existance of things that are not.

There is no "smallest" x > 0, at least not in the finite numbers.

That is just like the Barber that shaves everyone who does not shave
himself.

Your logic has just fallen into Russel's paradox.

zelos...@gmail.com schrieb am Montag, 8. Januar 2024 um 07:37:56 UTC+1:
> On Saturday 6 January 2024 at 03:17:23 UTC+1, mitchr...@gmail.com wrote:
> > zero is not a quantity so it would be absolute
> > But above zero is first quantity or infinitesimal
> > of the calculus. That is sub finite.
> doesn't exist you nimrod

There are ℵ real numbers in every interval (0, eps) for every eps > 0
that you can choose. That proves that ℵ real numbers cannot be chosen
individually but only collectively as I just described them. They are
dark.

Regards, WM

On 1/8/24 7:23 AM, WM wrote:
> zelos...@gmail.com schrieb am Montag, 8. Januar 2024 um 07:37:56 UTC+1:
>> On Saturday 6 January 2024 at 03:17:23 UTC+1, mitchr...@gmail.com
>> wrote: > zero is not a quantity so it would be absolute > But above
>> zero is first quantity or infinitesimal > of the calculus. That is sub
>> finite.
>> doesn't exist you nimrod
>
> There are ℵ real numbers in every interval (0, eps) for every eps > 0
> that you can choose. That proves that ℵ real numbers cannot be chosen
> individually but only collectively as I just described them. They are dark.
>
> Regards, WM
>
>

No, your claim is essentially that an infinite number is not also a
finite number.

It doesn't say the numbers are dark, just too numerous list all of them
at once on finite paper.

Your logic just doens't handle inifinites.

Le 08/01/2024 à 13:21, Richard Damon a écrit :
> On 1/8/24 5:56 AM, WM wrote:

>> I understand that every real x > 0 that you can name has ℵ smaller
>> positive reals ℵ of which you cannot name.
>
> why do you think I can not name any of them?

You can name many of them, but every attempt fails to reduce the remainder
below ℵ. It can be reduced however to the empty set collectively.

Regards, WM

On Sunday, January 7, 2024 at 9:15:27 PM UTC-8, Ross Finlayson wrote:
> On Sunday, January 7, 2024 at 7:32:09 PM UTC-8, Chris M. Thomasson wrote:
> > On 1/7/2024 7:21 PM, Ross Finlayson wrote:
> > > On Sunday, January 7, 2024 at 7:15:12 PM UTC-8, Chris M. Thomasson wrote:
> > >> On 1/7/2024 4:24 PM, Ross Finlayson wrote:
> > >>> On Sunday, January 7, 2024 at 3:19:25 PM UTC-8, Chris M. Thomasson wrote:
> > >>>> On 1/7/2024 2:34 PM, WM wrote:
> > >>>>> mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
> > >>>>>> zero is not a quantity so it would be absolute
> > >>>>>> But above zero is first quantity or infinitesimal
> > >>>>>> of the calculus. That is sub finite.
> > >>>>>
> > >>>>> The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point.) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).
> > >>>>>
> > >>>> There is no _first_ real number above zero.
> > >>>
> > >>> Real numbers, sets of, each, are generally considered "continuous domains"
> > >>> of "elements of a linear continuum". There are at least three models of these,
> > >>> line reals with infinitesimals [0,1], field reals the complete ordered field, and
> > >>> signal reals even/odd infinitesimals.
> > >>>
> > >>> integer continuum -> line reals <-> field reals <-> signal reals <- long-line continuum
> > >>>
> > >>> So, in line reals there _is- a first number above zero.
> > >> There is no first real number greater than zero. Which one is first?
> > >>
> > >> .001, or .01?
> > >>
> > >> Oh don't forget .00001. Not to mention .00000000001
> > >>
> > >> These are all reals greater than one... Now, if I define a normal unit,
> > >> say 1 / 42, then we can have a first one because I simply defined the
> > >> normal unit. Is not infinite anymore. See?
> > >>> This is called, for each "definition of continuity", a "completenes", all, a "repleteness".
> > >>>
> > >>> That is all, ....
> > >> We can plot a line in n-ary p0, and p1.
> > >>
> > >> dif = p1 - p0;
> > >>
> > >> Now we have our difference. We can define a normal, say:
> > >>
> > >> n = 2048;
> > >> normal_base = 1 / n;
> > >>
> > >> Then we can plot the points:
> > >>
> > >> for (i = 0; i < n; ++i)
> > >> {
> > >> normal = normal_base * i;
> > >> point = p0 + dif * normal;
> > >> plot_point(point);
> > >> }
> > >>
> > >>
> > >> Notice that as soon as we define n, its no longer infinite...
> > >
> > > What you do is you get a compass and you get an edge,
> > > and you throw away the compass, and get some chalk,
> > > then you mark down the edge, the chalk, and,
> > > you just wrote infinity points in a row.
> > >
> > >
> > > It's sort of similar with fixed point, it's just, you know,
> > > 1, with however deep a radix in all the storage there is.
> > >
> > > I mean, your real numbers have 1/3, right. They have n/d,
> > > for unsigned n, d. Then just let n go from zero to d,
> > > as sizeof(typeof(d)), or in accords with types, goes to infinity.
> > >
> > > It's called the equivalency function, its range has a sigma algebra.
> > >
> > > See also Ross Finlayson's 10,000's posts to sci.math,
> > > particularly his, "slates".
> > Just keep in mind that the concept of a "first" real number greater than
> > zero doesn't exist within the framework of real numbers...
> ... "Standard", "standard" real numbers, indeed, the entire idea of
> the "non-standard", real numbers, is these sorts things.
>
> Then it's what's called "continuous domains", and that on the "linear continuum",
> which are all, only, and everywhere real numbers, are these standard and non-standard,
> real numbers.
>
> For example if you look into Jordan measure it's for a line integral, in the line,
> and the Dirichlet function is, half the real numbers, even/odd, or alternating.
>
> Then, it's very simple, real simple, that n/d for n, d natural as n -> d, and d -> oo,
> results a model of a "not-a-real-function" a "not-a-standard-real-function",
> that for example has least upper bound property, very trivially,
> that for example "zooming" never reaches, because it's the ordered field,
> not the complete ordered field, which only follows axiomatization of least-upper-bound property.
> It has sigma algebras for emasure and LUB for gaplessness so the fundamental
> theorems of calculus follow, it's a continuous domain.
>
> Other "not-a-real-function's" of interest include the Dirac delta.
> (As modeled by real functions and only true in the limit.)
>
> So, in the wider framework of continuous domains, and real numbers,
> are three different sets of real numbers, or frameworks of functions,
> that make the equivalency function that is not a Cartesian function,
> a continuous domain, and, the usual standard complete ordered field,
> which is equivalence classes of sequences that are Cauchy, and for signal reals,
> functions in Dirichlet functions that subset the continuous domain while
> keeping rational analytical character.
>
> https://www.youtube.com/@rossfinlayson
>
>
> Then "line-drawing" is called "line-drawing" and it's a fundamental geometrical act.
>
> So, in line reals there is a first number after f(0) = 0, it's f(1), in "iota-values".
>
> It's real valued.
>
> ...

See, with such deliberate clarifications of definition, many of the intuitions of
posters like Mitch and WM can be corrected, illustrating framings where their
ideas of the objects of mathematics exist.

It takes really not so much, and indeed it's pretty simple, to arrive at these
mutually concordant definitions of repleteness in continuity.

That cranks like Mitch (the lesser, vis-a-vis stronger Mitch, fom, on sci.logic),
and WM the Hodges-hog, can simply have their definitions fixed and separated
from the usual and arrive at saying things that are actually true, is why it's
called "correcting papers" not just "marking mistakes".

So anyways there are at least three definitions of continuity, and there is
a first and central definition after line continuity, in the objects of mathematics,
where there's always a continuum.

Then, one of these models of real numbers, line reals, has each real number has
an integer part and a non-integer part, and the non-integer part ranges [0,1),
and it is a set of, "iota-values", which are real-valued, and valid in real-valued formulae.

Under my youtube's there's Philosophical Foreground, then "reading Weyl", is all standard,
then the next for "continuum infinitesimal analysis, after the first half details these things.

I.e., "Nonstandard analysis" for any kind of infinitesimals, then is for "continuum infinitesimal
analysis", nonstandard analysis with real analytical character.

--

I have a mathematics degree, and I earned one, too. "A defense of mathematics."

WM used his keyboard to write :
> zelos...@gmail.com schrieb am Montag, 8. Januar 2024 um 07:37:56 UTC+1:
>> On Saturday 6 January 2024 at 03:17:23 UTC+1, mitchr...@gmail.com wrote:
>> > zero is not a quantity so it would be absolute But above zero is first
>> quantity or infinitesimal of the calculus. That is sub finite.
>> doesn't exist you nimrod
>
> There are ℵ real numbers in every interval (0, eps) for every eps > 0 that
> you can choose.

At least that many, because there are that many as rational numbers.

> That proves that ℵ real numbers cannot be chosen individually
> but only collectively as I just described them. They are dark.

No it doesn't.

On Monday, January 8, 2024 at 3:06:10 PM UTC-8, Fritz Feldhase wrote:
> On Monday, January 8, 2024 at 1:29:37 PM UTC+1, WM wrote:
>
> > ℵ
>
> Kannst Du Trottel "ℵ" auch definieren?

Ordnungs sind nich Machtigkeits, under aber so.

Unendlichkeit, Umbegrentzheit, ..., mehre Worten machts
fur die Scalar Umbegrentzheit.

Mann macht nich die Wortschatz schlect.

The laws (of grammar) aren't all powerset, and vice versa.

Unending, unbounded, ..., more words make
for the scalar infinity.

One doesn't pollute the vocabulary.

On Monday, January 8, 2024 at 9:55:47 AM UTC-8, mitchr...@gmail.com wrote:
> On Sunday, January 7, 2024 at 3:19:25 PM UTC-8, Chris M. Thomasson wrote:
> > On 1/7/2024 2:34 PM, WM wrote:
> > > mitchr...@gmail.com schrieb am Samstag, 6. Januar 2024 um 03:17:23 UTC+1:
> > >> zero is not a quantity so it would be absolute
> > >> But above zero is first quantity or infinitesimal
> > >> of the calculus. That is sub finite.
> > >
> > > The first real numbers above zero are dark in that they can only be used collectively. (But some of them must be there below any defined point.) Moreover, between any two defined points on the real line there are ℵ dark points (if actual infinity exists) or nothing (if infinity is potential only).
> > >
> > There is no _first_ real number above zero.
> It is the infinitesimal above zero...

There are lots of kinds of notions of infinitesimals, just like infinities.

So, you are pointing to a particular kind, like "iota-values".

On 1/8/24 7:29 AM, WM wrote:
> Le 08/01/2024 à 13:21, Richard Damon a écrit :
>> On 1/8/24 5:56 AM, WM wrote:
>
>>> I understand that every real x > 0 that you can name has ℵ smaller
>>> positive reals ℵ of which you cannot name.
>>
>> why do you think I can not name any of them?
>
> You can name many of them, but every attempt fails to reduce the
> remainder below ℵ. It can be reduced however to the empty set collectively.
>
> Regards, WM
>

Why do you expect otherwise?

We have an infinite set.

Individually naming is a finite operation, so can only name a finite
number of them at a time.

Thus there will alway be an infinite number that we haven't produced the
name.

That doesn't mean the rest can't be named.

"naming" is the opposite of your "dark", you can only do it
individually, but not to an infinite group.

It doesn't mean that any particular ones left can't be named.

It seems your problem is you can't keep the right domain in focus.

Your "dark" numbers have some of the properties of the transfinite
numbers, except those are individually namable, and are not members of
the finite numbers.

On Monday 8 January 2024 at 13:23:14 UTC+1, WM wrote:
> zelos...@gmail.com schrieb am Montag, 8. Januar 2024 um 07:37:56 UTC+1:
> > On Saturday 6 January 2024 at 03:17:23 UTC+1, mitchr...@gmail.com wrote:
> > > zero is not a quantity so it would be absolute
> > > But above zero is first quantity or infinitesimal
> > > of the calculus. That is sub finite.
> > doesn't exist you nimrod
> There are ℵ real numbers in every interval (0, eps) for every eps > 0
> that you can choose. That proves that ℵ real numbers cannot be chosen
> individually but only collectively as I just described them. They are
> dark.
>
> Regards, WM
We have been through this time and time again, all of that "individually" is meaningless.

Le 09/01/2024 à 03:31, Richard Damon a écrit :
> On 1/8/24 7:29 AM, WM wrote:
>> Le 08/01/2024 à 13:21, Richard Damon a écrit :
>>> On 1/8/24 5:56 AM, WM wrote:
>>
>>>> I understand that every real x > 0 that you can name has ℵ smaller
>>>> positive reals ℵ of which you cannot name.
>>>
>>> why do you think I can not name any of them?
>>
>> You can name many of them, but every attempt fails to reduce the
>> remainder below ℵ. It can be reduced however to the empty set collectively.
>
> Why do you expect otherwise?

I don't. I only note that fact.
>
> We have an infinite set.
>
> Individually naming is a finite operation, so can only name a finite
> number of them at a time.

But if you could name every number, then you could name one with lesser
successors.
>
> Thus there will alway be an infinite number that we haven't produced the
> name.
>
> That doesn't mean the rest can't be named.

ℵ will always be without name. But they can be removed such that none
remains, collectively.
>
> "naming" is the opposite of your "dark", you can only do it
> individually, but not to an infinite group.

So it is. And not for the numbers belonging to the last ℵ.
>
> It doesn't mean that any particular ones left can't be named.

It does. The last ℵ numbers will remain.
>
> It seems your problem is you can't keep the right domain in focus.
>
> Your "dark" numbers have some of the properties of the transfinite
> numbers,

Yes, they are also dark because they have no finite initial segment of
natural numbers.

Regards, WM

On 1/9/2024 2:03 AM, zelos...@gmail.com wrote:
> On Monday 8 January 2024 at 13:23:14 UTC+1, WM wrote:
>> zelos...@gmail.com schrieb am Montag, 8. Januar 2024 um 07:37:56 UTC+1:
>>> On Saturday 6 January 2024 at 03:17:23 UTC+1, mitchr...@gmail.com wrote:
>>>> zero is not a quantity so it would be absolute
>>>> But above zero is first quantity or infinitesimal
>>>> of the calculus. That is sub finite.
>>> doesn't exist you nimrod
>> There are ℵ real numbers in every interval (0, eps) for every eps > 0
>> that you can choose. That proves that ℵ real numbers cannot be chosen
>> individually but only collectively as I just described them. They are
>> dark.
>>
>> Regards, WM
> We have been through this time and time again, all of that "individually" is meaningless.

The parrot is dead! ;^)

On Tuesday, January 9, 2024 at 3:02:01 PM UTC-8, Chris M. Thomasson wrote:
> On 1/9/2024 2:03 AM, zelos...@gmail.com wrote:
> > On Monday 8 January 2024 at 13:23:14 UTC+1, WM wrote:
> >> zelos...@gmail.com schrieb am Montag, 8. Januar 2024 um 07:37:56 UTC+1:
> >>> On Saturday 6 January 2024 at 03:17:23 UTC+1, mitchr...@gmail.com wrote:
> >>>> zero is not a quantity so it would be absolute
> >>>> But above zero is first quantity or infinitesimal
> >>>> of the calculus. That is sub finite.
> >>> doesn't exist you nimrod
> >> There are ℵ real numbers in every interval (0, eps) for every eps > 0
> >> that you can choose. That proves that ℵ real numbers cannot be chosen
> >> individually but only collectively as I just described them. They are
> >> dark.
> >>
> >> Regards, WM
> > We have been through this time and time again, all of that "individually" is meaningless.
> The parrot is dead! ;^)

It's just resting, ....

....

On 1/9/2024 3:23 PM, Ross Finlayson wrote:
> On Tuesday, January 9, 2024 at 3:02:01 PM UTC-8, Chris M. Thomasson wrote:
>> On 1/9/2024 2:03 AM, zelos...@gmail.com wrote:
>>> On Monday 8 January 2024 at 13:23:14 UTC+1, WM wrote:
>>>> zelos...@gmail.com schrieb am Montag, 8. Januar 2024 um 07:37:56 UTC+1:
>>>>> On Saturday 6 January 2024 at 03:17:23 UTC+1, mitchr...@gmail.com wrote:
>>>>>> zero is not a quantity so it would be absolute
>>>>>> But above zero is first quantity or infinitesimal
>>>>>> of the calculus. That is sub finite.
>>>>> doesn't exist you nimrod
>>>> There are ℵ real numbers in every interval (0, eps) for every eps > 0
>>>> that you can choose. That proves that ℵ real numbers cannot be chosen
>>>> individually but only collectively as I just described them. They are
>>>> dark.
>>>>
>>>> Regards, WM
>>> We have been through this time and time again, all of that "individually" is meaningless.
>> The parrot is dead! ;^)
>
> It's just resting, ....
>
> ...

LOL!!! Its stunned! ROFL!!!!