On 1/26/24 4:48 AM, WM wrote:
> Le 25/01/2024 à 14:06, FromTheRafters a écrit :
>> WM formulated the question :
>>> Le 25/01/2024 à 13:22, Richard Damon a écrit :
>>>> On 1/25/24 6:15 AM, WM wrote:
>>>>> Le 25/01/2024 à 02:05, Richard Damon a écrit :
>>>>>> On 1/24/24 10:46 AM, WM wrote:
>>>>>
>>>>>>> They end at zero. Therefore they begin above zero.
>>>>>>
>>>>>> But there is no "begin" as the values are dense there.
>>>>>
>>>>> They are not dense but all have distances.
>>>>
>>>> Not the Reals or the Rationals.
>>>
>>> We talk about unit fractions.
>>
>> As embedded in the reals.
>
> Irrelevant for the statement to be wrong.
>
> Regards, WM
>
>

But you still don't understand that there is no smallest unit fraction,
so you can't start counting at it.

You talk about all have distance BETWEEN them, but "first" isn't between
two of them.

I have shown that for ANY value 1/n, there is room for at least n more
unit fractions below it, so your "finite gap" idea doesn't show there
must be an smallest, but that there CAN'T be a smallest.

You are just using bad logic, because you logic doesn't actually have
anything to reason with.

Fritz Feldhase schrieb am Freitag, 26. Januar 2024 um 13:11:55 UTC+1:
> On Friday, January 26, 2024 at 10:52:07 AM UTC+1, WM wrote:
> > Le 25/01/2024 à 15:57, Jim Burns a écrit :
> > >
> > > ∀x ∈ (0,1]: NUF(x) = ℵ₀
> > >
> > ℵ₀ unit fractions [...], do not fit between 0 and every x > 0.
>
> Hint:
>
> ∀ x > 0: E^ℵ₀ u ∈ UF: 0 < u < x (true).
>
> E^ℵ₀ u ∈ UF: ∀ x > 0: 0 < u < x (false).
>
> The correct order of quantifiers is essential.

Not for true statements like this:

∀ x > 0, ∃^oo y < 0: y < x .
∃^oo y < 0, ∀ x > 0: y < x .

> (Hint: We are dealing with two completely different claims here.)
>
Not if the first statement was true. But it is not.
You cannot *find* an x > 0 withput less unit fractions.
If there *were* no x > 0 with less unit fractions, then the reversal could
be done.

Regards, WM

On Friday, January 26, 2024 at 4:35:29 AM UTC-8, Richard Damon wrote:
> On 1/26/24 4:48 AM, WM wrote:
> > Le 25/01/2024 à 14:06, FromTheRafters a écrit :
> >> WM formulated the question :
> >>> Le 25/01/2024 à 13:22, Richard Damon a écrit :
> >>>> On 1/25/24 6:15 AM, WM wrote:
> >>>>> Le 25/01/2024 à 02:05, Richard Damon a écrit :
> >>>>>> On 1/24/24 10:46 AM, WM wrote:
> >>>>>
> >>>>>>> They end at zero. Therefore they begin above zero.
> >>>>>>
> >>>>>> But there is no "begin" as the values are dense there.
> >>>>>
> >>>>> They are not dense but all have distances.
> >>>>
> >>>> Not the Reals or the Rationals.
> >>>
> >>> We talk about unit fractions.
> >>
> >> As embedded in the reals.
> >
> > Irrelevant for the statement to be wrong.
> >
> > Regards, WM
> >
> >
> But you still don't understand that there is no smallest unit fraction,
> so you can't start counting at it.
>
> You talk about all have distance BETWEEN them, but "first" isn't between
> two of them.
>
> I have shown that for ANY value 1/n, there is room for at least n more
> unit fractions below it, so your "finite gap" idea doesn't show there
> must be an smallest, but that there CAN'T be a smallest.
>
> You are just using bad logic, because you logic doesn't actually have
> anything to reason with.

Instead what you do is model a function as an infinite limit of functions,
the numerators over the denominators in the naturals,
so it arrives that there's

extent [0,1]
density [0,1]
completeness [0,1]
measure [0,1]

and then the values from zero to one are called "iota-values",
which is nice because iota already has the natural meaning
"infinitesimal", and probably since Cavalieri in the time of Galileo,
it's Aristotle's continuum, then for re-Vitali-izing measure theory
with doubling-spaces among continuous domains.

https://en.wikipedia.org/wiki/Bonaventura_Cavalieri

De Santillana has a great passage about Galileo in the
chapter "The Years of Silence".

Page 157:

"Through all this he was collecting himself. As the diver descending into
the dark depths feels his strength rise from the burning of hix oxygen under
pressure, so the scientist's mind, as it receded into isolation, burned brighter
to itself. The past compromises and loose adjustments of though discarded,
Galileo seems to be facing more decisively the full implications if his theory.

It is an inhuman wrench for the mind to tear itself loose from the instinctive
world of the senses around us, so "full of a number of things", peopled with
familiar substances which reassuringly go about subsisting in their own
several ways as if they knew how -- and to uncover instead everywhere the
mysterious realization of abstract laws; to conceive of mathematical function
determining each point of being in a dizzying tension between the infinite
and the infinitesimal, where the lonely intellect has to feel its way with new
instruments of rigorous analysis. It implies a dedication; a new feeling not
only of nature but also of the divine. When the time for expression comes again,
the quiet prose of the physicist will rise of itself to compelling heights, spark
and crackle with contained metaphysical power. But, for many years yet,
silence was in order."

-- de Santillana, 1954

I read this yesterday and really identified, it reminded me one time walking the
quad with a fellow and a professor, and I said "I've been studying infinity"
and he said "studying infinity makes people nuts" and I said "ayup".

Page 156:

"The epistolary activity of Galileo in those years was immense, for he kept
in constant touch with his disciples, whose fame was growing, and he was
never weary of assisting their thought in a vast range of subjects. Young
Father Cavalieri had taken up from him the study of infinitesimals, from which
was to come his _Geometry of Indivisibles_."

Yeah, here's those are just called "standard infinitesimals: iota-values".
There are iota-sums and iota-multiples and they're basically different
in arithmetic.

It's very geometrical and arithmetical, of course.

Its "base" is very simple: n/d, n->d, d-> oo.
It sort of precedes there being a gapless domain.

Le 26/01/2024 à 13:31, Richard Damon a écrit :
> On 1/26/24 5:07 AM, WM wrote:

> I never said the distance goes becomes 0,

Then NUF cannot increase by more than 1 at a point.

> Below ANY unit fraction are more, so there is no smallest unit fraction.

If the function can't start, it cannot count.

> And all those smaller are definable, so don't need to be "dark"

There is no point in (0, 1] without smaller unit fractions. Hence, they
cannot sit in (0, 1] because every point there is at the right-hand side.
Even if they all would fit into one point, there is none in (0, 1] to
accomodate them.
>
> Point of note, you claim that there is only "One Mathematics", because
> you seem to be a "Naturalist" for mathematics, using the properties
> observed in ordinary math.

Of course.
>
> The problem with this method is that it can only define the mathematics
> for the numbers we "see", and thus can't HAVE "dark" numbers.

Nevertheless I have proved their existence under the assumption that the
real line is complete without gaps. More cannot be done.
>
> Your "dark" numbers are just numbers that you haven't discovered the
> actual properties that make them do what you see happening.

Most cannot be disco vered as individuals because otherwise you could find
the smallest unit fraction.
>
> And the problem is that YOU apparently can't see the nature of unbounded
> sets, and thus you can't use them in this mathematics.

The sets of unit fractions and hence of natural numbers are bounded. With
them also all other sets of numbers are bounded. Infinity means only the
fact that the end is invisible and that visible numbers are potentially
infinite.

Regards, WM

Le 26/01/2024 à 13:32, Richard Damon a écrit :
> On 1/26/24 5:05 AM, WM wrote:

>>
>> NUF counts the unit fractions.
>
> And thus jumps to aleph0 for all x > 0 since there is no smallest unit
> fraction.

Wrong by mathematics. It cannot jump before (0, 1] because there are no
unit fractions and it can only jump by 1 because all unit fractions have
distances.
>
>>>
>>> So, your "dark" numbers are numbers that can't be described as a set,
>>> but only individually.
>>
>> The other way round.
>
> So you say, except that you have shown that you can't actually define
> them as a set, but at least some of them have individual descriptions,
> they are the points x where NUF(x) has the various finite values.

These points have no descriptions and can never be described.

Regards, WM

Le 26/01/2024 à 13:35, Richard Damon a écrit :
> On 1/26/24 4:48 AM, WM wrote:
> But you still don't understand that there is no smallest unit fraction,
> so you can't start counting at it.

Real points on the real line must be existing.

> You talk about all have distance BETWEEN them, but "first" isn't between
> two of them.

It is one of them.
>
> I have shown that for ANY value 1/n, there is room for at least n more
> unit fractions below it, so your "finite gap" idea doesn't show there
> must be an smallest, but that there CAN'T be a smallest.

That holds for definable n only.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
You have never exhausted the set. But that is possible, collectively:
|ℕ \ {1, 2, 3, ...}| = 0

> You are just using bad logic, because you logic doesn't actually have
> anything to reason with.

I am avoiding matheology.

Regards, WM

On Friday, January 26, 2024 at 9:46:29 AM UTC-8, Ross Finlayson wrote:
> On Friday, January 26, 2024 at 4:35:29 AM UTC-8, Richard Damon wrote:
> > On 1/26/24 4:48 AM, WM wrote:
> > > Le 25/01/2024 à 14:06, FromTheRafters a écrit :
> > >> WM formulated the question :
> > >>> Le 25/01/2024 à 13:22, Richard Damon a écrit :
> > >>>> On 1/25/24 6:15 AM, WM wrote:
> > >>>>> Le 25/01/2024 à 02:05, Richard Damon a écrit :
> > >>>>>> On 1/24/24 10:46 AM, WM wrote:
> > >>>>>
> > >>>>>>> They end at zero. Therefore they begin above zero.
> > >>>>>>
> > >>>>>> But there is no "begin" as the values are dense there.
> > >>>>>
> > >>>>> They are not dense but all have distances.
> > >>>>
> > >>>> Not the Reals or the Rationals.
> > >>>
> > >>> We talk about unit fractions.
> > >>
> > >> As embedded in the reals.
> > >
> > > Irrelevant for the statement to be wrong.
> > >
> > > Regards, WM
> > >
> > >
> > But you still don't understand that there is no smallest unit fraction,
> > so you can't start counting at it.
> >
> > You talk about all have distance BETWEEN them, but "first" isn't between
> > two of them.
> >
> > I have shown that for ANY value 1/n, there is room for at least n more
> > unit fractions below it, so your "finite gap" idea doesn't show there
> > must be an smallest, but that there CAN'T be a smallest.
> >
> > You are just using bad logic, because you logic doesn't actually have
> > anything to reason with.
> Instead what you do is model a function as an infinite limit of functions,
> the numerators over the denominators in the naturals,
> so it arrives that there's
>
> extent [0,1]
> density [0,1]
> completeness [0,1]
> measure [0,1]
>
> and then the values from zero to one are called "iota-values",
> which is nice because iota already has the natural meaning
> "infinitesimal", and probably since Cavalieri in the time of Galileo,
> it's Aristotle's continuum, then for re-Vitali-izing measure theory
> with doubling-spaces among continuous domains.
>
>
> https://en.wikipedia.org/wiki/Bonaventura_Cavalieri
>
>
> De Santillana has a great passage about Galileo in the
> chapter "The Years of Silence".
>
> Page 157:
>
> "Through all this he was collecting himself. As the diver descending into
> the dark depths feels his strength rise from the burning of hix oxygen under
> pressure, so the scientist's mind, as it receded into isolation, burned brighter
> to itself. The past compromises and loose adjustments of though discarded,
> Galileo seems to be facing more decisively the full implications if his theory.
>
> It is an inhuman wrench for the mind to tear itself loose from the instinctive
> world of the senses around us, so "full of a number of things", peopled with
> familiar substances which reassuringly go about subsisting in their own
> several ways as if they knew how -- and to uncover instead everywhere the
> mysterious realization of abstract laws; to conceive of mathematical function
> determining each point of being in a dizzying tension between the infinite
> and the infinitesimal, where the lonely intellect has to feel its way with new
> instruments of rigorous analysis. It implies a dedication; a new feeling not
> only of nature but also of the divine. When the time for expression comes again,
> the quiet prose of the physicist will rise of itself to compelling heights, spark
> and crackle with contained metaphysical power. But, for many years yet,
> silence was in order."
>
> -- de Santillana, 1954
>
>
> I read this yesterday and really identified, it reminded me one time walking the
> quad with a fellow and a professor, and I said "I've been studying infinity"
> and he said "studying infinity makes people nuts" and I said "ayup".
>
>
> Page 156:
>
> "The epistolary activity of Galileo in those years was immense, for he kept
> in constant touch with his disciples, whose fame was growing, and he was
> never weary of assisting their thought in a vast range of subjects. Young
> Father Cavalieri had taken up from him the study of infinitesimals, from which
> was to come his _Geometry of Indivisibles_."
>
>
>
> Yeah, here's those are just called "standard infinitesimals: iota-values"..
> There are iota-sums and iota-multiples and they're basically different
> in arithmetic.
>
> It's very geometrical and arithmetical, of course.
>
> Its "base" is very simple: n/d, n->d, d-> oo.
> It sort of precedes there being a gapless domain.

"iota-values"

On 1/26/24 1:03 PM, WM wrote:
> Le 26/01/2024 à 13:32, Richard Damon a écrit :
>> On 1/26/24 5:05 AM, WM wrote:
>
>>>
>>> NUF counts the unit fractions.
>>
>> And thus jumps to aleph0 for all x > 0 since there is no smallest unit
>> fraction.
>
> Wrong by mathematics. It cannot jump before (0, 1] because there are no
> unit fractions and it can only jump by 1 because all unit fractions have
> distances.

WHy? Not, "By Mathematics" you can't define NUF(x) as ordinary (not the
axiometric Mathologies) doesn't have values for NUF(x) for positive
values, as it doesn't actually have Aleph as a value (or omega), those
are only the product of your "rejected" axiomatic mathologies.

"Mathematics" as an un-axiomatic concept, starts with counting the
numbers 0, 1, 2, 3, 4, 5, and so on through the finite values maybe to a
concept of "Many" being more than we have counted (but could) and then
maybe to a concept of infinity as more than we could count to, so only a
nebulous concept and not a "value".

From this we can develop Addition and Subtraction, then to
Multiplication and then to Division (being careful not to divide by 0,
as that can't be defined), and perhaps on to higher arithmetic like
powers and the such.

We can get the Rationals by Division, and SOME of the Reals as results
of equations.

You can't actually get to infinity, as that doesn't exist in the
physical observable universe. Only "Many" or "Big" as beyond what we can
practically measure, but in our mind, we could possibly measure.

You can't get to "Dark" numbers, as those are just things beyond the
border of what you have explored. You can't get them by "Subtracting" a
finite collection from the infinite set, as you can't actually define
that infinite set, it just is more than you can do.

Mathematics understands that there is no "biggest" numbers, or "Smallest
Positive Number" as its math shows that given any number you can make a
bigger one, or one closer to zero.

So, if you REALLY want to claim you are just using "Mathematics: and not
an axiometic Mathology, you need to stop making claims that it just
doesn't handle.

Now, if you want to work with Aleph or Omega, you need to accept that
you ARE using Axioms, and figure out which ones you are using, and stick
with it.

>>
>>>>
>>>> So, your "dark" numbers are numbers that can't be described as a
>>>> set, but only individually.
>>>
>>> The other way round.
>>
>> So you say, except that you have shown that you can't actually define
>> them as a set, but at least some of them have individual descriptions,
>> they are the points x where NUF(x) has the various finite values.
>
> These points have no descriptions and can never be described.

Only because they DON'T EXIST in your number system.

>
> Regards, WM
>
>

On 1/26/24 1:09 PM, WM wrote:
> Le 26/01/2024 à 13:35, Richard Damon a écrit :
>> On 1/26/24 4:48 AM, WM wrote:
>
>> But you still don't understand that there is no smallest unit
>> fraction, so you can't start counting at it.
>
> Real points on the real line must be existing.

Yes. But there is no "Smallest Positive"

>
>> You talk about all have distance BETWEEN them, but "first" isn't
>> between two of them.
>
> It is one of them.

Nope, there is no "Smallest Positive".

You can make up descriptions for lots of numbers tht don't actuallyly exist.

>>
>> I have shown that for ANY value 1/n, there is room for at least n more
>> unit fractions below it, so your "finite gap" idea doesn't show there
>> must be an smallest, but that there CAN'T be a smallest.
>
> That holds for definable n only.
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

So, you admit to using an Axiometric Mathology?

Your N_def isn't defined universally.

Yes, If N_def is defined as the set of the initial sequence below a
GIVEN natural number, you get an infinite set above it, but that isn't
all the "definable" numbers, as n+1 is defin

> You have never exhausted the set. But that is possible, collectively:
> |ℕ \ {1, 2, 3, ...}| = 0

So.

>
>> You are just using bad logic, because you logic doesn't actually have
>> anything to reason with.
>
> I am avoiding matheology.

No, you are not, you are USING it to talk about ℵo.

WHere do you get it without an axiometric matheology?

ℵo is NOT part of self-obvious mathematics, but a product of axioms,
which you are calling matheology.

Maybe you are a hidden adict.

>
> Regards, WM
>
>

On 1/26/24 1:00 PM, WM wrote:
> Le 26/01/2024 à 13:31, Richard Damon a écrit :
>> On 1/26/24 5:07 AM, WM wrote:
>
>> I never said the distance goes becomes 0,
>
> Then NUF cannot increase by more than 1 at a point.

But where it increase isn't AT a point.

>
>> Below ANY unit fraction are more, so there is no smallest unit fraction.
>
> If the function can't start, it cannot count.

Right, since ther is no spot to start, you can't actually count from
that end.

>
>> And all those smaller are definable, so don't need to be "dark"
>
> There is no point in (0, 1] without smaller unit fractions. Hence, they
> cannot sit in (0, 1] because every point there is at the right-hand
> side. Even if they all would fit into one point, there is none in (0, 1]
> to accomodate them.
Right, there is no unit fraction without smaller unit fractions, so
there is no unit fraction that NUF(x) CAN be 1 at, so its definition is
just broken.

>>
>> Point of note, you claim that there is only "One Mathematics", because
>> you seem to be a "Naturalist" for mathematics, using the properties
>> observed in ordinary math.
>
> Of course.

Except that you use things that are not actually true. Like the PROVEN
FALSEHOOD that t

>>
>> The problem with this method is that it can only define the
>> mathematics for the numbers we "see", and thus can't HAVE "dark" numbers.
>
> Nevertheless I have proved their existence under the assumption that the
> real line is complete without gaps. More cannot be done.

Really? How do you define this? Does it understand that because there
are no gaps, there are no "adjacent" points, as two points that are
"different" have a finite distance between them, so a value exists
between them.

So "No Gaps" does NOT mean, there is a "Next".

The concept of "Next" requires that the things have a minimum finite
distance between them.

Things that can be "dense" (or at a point of accumulation) do not have a
"next" property. Logic that claims otherwise is just inconsstant.

>>
>> Your "dark" numbers are just numbers that you haven't discovered the
>> actual properties that make them do what you see happening.
>
> Most cannot be disco vered as individuals because otherwise you could
> find the smallest unit fraction.

Nope.

They can't be discovered individually because they just don't exist.

>>
>> And the problem is that YOU apparently can't see the nature of
>> unbounded sets, and thus you can't use them in this mathematics.
>
> The sets of unit fractions and hence of natural numbers are bounded.
> With them also all other sets of numbers are bounded. Infinity means
> only the fact that the end is invisible and that visible numbers are
> potentially infinite.
>

Really? What is the BOUND of the Natural Numbers? What number is at both
ends of them? And end that is "invisible" doesn't exist.

You just don't understand what the Natural Numbers are, because your
logic is just incapable of handling them.

Your idea of "potentially" infinite is just bad logic.

You need to stay out of axiomatic systems, as if you take on axioms,
your definition is proven incorrect, but without a system of axioms, you
can't make your arguments as "infinity" loses the power you need of it
to maky your point.

You confuse "Many"/"Big" with infinite.

> Regards, WM
>
>

On 1/26/2024 4:32 AM, Richard Damon wrote:
[...]

I hope you are taking the proper brain coolants when conversing with
WM... A circular firing squad comes to mind... ;^)

On 1/26/2024 4:35 AM, Richard Damon wrote:
> On 1/26/24 4:48 AM, WM wrote:
>> Le 25/01/2024 à 14:06, FromTheRafters a écrit :
>>> WM formulated the question :
>>>> Le 25/01/2024 à 13:22, Richard Damon a écrit :
>>>>> On 1/25/24 6:15 AM, WM wrote:
>>>>>> Le 25/01/2024 à 02:05, Richard Damon a écrit :
>>>>>>> On 1/24/24 10:46 AM, WM wrote:
>>>>>>
>>>>>>>> They end at zero. Therefore they begin above zero.
>>>>>>>
>>>>>>> But there is no "begin" as the values are dense there.
>>>>>>
>>>>>> They are not dense but all have distances.
>>>>>
>>>>> Not the Reals or the Rationals.
>>>>
>>>> We talk about unit fractions.
>>>
>>> As embedded in the reals.
>>
>> Irrelevant for the statement to be wrong.
>>
>> Regards, WM
>>
>>
>
> But you still don't understand that there is no smallest unit fraction,
> so you can't start counting at it.
>
> You talk about all have distance BETWEEN them, but "first" isn't between
> two of them.
>
> I have shown that for ANY value 1/n, there is room for at least n more
> unit fractions below it, so your "finite gap" idea doesn't show there
> must be an smallest, but that there CAN'T be a smallest.
>
> You are just using bad logic, because you logic doesn't actually have
> anything to reason with.

WM is dip-shit-icus moron-icus ?

On 1/26/24 8:09 PM, Chris M. Thomasson wrote:
> On 1/26/2024 4:32 AM, Richard Damon wrote:
> [...]
>
> I hope you are taking the proper brain coolants when conversing with
> WM... A circular firing squad comes to mind... ;^)

It provides useful practice for handling non-logical idiots.

On 1/26/2024 5:22 PM, Richard Damon wrote:
> On 1/26/24 8:09 PM, Chris M. Thomasson wrote:
>> On 1/26/2024 4:32 AM, Richard Damon wrote:
>> [...]
>>
>> I hope you are taking the proper brain coolants when conversing with
>> WM... A circular firing squad comes to mind... ;^)
>
> It provides useful practice for handling non-logical idiots.

Hummm... Touche! :^)

On Friday, January 26, 2024 at 11:47:08 AM UTC-8, Fritz Feldhase wrote:
> On Friday, January 26, 2024 at 7:10:07 PM UTC+1, WM wrote:
>
> > I am avoiding matheology.
>
> What you are avoiding is _sanity_, Mückenheim.

Ah, "mathematical conscientiousness", according
to and accordingly "mathematical rigor" in
"mathematical formalism" with "free mathematical inference"
about "the language of mathematics" in "the philosophy of mathematics"
for "the foundations of mathematics", is all that would matter.

I.e., Hilbert's infinite living museum of mathematics,
has various wings, but the central exhibit has a home
for "the mathematical conscience", writ large.

All one consistent theory, ....

On Friday, January 26, 2024 at 5:22:51 PM UTC-8, Richard Damon wrote:
> On 1/26/24 8:09 PM, Chris M. Thomasson wrote:
> > On 1/26/2024 4:32 AM, Richard Damon wrote:
> > [...]
> >
> > I hope you are taking the proper brain coolants when conversing with
> > WM... A circular firing squad comes to mind... ;^)
> It provides useful practice for handling non-logical idiots.

Your pun though reads, "properly logical, idiots", ....

Like, if you quantify over sets-that-don't-contain-themselves,
only finite ones where ordinals are cardinals, and their collection
does, contain itself, and, Russell, "properly, non-logically", defines it away,
making an ignorance of it, you don't see that as reflecting
on your own riposte?

Have you heard of set theories with universes?
Nonstandard analysis? Point at infinity?
Banach-Tarski and Vitali-Hausdorff doubling-spaces?

Cantor's Domain Principle was a universe existed,
so, it doesn't in the ordinary, because it would be its own powerset,
but somebody else's Domain Principle does, in their theory
with a universe of objects,

Arithmetic can be implemented variously, and in ways that
do and don't fall together with algebra in the middle.

So, you seem unawares of the entire world of anti-foundational
set theory, non-standard analysis, number theory and set
theory with universes or a point at infinity, usual consequences
of quantification like set-of-all-sets, order-type-of-all-ordinals,
group-of-all-groups, and these things that matter to foundations,
not just the brief, uncomplicated, schoolboy, linear curriculum,
instead _foundations_.

So, if you're not helping, then, nobody needs your help.
For people who already have the linear curriculum,
and it's expected everybody here does,
anything else getsa generous reading as wishful thinking,
of the mathematical variety.

Soul Calibur Four-Five-Six / Legendary /
Campaign Hard / No Assist /
Gran Turismo 3-4-5-6-7 / FinnEcces /
Red Alert / Briefly Top 10 /
Civilization V-VI / 0 AD - 90% Emporer /
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Le 27/01/2024 à 01:50, Richard Damon a écrit :
> On 1/26/24 1:03 PM, WM wrote:
>> Le 26/01/2024 à 13:32, Richard Damon a écrit :
>>> On 1/26/24 5:05 AM, WM wrote:
>>
>>>>
>>>> NUF counts the unit fractions.
>>>
>>> And thus jumps to aleph0 for all x > 0 since there is no smallest unit
>>> fraction.
>>
>> Wrong by mathematics. It cannot jump before (0, 1] because there are no
>> unit fractions and it can only jump by 1 because all unit fractions have
>> distances.
>
> WHy?

It cannot jump before (0, 1] because there are no unit fractions and it
can only jump by 1 because all unit fractions have distances.

> You can't actually get to infinity, as that doesn't exist in the
> physical observable universe.

But it can be assumed between every pair of definable real numbers.

> You can't get to "Dark" numbers, as those are just things beyond the
> border of what you have explored. You can't get them by "Subtracting" a
> finite collection from the infinite set, as you can't actually define
> that infinite set, it just is more than you can do.

Dark numbers can be manipulated collectively.
>
> Mathematics understands that there is no "biggest" numbers, or "Smallest
> Positive Number" as its math shows that given any number you can make a
> bigger one, or one closer to zero.
>
> So, if you REALLY want to claim you are just using "Mathematics: and not
> an axiometic Mathology, you need to stop making claims that it just
> doesn't handle.
>
> Now, if you want to work with Aleph or Omega, you need to accept that
> you ARE using Axioms, and figure out which ones you are using, and stick
> with it.

I use only the numbers existing under the premise of actul infinity.

Regards, WM

Le 27/01/2024 à 01:50, Richard Damon a écrit :
> On 1/26/24 1:09 PM, WM wrote:
>> Le 26/01/2024 à 13:35, Richard Damon a écrit :
>>> On 1/26/24 4:48 AM, WM wrote:
>>
>>> But you still don't understand that there is no smallest unit
>>> fraction, so you can't start counting at it.
>>
>> Real points on the real line must be existing.
>
> Yes. But there is no "Smallest Positive"

If actual infinity exists, then there is a smallest positive. If not, then
there is a gap.
>
> Your N_def isn't defined universally.

No, it changes.
>
> Yes, If N_def is defined as the set of the initial sequence below a
> GIVEN natural number, you get an infinite set above it, but that isn't
> all the "definable" numbers, as n+1 is defin

Hard to accept, but fact.
>
>> You have never exhausted the set. But that is possible, collectively:
>> |ℕ \ {1, 2, 3, ...}| = 0
>
> So.
>>
>>> You are just using bad logic, because you logic doesn't actually have
>>> anything to reason with.
>>
>> I am avoiding matheology.
>
> No, you are not, you are USING it to talk about ℵo.

I use ℵo or better ℵ only as an expression for infinitely many.

Regards, WM

Le 27/01/2024 à 01:50, Richard Damon a écrit :
> On 1/26/24 1:00 PM, WM wrote:
>> Le 26/01/2024 à 13:31, Richard Damon a écrit :
>>> On 1/26/24 5:07 AM, WM wrote:
>>
>>> I never said the distance goes becomes 0,
>>
>> Then NUF cannot increase by more than 1 at a point.
>
> But where it increase isn't AT a point.

It can only increase at points 1/n.
>
>>
>>> Below ANY unit fraction are more, so there is no smallest unit fraction.
>>
>> If the function can't start, it cannot count.
>
> Right, since ther is no spot to start, you can't actually count from
> that end.

I can't but the points 1/n must be there.

Regards, WM

Chris M. Thomasson wrote:

> On 1/26/2024 5:22 PM, Richard Damon wrote:
>> On 1/26/24 8:09 PM, Chris M. Thomasson wrote:
>>> On 1/26/2024 4:32 AM, Richard Damon wrote:
>>> [...]
>>>
>>> I hope you are taking the proper brain coolants when conversing with
>>> WM... A circular firing squad comes to mind... ;^)
>>
>> It provides useful practice for handling non-logical idiots.
>
> Hummm... Touche! :^)

here is how to do it, to ignore that criminal polluter. Likely an
unsatisfied dual citizenship holder 𝗸𝗵𝗮𝘇𝗮𝗿_𝗴𝗼𝘆 from amrica.

add to 𝗦𝘂𝗯𝗷𝗲𝗰𝘁 (and from/author) 𝗠𝗮𝘁𝗰𝗵𝗶𝗻𝗴_𝗥𝗲𝗴𝗲𝘅

^.*[\p{Thai}].*$

or you can also remove the start ^ and end $, if it's multiline.

On 1/27/24 6:07 AM, WM wrote:
> Le 27/01/2024 à 01:50, Richard Damon a écrit :
>> On 1/26/24 1:03 PM, WM wrote:
>>> Le 26/01/2024 à 13:32, Richard Damon a écrit :
>>>> On 1/26/24 5:05 AM, WM wrote:
>>>
>>>>>
>>>>> NUF counts the unit fractions.
>>>>
>>>> And thus jumps to aleph0 for all x > 0 since there is no smallest
>>>> unit fraction.
>>>
>>> Wrong by mathematics. It cannot jump before (0, 1] because there are
>>> no unit fractions and it can only jump by 1 because all unit
>>> fractions have distances.
>>
>> WHy?
>
> It cannot jump before (0, 1] because there are no unit fractions and it
> can only jump by 1 because all unit fractions have distances.

WHy can't it "Jump" since there is a boundry of domain that it crossed.

>
>> You can't actually get to infinity, as that doesn't exist in the
>> physical observable universe.
>
> But it can be assumed between every pair of definable real numbers.

What "Infinity" can be assumed?

Infinity, in natural math, is a value farther than you can actually get to.

Between every definable real number is another definable real number, we
don't need to invoke "infinity". Between any X1 and X2 is (X1+X2)/2,
which is definable if X1 and X2 are.

You are just assuming an infinity there that isn't needed to try to get
gaps to put the dark numbers in there.

>
>> You can't get to "Dark" numbers, as those are just things beyond the
>> border of what you have explored. You can't get them by "Subtracting"
>> a finite collection from the infinite set, as you can't actually
>> define that infinite set, it just is more than you can do.
>
> Dark numbers can be manipulated collectively.

The give me a collection that is JUST dark numbers. You say that can be
maninpulatd collectivily, so do it.

>>
>> Mathematics understands that there is no "biggest" numbers, or
>> "Smallest Positive Number" as its math shows that given any number you
>> can make a bigger one, or one closer to zero.
>>
>> So, if you REALLY want to claim you are just using "Mathematics: and
>> not an axiometic Mathology, you need to stop making claims that it
>> just doesn't handle.
>>
>> Now, if you want to work with Aleph or Omega, you need to accept that
>> you ARE using Axioms, and figure out which ones you are using, and
>> stick with it.
>
> I use only the numbers existing under the premise of actul infinity.
> Regards, WM
>

What is that premise? Sounds like you are using an actual set of axioms,
but you won't express them explicitly as then someone could show your
errors.

Note, there ARE ACTUALLY an infinite number of Natural Numbers (and thus
also Rationals and Reals). They aren't just "potentially" infinite sets,
they are ACTUALLY infinite sets (and the Reals are even bigger than the
others).

On 1/27/24 6:17 AM, WM wrote:
> Le 27/01/2024 à 01:50, Richard Damon a écrit :
>> On 1/26/24 1:09 PM, WM wrote:
>>> Le 26/01/2024 à 13:35, Richard Damon a écrit :
>>>> On 1/26/24 4:48 AM, WM wrote:
>>>
>>>> But you still don't understand that there is no smallest unit
>>>> fraction, so you can't start counting at it.
>>>
>>> Real points on the real line must be existing.
>>
>> Yes. But there is no "Smallest Positive"
>
> If actual infinity exists, then there is a smallest positive. If not,
> then there is a gap.

Says what.

The

>>
>> Your N_def isn't defined universally.
>
> No, it changes.

So it isn't defined. Defined sets (that aren't made a "function" of
something) don't change.

>>
>> Yes, If N_def is defined as the set of the initial sequence below a
>> GIVEN natural number, you get an infinite set above it, but that isn't
>> all the "definable" numbers, as n+1 is defin
>
> Hard to accept, but fact.

So you are stating it is a FACT that there is no actual finite set of
"Defined" Natural Numbers, as any such proposed set leaves out a defined
Natural number.

Good that you accept the "hard" fact that your ideas are based on
falsehoods.

>>
>>> You have never exhausted the set. But that is possible, collectively:
>>> |ℕ \ {1, 2, 3, ...}| = 0
>>
>> So.
>>>
>>>> You are just using bad logic, because you logic doesn't actually
>>>> have anything to reason with.
>>>
>>> I am avoiding matheology.
>>
>> No, you are not, you are USING it to talk about ℵo.
>
> I use ℵo or better ℵ only as an expression for infinitely many.

Then you should use the right symbol: ∞

ℵ is a symbol from a specific group of axiometric theories, and includes
implications of properties that don't come out of "natural mathematics".

>
> Regards, WM
>
>

On 1/27/24 6:20 AM, WM wrote:
> Le 27/01/2024 à 01:50, Richard Damon a écrit :
>> On 1/26/24 1:00 PM, WM wrote:
>>> Le 26/01/2024 à 13:31, Richard Damon a écrit :
>>>> On 1/26/24 5:07 AM, WM wrote:
>>>
>>>> I never said the distance goes becomes 0,
>>>
>>> Then NUF cannot increase by more than 1 at a point.
>>
>> But where it increase isn't AT a point.
>
> It can only increase at points 1/n.

But if there isn't a "first" point to start from, you don't have a spot
to start counting at 1.

So, it just starts at infinity.

>>
>>>
>>>> Below ANY unit fraction are more, so there is no smallest unit
>>>> fraction.
>>>
>>> If the function can't start, it cannot count.
>>
>> Right, since ther is no spot to start, you can't actually count from
>> that end.
>
> I can't but the points 1/n must be there.

Right, all the 1/n points are there, all infinite number of them, so no
"first" (aka lowest) spot to start.

You can't define a count from a point that doesn't eist.

>
> Regards, WM

On Saturday, January 27, 2024 at 7:26:25 AM UTC-8, Richard Damon wrote:
> On 1/27/24 6:20 AM, WM wrote:
> > Le 27/01/2024 à 01:50, Richard Damon a écrit :
> >> On 1/26/24 1:00 PM, WM wrote:
> >>> Le 26/01/2024 à 13:31, Richard Damon a écrit :
> >>>> On 1/26/24 5:07 AM, WM wrote:
> >>>
> >>>> I never said the distance goes becomes 0,
> >>>
> >>> Then NUF cannot increase by more than 1 at a point.
> >>
> >> But where it increase isn't AT a point.
> >
> > It can only increase at points 1/n.
> But if there isn't a "first" point to start from, you don't have a spot
> to start counting at 1.
>
> So, it just starts at infinity.
> >>
> >>>
> >>>> Below ANY unit fraction are more, so there is no smallest unit
> >>>> fraction.
> >>>
> >>> If the function can't start, it cannot count.
> >>
> >> Right, since ther is no spot to start, you can't actually count from
> >> that end.
> >
> > I can't but the points 1/n must be there.
> Right, all the 1/n points are there, all infinite number of them, so no
> "first" (aka lowest) spot to start.
>
> You can't define a count from a point that doesn't eist.
>
> >
> > Regards, WM

So what, "iota-values" are a valid thing.

That MW/WM is wrong is straw-man, and a
variety of other rhetorical fallacies.

So, "iota-values" fulfill most people's
notions of standard infinitesimals,
indivisible individua of a continuum, [0,1].

And, they result from simply a second course
of the usual notion of limit, the infinite limit,
the continuum limit.

People often conflate counts with numbers,
but, a count is after an act, while a number's as a name.

In a theory with ubiquitous ordinals, like if the set-theoretic
universe is rulial with the well-ordering principle,
it's kind of this way.

On 1/27/2024 3:17 AM, WM wrote:
> Le 27/01/2024 à 01:50, Richard Damon a écrit :
>> On 1/26/24 1:09 PM, WM wrote:
>>> Le 26/01/2024 à 13:35, Richard Damon a écrit :
>>>> On 1/26/24 4:48 AM, WM wrote:
>>>
>>>> But you still don't understand that there is no smallest unit
>>>> fraction, so you can't start counting at it.
>>>
>>> Real points on the real line must be existing.
>>
>> Yes. But there is no "Smallest Positive"
>
> If actual infinity exists, then there is a smallest positive. If not,
> then there is a gap.

[...]

Oh my god, you really are a massive idiotic moron. Not just a moron, but
a hardcore one, King of Stupid?