On Tuesday, January 9, 2024 at 6:21:38 PM UTC-8, mitchr...@gmail.com wrote:
> On Tuesday, January 9, 2024 at 4:31:22 PM UTC-8, Chris M. Thomasson wrote:
> > 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
> Above zero is first non zero fundamental infinitesimal.

It's "nonstandard real", but, "standard infinitesimal".

On 1/9/24 12:23 PM, WM wrote:
> 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.

Why?

If you could name a number with a finite number of points after it, you
could find the last number, but since no number is the last, you can't
do that.

There is no problem with having infinite sets that don't have an end,
that is just the nature of unbounded numbers.

>>
>> 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.

Nope, they all have a name, we just can't express them all at once.

>>
>> "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.

No, we didn't "use up" the names, so we still have names for them all.

>>
>> It doesn't mean that any particular ones left can't be named.
>
> It does. The last ℵ numbers will remain.

Nope. I gave the formula to name any of them individually.

>>
>> 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
>

Nope. Take ANY of your "dark numbers", if is IS finite, then make you
finite initail segment starting with the number + 1.

If it isn't finite, then it wasn't one of the Natural Numbers after the
last one you previously named.

So, you logic is just broken, because it can't handle unbounded sets.

EVERY element of the Natural Numbers is a FINITE number, and thus has a
FINITE name, and is namable.

They are unbounded, so there is no "last" number.

They set of them is INFINITE, so it isn't surprizing that we can't name
them all a once (or even more than just a few comparatively). That is
just the nature of infinite sets, and any logic that doesn't take that
into account is just incapable of working with such sets.

On 1/9/2024 6:53 PM, Ross Finlayson wrote:
> On Tuesday, January 9, 2024 at 6:21:38 PM UTC-8, mitchr...@gmail.com wrote:
>> On Tuesday, January 9, 2024 at 4:31:22 PM UTC-8, Chris M. Thomasson wrote:
>>> 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
>> Above zero is first non zero fundamental infinitesimal.
>
> It's "nonstandard real", but, "standard infinitesimal".
>

infinitesimal is small, very small. So small that it get smaller before
you can observe it... ;^) lol. Just some jest.

Le 10/01/2024 à 03:58, Richard Damon a écrit :
> On 1/9/24 12:23 PM, WM wrote:
>> 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.
>
> Why?

Because they are existing.
>
> If you could name a number with a finite number of points after it, you
> could find the last number, but since no number is the last, you can't
> do that.

There is a smallest unit fraction with no dount because
∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0
is valid for all unit fractions.
But you cannot name it.
>
> There is no problem with having infinite sets that don't have an end,
> that is just the nature of unbounded numbers.

It is the nature of all visible numbers.
>>
>> ℵ will always be without name. But they can be removed such that none
>> remains, collectively.
>
> Nope, they all have a name, we just can't express them all at once.

You can remove all. But you cannot name those with less than ℵo
successors. So you cannot name all.
>
>>>
>>> "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.
>
> No, we didn't "use up" the names, so we still have names for them all.

Apply them. Nevertheless ℵo will remain without name.
>
>>> It doesn't mean that any particular ones left can't be named.
>>
>> It does. The last ℵ numbers will remain.
>
> Nope. I gave the formula to name any of them individually.

But you cannot give the names.

Regards, WM

mitchr...@gmail.com schrieb am Mittwoch, 10. Januar 2024 um 03:21:38
UTC+1:

> Above zero is first non zero fundamental infinitesimal.

With no doubt, there is a first unit fraction because a liner chain with
gaps has a first element:
∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.

This 1/n is not infinitesimal like its n is not infinite. But it is dark.
The smallest ℵo unit fractions and the largest ℵo natural numbers
cannot be used as individuals. Every defined positive real number does not
belong to the smallest ℵo unit fractions or to the largest ℵo natural
numbers.

Regards, WM

Le 08/01/2024 à 17:37, FromTheRafters a écrit :
> WM used his keyboard to write :

>> 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.

Every defined number does not belong to the domain covered by the smallest
ℵo unit fractions or by the largest ℵo natural numbers.

Regards, WM

On 1/10/24 4:51 PM, WM wrote:
> Le 10/01/2024 à 03:58, Richard Damon a écrit :
>> On 1/9/24 12:23 PM, WM wrote:
>>> 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.
>>
>> Why?
>
> Because they are existing.

What Natural number exists with only a finite number of numbers above it?

Finite sets are always namable, so you should be able to name it.

>>
>> If you could name a number with a finite number of points after it,
>> you could find the last number, but since no number is the last, you
>> can't do that.
>
> There is a smallest unit fraction with no dount because
> ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0
> is valid for all unit fractions.
> But you cannot name it.

That doesn't say there is a smallest number, after all, for every one of
these n, the gap below it is smaller than its spacing from 0, and in
fact there is obviously room for at least n more unit fractions below 1/n.

>>
>> There is no problem with having infinite sets that don't have an end,
>> that is just the nature of unbounded numbers.
>
> It is the nature of all visible numbers.

Really? says what principle?

>>>
>>> ℵ will always be without name. But they can be removed such that none
>>> remains, collectively.
>>
>> Nope, they all have a name, we just can't express them all at once.
>
> You can remove all. But you cannot name those with less than ℵo
> successors. So you cannot name all.

But none of them exist, so I don't need to name them.

You of course can't name that which doesn't exist.

I guess your "dark" numbers are just figments of your imagination.

>>
>>>>
>>>> "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.
>>
>> No, we didn't "use up" the names, so we still have names for them all.
>
> Apply them. Nevertheless ℵo will remain without name.

Where did I stop naming them?

>>
>>>> It doesn't mean that any particular ones left can't be named.
>>>
>>> It does. The last ℵ numbers will remain.
>>
>> Nope. I gave the formula to name any of them individually.
>
> But you cannot give the names.

But I did.
>
> Regards, WM

On 1/10/2024 3:30 PM, mitchr...@gmail.com wrote:
> On Tuesday, January 9, 2024 at 11:48:00 PM UTC-8, Chris M. Thomasson wrote:
>> On 1/9/2024 6:53 PM, Ross Finlayson wrote:
>>> On Tuesday, January 9, 2024 at 6:21:38 PM UTC-8, mitchr...@gmail.com wrote:
>>>> On Tuesday, January 9, 2024 at 4:31:22 PM UTC-8, Chris M. Thomasson wrote:
>>>>> 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
>>>> Above zero is first non zero fundamental infinitesimal.
>>>
>>> It's "nonstandard real", but, "standard infinitesimal".
>>>
>> infinitesimal is small, very small. So small that it get smaller before
>> you can observe it... ;^) lol. Just some jest.
>
> There is no smaller than inverse infinity.
> Zero has no size at all. It is not a smaller.
> As small it doesn't exist.

Small as is something that is greater than zero, but still small... Tack
on a negative sign, and is still small, or close to zero so to speak...
;^D lol.

By the time you take a look at .01, well, it went to .0000001,
..00000000000000000000001, on and on:

https://youtu.be/oRrlRbGT-LU

On 1/10/2024 1:51 PM, WM wrote:
> Le 10/01/2024 à 03:58, Richard Damon a écrit :
>> On 1/9/24 12:23 PM, WM wrote:
>>> 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.
>>
>> Why?
>
> Because they are existing.
>>
>> If you could name a number with a finite number of points after it,
>> you could find the last number, but since no number is the last, you
>> can't do that.
>
> There is a smallest unit fraction with no dount because

Are you sure about that? There is no smallest unit fraction...

> ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0
> is valid for all unit fractions.
> But you cannot name it.
>>
>> There is no problem with having infinite sets that don't have an end,
>> that is just the nature of unbounded numbers.
>
> It is the nature of all visible numbers.
>>>
>>> ℵ will always be without name. But they can be removed such that none
>>> remains, collectively.
>>
>> Nope, they all have a name, we just can't express them all at once.
>
> You can remove all. But you cannot name those with less than ℵo
> successors. So you cannot name all.
>>
>>>>
>>>> "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.
>>
>> No, we didn't "use up" the names, so we still have names for them all.
>
> Apply them. Nevertheless ℵo will remain without name.
>>
>>>> It doesn't mean that any particular ones left can't be named.
>>>
>>> It does. The last ℵ numbers will remain.
>>
>> Nope. I gave the formula to name any of them individually.
>
> But you cannot give the names.
>
> 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.
>
> ...

In the above that's "measure" not "emasure",
wouldn't want it to be mistaken at all.

I intentionally spell my own words,
and know them, any mistakes are my own,
and, there aren't any.

Here then let's talk about mathematics
where light" and "dark" are simply complements,
because worldly concerns, while matters of
Man and Mind and Animal and Machine,
are only peripheral to the theory of mathematics.

Of course then mathematics upholds mathematical sciences,
according to philosophy of science.

(That "dark matter" contra "luminous matter" is just a non-scientific
placeholder in modern cosmology is matters of metaphor.)

Warm regards, and for good will toward Man, and for that
"I am not my brother's keeper, I am my brother's brother."

(Here Man is the gender neutral nominative, not the neutered androgenous,
and the filial and fraternal and paternal reflects relation and origin.)

Universe is one big dot

Le 11/01/2024 à 01:49, Richard Damon a écrit :

> What Natural number exists with only a finite number of numbers above it?
>
> Finite sets are always namable, so you should be able to name it.

Not within the dark domain.

Regards, WM

On 1/11/24 4:33 PM, WM wrote:
> Le 11/01/2024 à 01:49, Richard Damon a écrit :
>
>> What Natural number exists with only a finite number of numbers above it?
>>
>> Finite sets are always namable, so you should be able to name it.
>
> Not within the dark domain.
>
> Regards, WM

Why?

I guess this shows that dark numbers aren't actually numbers.

It seems your problem is just that your logic system blew up on you from
misuse, and you can't see what happened.

Le 12/01/2024 à 04:11, Richard Damon a écrit :
> On 1/11/24 4:33 PM, WM wrote:
>> Le 11/01/2024 à 01:49, Richard Damon a écrit :
>>
>>> What Natural number exists with only a finite number of numbers above it?
>>>
>>> Finite sets are always namable, so you should be able to name it.
>>
>> Not within the dark domain.
>
> Why?
>
You must try it. The last ℵo natural numbers and the first ℵo unit
fractions cannot be subdivided.

Regards, WM

On 1/12/2024 6:17 AM, WM wrote:
> Le 12/01/2024 à 04:11, Richard Damon a écrit :
>> On 1/11/24 4:33 PM, WM wrote:
>>> Le 11/01/2024 à 01:49, Richard Damon a écrit :
>>>
>>>> What Natural number exists with only a finite number of numbers
>>>> above it?
>>>>
>>>> Finite sets are always namable, so you should be able to name it.
>>>
>>> Not within the dark domain.
>>
>> Why?
>>
> You must try it. The last ℵo natural numbers and the first ℵo unit
> fractions cannot be subdivided.

There are no last natural numbers because they go on forever.

On 1/12/24 9:17 AM, WM wrote:
> Le 12/01/2024 à 04:11, Richard Damon a écrit :
>> On 1/11/24 4:33 PM, WM wrote:
>>> Le 11/01/2024 à 01:49, Richard Damon a écrit :
>>>
>>>> What Natural number exists with only a finite number of numbers
>>>> above it?
>>>>
>>>> Finite sets are always namable, so you should be able to name it.
>>>
>>> Not within the dark domain.
>>
>> Why?
>>
> You must try it. The last ℵo natural numbers and the first ℵo unit
> fractions cannot be subdivided.
>
> Regards, WM
>
>

Why not?

WHere is the actual property derivable from the actual definition of
Natural Numbers that creates these non-subdividable numbers?

I think it is all just in your head, caused by using faulty logic.

On 1/12/2024 3:46 PM, Richard Damon wrote:
> On 1/12/24 9:17 AM, WM wrote:
>> Le 12/01/2024 à 04:11, Richard Damon a écrit :
>>> On 1/11/24 4:33 PM, WM wrote:
>>>> Le 11/01/2024 à 01:49, Richard Damon a écrit :
>>>>
>>>>> What Natural number exists with only a finite number of numbers
>>>>> above it?
>>>>>
>>>>> Finite sets are always namable, so you should be able to name it.
>>>>
>>>> Not within the dark domain.
>>>
>>> Why?
>>>
>> You must try it. The last ℵo natural numbers and the first ℵo unit
>> fractions cannot be subdivided.
>>
>> Regards, WM
>>
>>
>
> Why not?
>
> WHere is the actual property derivable from the actual definition of
> Natural Numbers that creates these non-subdividable numbers?
>
> I think it is all just in your head, caused by using faulty logic.

Faulty, cryptic, negligence, he is a teacher, right? Yikes!!!!

On 1/10/2024 1:59 PM, WM wrote:
> mitchr...@gmail.com schrieb am Mittwoch, 10. Januar 2024 um 03:21:38 UTC+1:
>
>> Above zero is first non zero fundamental infinitesimal.
>
> With no doubt, there is a first unit fraction because a liner chain with
> gaps has a first element:
> ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.
>
> This 1/n is not infinitesimal like its n is not infinite. But it is
> dark. The smallest ℵo unit fractions and the largest ℵo natural numbers
> cannot be used as individuals. Every defined positive real number does
> not belong to the smallest ℵo unit fractions or to the largest ℵo
> natural numbers.
>
1/1 is first, try to find a last smaller one...

On 1/12/2024 9:14 PM, Chris M. Thomasson wrote:
> On 1/10/2024 1:59 PM, WM wrote:
>> mitchr...@gmail.com schrieb am Mittwoch, 10. Januar 2024 um 03:21:38
>> UTC+1:
>>
>>> Above zero is first non zero fundamental infinitesimal.
>>
>> With no doubt, there is a first unit fraction because a liner chain
>> with gaps has a first element:
>> ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.
>>
>> This 1/n is not infinitesimal like its n is not infinite. But it is
>> dark. The smallest ℵo unit fractions and the largest ℵo natural
>> numbers cannot be used as individuals. Every defined positive real
>> number does not belong to the smallest ℵo unit fractions or to the
>> largest ℵo natural numbers.
>>
> 1/1 is first, try to find a last smaller one...

The smallest one is trying to feel you:

https://youtu.be/7SSzpcI9Yzc

lol! Keep in mind that there is no smallest one.

Le 13/01/2024 à 00:46, Richard Damon a écrit :
> On 1/12/24 9:17 AM, WM wrote:
>> Le 12/01/2024 à 04:11, Richard Damon a écrit :
>>> On 1/11/24 4:33 PM, WM wrote:
>>>> Le 11/01/2024 à 01:49, Richard Damon a écrit :
>>>>
>>>>> What Natural number exists with only a finite number of numbers
>>>>> above it?
>>>>>
>>>>> Finite sets are always namable, so you should be able to name it.
>>>>
>>>> Not within the dark domain.
>>>
>>> Why?
>>>
>> You must try it. The last ℵo natural numbers and the first ℵo unit
>> fractions cannot be subdivided.
>
> Why not?

Because otherwise the last one could be identified. But it proves
impossible like the splitting of a quark. There remain always ℵ
elements. Therefore they are called dark.
>
> WHere is the actual property derivable from the actual definition of
> Natural Numbers that creates these non-subdividable numbers?

The axioms only create the definable natnumbers.
>
> I think it is all just in your head, caused by using faulty logic.

It is caused by your inability to reduce the amounts to less than ℵ.

Regards, WM

On 1/13/24 5:23 AM, WM wrote:
> Le 13/01/2024 à 00:46, Richard Damon a écrit :
>> On 1/12/24 9:17 AM, WM wrote:
>>> Le 12/01/2024 à 04:11, Richard Damon a écrit :
>>>> On 1/11/24 4:33 PM, WM wrote:
>>>>> Le 11/01/2024 à 01:49, Richard Damon a écrit :
>>>>>
>>>>>> What Natural number exists with only a finite number of numbers
>>>>>> above it?
>>>>>>
>>>>>> Finite sets are always namable, so you should be able to name it.
>>>>>
>>>>> Not within the dark domain.
>>>>
>>>> Why?
>>>>
>>> You must try it. The last ℵo natural numbers and the first ℵo unit
>>> fractions cannot be subdivided.
>>
>> Why not?
>
> Because otherwise the last one could be identified. But it proves
> impossible like the splitting of a quark. There remain always ℵ
> elements. Therefore they are called dark.

But there is no difference between the "dark" numbers and the identified
numbers.

"The Last" in non-existant, so if that is what you mean by it, you are
lying as there are ZERO dark numbers, not Aleph_0 of them.

>>
>> WHere is the actual property derivable from the actual definition of
>> Natural Numbers that creates these non-subdividable numbers?
>
> The axioms only create the definable natnumbers.

But since the axioms create ALL the Natural Numbers, you are just
admitting that your "Dark Numbers" are not part of them.

So, you are just admitting you are just a liar. Your Dark numbers are
defined to be part of the Natural Numbers, but are not created by the
method that creates the Natural Numbers, so are not.

>>
>> I think it is all just in your head, caused by using faulty logic.
>
> It is caused by your inability to reduce the amounts to less than ℵ.

And why do I need to?

That seems to be your problem, you are assuming you can do what isn't
allowed, and therefore make up something that doesn't exist to attempt it.

>
> Regards, WM
>
>

On Saturday, January 13, 2024 at 8:56:42 AM UTC-8, Fritz Feldhase wrote:
> On Saturday, January 13, 2024 at 2:22:59 PM UTC+1, Richard Damon wrote:
> > On 1/13/24 5:23 AM, WM wrote:
> > >
> > > The axioms only create the definable natnumbers.
> Nonsense. Axioms don't "create" any numbers.
> > But since the axioms create ALL the Natural Numbers
> Nonsense. Axioms don't "create" any numbers.
>
> Though the axiom of infinity (in ZFC) ensures for the existence of an infinite set (which is the basis for the set IN of natural numbers).

You mean like axiomatizing a well-founded infinite inductive set?

It sort of has its reasoning, ....

It has its reasoning that it would be an inconsistent multiplicity
to simply build one logically with quantification.

Number theorists, for example, have a point at infinity. Geometry,
for example, has a point at infinity. It's not in all theories, but,
it is in the theory.

I hope you can recognize what would be your own hypocrisy otherwise,
that "a well-founded infinite inductive set", is a non-logical constant,
and as of a restriction of comprehension, sort of like "a unique empty set"..

You know, for not being wrong, ....

You know what else are objects in Peano's theories? Infinitesimals.

It seems Russell sort of laughs at Frege's honesty, and having duped most,
it's sort of insufferable. Sort of like a whimsical fairy.

The "objects of mathematics" include ordinary and extra-ordinary inductive sets,
with/without "no infinite descending epsilon chains".

When you read Cohen's Continuum Hypothesis Independence proof,
you're going to be surprised, what he finds, from what was ZFC.

On Saturday, January 13, 2024 at 10:11:02 AM UTC-8, Ross Finlayson wrote:
> On Saturday, January 13, 2024 at 8:56:42 AM UTC-8, Fritz Feldhase wrote:
> > On Saturday, January 13, 2024 at 2:22:59 PM UTC+1, Richard Damon wrote:
> > > On 1/13/24 5:23 AM, WM wrote:
> > > >
> > > > The axioms only create the definable natnumbers.
> > Nonsense. Axioms don't "create" any numbers.
> > > But since the axioms create ALL the Natural Numbers
> > Nonsense. Axioms don't "create" any numbers.
> >
> > Though the axiom of infinity (in ZFC) ensures for the existence of an infinite set (which is the basis for the set IN of natural numbers).
> You mean like axiomatizing a well-founded infinite inductive set?
>
> It sort of has its reasoning, ....
>
> It has its reasoning that it would be an inconsistent multiplicity
> to simply build one logically with quantification.
>
> Number theorists, for example, have a point at infinity. Geometry,
> for example, has a point at infinity. It's not in all theories, but,
> it is in the theory.
>
>
> I hope you can recognize what would be your own hypocrisy otherwise,
> that "a well-founded infinite inductive set", is a non-logical constant,
> and as of a restriction of comprehension, sort of like "a unique empty set".
>
> You know, for not being wrong, ....
>
>
> You know what else are objects in Peano's theories? Infinitesimals.
>
> It seems Russell sort of laughs at Frege's honesty, and having duped most,
> it's sort of insufferable. Sort of like a whimsical fairy.
>
> The "objects of mathematics" include ordinary and extra-ordinary inductive sets,
> with/without "no infinite descending epsilon chains".
>
> When you read Cohen's Continuum Hypothesis Independence proof,
> you're going to be surprised, what he finds, from what was ZFC.

https://www.youtube.com/watch?v=njSqmjkj0gQ&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=7

Now, when I say "Thomas Hardy" I pretty much always mean G.H. Hardy who
wrote "A Course of Pure Mathematics", kind of like when I say "Ernst" for "Kurt", Goedel.

Also I'm pretty sure "didactive" wasn't a word, after "didactic", but, maybe so, ....

"Diffraction", that's a tough one. "Diff-Fraction."

Le 13/01/2024 à 14:22, Richard Damon a écrit :
> On 1/13/24 5:23 AM, WM wrote:

>> The axioms only create the definable natnumbers.
>
> But since the axioms create ALL the Natural Numbers, you are just
> admitting that your "Dark Numbers" are not part of them.

No, the matter is not so easy. Peano's axioms create the natural numbers,
a potentially infinite sequence or collection, not a set.Only Zermelo's
axioms, which are adopted from Peano or Dedekind, create all natural
numbers because in set theory there is the complete set.
>>
>> It is caused by your inability to reduce the amounts to less than ℵ.
>
> And why do I need to?

In order to prove that you could.
>
> That seems to be your problem, you are assuming you can do what isn't
> allowed,

It is not only disallowed (a revolutionary spirit would violate this
prohibition), but it is impossible. Why? Because it is impossible. I call
that dark. But the notion is irrelevant. The fact is what counts (and
hinders the counting).

Regards, WM

On 1/14/24 7:35 AM, WM wrote:
> Le 13/01/2024 à 14:22, Richard Damon a écrit :
>> On 1/13/24 5:23 AM, WM wrote:
>
>>> The axioms only create the definable natnumbers.
>>
>> But since the axioms create ALL the Natural Numbers, you are just
>> admitting that your "Dark Numbers" are not part of them.
>
> No, the matter is not so easy. Peano's axioms create the natural
> numbers, a potentially infinite sequence or collection, not a set.Only
> Zermelo's axioms, which are adopted from Peano or Dedekind, create all
> natural numbers because in set theory there is the complete set.

Really? The Peano Axioms (at least one formulation of them)

1. Zero is a number.

2. If a is a number, the successor of a is a number.

3. zero is not the successor of a number.

4. Two numbers of which the successors are equal are themselves equal.

5. If a set S of numbers contains zero and also the successor of every
number in S, then every number is in S.

Seems that Axiom 5 generates the set.

Perhaps you are thinking of Peano Arithmetic, which remove the last
Axiom (the Axiom of Induction, which is second order) with some first
order logic rules.

>>>
>>> It is caused by your inability to reduce the amounts to less than ℵ.
>>
>> And why do I need to?
>
> In order to prove that you could.

Why do I need to? My claim, and that of the mathematics is that it is an
infinite set, and no infinite set can have a finite number of elements
removed and result in a finite set.

>>
>> That seems to be your problem, you are assuming you can do what isn't
>> allowed,
>
> It is not only disallowed (a revolutionary spirit would violate this
> prohibition), but it is impossible. Why? Because it is impossible. I
> call that dark. But the notion is irrelevant. The fact is what counts
> (and hinders the counting).
>
> Regards, WM
>
>

So, your logic is based on the concept that since we can't divide an
infinite set into two finite sets, there must be elements that we can
not name in the set?

Where do you get that idea from?

I think you have the wrong definition of "infinity" or Unbounded.

Le 14/01/2024 à 19:42, Richard Damon a écrit :
> On 1/14/24 7:35 AM, WM wrote:

>>> That seems to be your problem, you are assuming you can do what isn't
>>> allowed,
>>
>> It is not only disallowed (a revolutionary spirit would violate this
>> prohibition), but it is impossible. Why? Because it is impossible. I
>> call that dark. But the notion is irrelevant. The fact is what counts
>> (and hinders the counting).

> So, your logic is based on the concept that since we can't divide an
> infinite set into two finite sets, there must be elements that we can
> not name in the set?

There *are* elements we cannot name, even almost all, namely ℵ, whereas
we can name only finitely many.

Regards, WM