On Saturday, February 11, 2023 at 5:51:34 PM UTC+1, WM wrote:

> Passing the [...] unit fractions the cursor will not reach zero.

Indeed! That's why Achill will never catch up with the turtle!

> There are always ℵo missing.

Exactly! (Except when the curser arrives at zero.)

> But the cursor will reach zero and then it will have passed all unit fractions.

Yeah, somehow Achill catches up with the turtle - no one knows how and why!

Fritz Feldhase schrieb am Samstag, 11. Februar 2023 um 20:19:07 UTC+1:
> On Saturday, February 11, 2023 at 5:51:34 PM UTC+1, WM wrote:
>
> > Passing the [...] unit fractions the cursor will not reach zero.
>
> Indeed! That's why Achill will never catch up with the turtle!
> > There are always ℵo missing.
> Exactly! (Except when the curser arrives at zero.)

But continuity forbids that ℵo points of the real line are placed between 0 and (0, 1].

> > But the cursor will reach zero and then it will have passed all unit fractions.
> Yeah, somehow Achill catches up with the turtle - no one knows how and why!

I know it.

Regards, WM

William schrieb am Samstag, 11. Februar 2023 um 19:21:47 UTC+1:
> On Saturday, February 11, 2023 at 2:06:21 PM UTC-4, WM wrote:
> > William schrieb am Samstag, 11. Februar 2023 um 18:42:40 UTC+1:
> > > You have agreed that a "dark natural number" is larger than any element of |N_P.
> > Of course.
> > > No element of |R is larger than any element of |N_P.
> > Almost all elements of |R and almost all elements of |N are larger than the elements of |N_P.
> Nope. No element of |R is larger than each element of |N_P. (Each element of R is followed by an infinite number of elements of |N_P none of which is larger than any element of |N_P)

Your opinion forces ℵo unit fractions to sit between 0 and (0, 1]. That is impossible.

> So a "dark natural number" is not an element of |R.

ℵo points of dark unit fractions sit in the interval (0, 1].

Regards, WM

On Saturday, February 11, 2023 at 4:08:26 PM UTC-4, WM wrote:
> William schrieb am Samstag, 11. Februar 2023 um 19:21:47 UTC+1:
> > On Saturday, February 11, 2023 at 2:06:21 PM UTC-4, WM wrote:
> > > William schrieb am Samstag, 11. Februar 2023 um 18:42:40 UTC+1:
> > > > You have agreed that a "dark natural number" is larger than any element of |N_P.
> > > Of course.
> > > > No element of |R is larger than any element of |N_P.
> > > Almost all elements of |R and almost all elements of |N are larger than the elements of |N_P.
> > Nope. No element of |R is larger than each element of |N_P. (Each element of R is followed by an infinite number of elements of |N_P none of which is larger than any element of |N_P)
> Your opinion

You agree than a "dark natural number" is larger than any element of N_P, and that any element of |R is followed by an infinite number of elements of |N_P. It is not just my opinion.

On Saturday, February 11, 2023 at 9:06:03 PM UTC+1, WM wrote:

> continuity forbids that ℵo points of the real line are placed between 0 and (0, 1].

If you say so. On the other hand, that's just fine, since there are no points "between" 0 and (0, 1]. With other words, there is no real number x such that for all y e (0, 1]: 0 < x < y.

On Saturday, February 11, 2023 at 12:27:51 PM UTC-8, Fritz Feldhase wrote:
> On Saturday, February 11, 2023 at 9:06:03 PM UTC+1, WM wrote:
>
> > continuity forbids that ℵo points of the real line are placed between 0 and (0, 1].
> If you say so. On the other hand, that's just fine, since there are no points "between" 0 and (0, 1]. With other words, there is no real number x such that for all y e (0, 1]: 0 < x < y.

No "standard" real number, but there are "only" real numbers between zero and one.

"Standard" enough "standard infinitesimals" make a continuous domain [0,1].

Did you know that there are some functions that are not real-functions, and
did you know that there are some functions that are "special" not "elementary"?

This is more than rudimentary.

On Saturday, February 11, 2023 at 1:10:28 PM UTC-8, Ross A. Finlayson wrote:
> On Saturday, February 11, 2023 at 12:27:51 PM UTC-8, Fritz Feldhase wrote:
> > On Saturday, February 11, 2023 at 9:06:03 PM UTC+1, WM wrote:
> >
> > > continuity forbids that ℵo points of the real line are placed between 0 and (0, 1].
> > If you say so. On the other hand, that's just fine, since there are no points "between" 0 and (0, 1]. With other words, there is no real number x such that for all y e (0, 1]: 0 < x < y.
> No "standard" real number, but there are "only" real numbers between zero and one.
>
> "Standard" enough "standard infinitesimals" make a continuous domain [0,1].
>
> Did you know that there are some functions that are not real-functions, and
> did you know that there are some functions that are "special" not "elementary"?
>
> This is more than rudimentary.

Have you ever heard of the meromorphic?

How about the Dirac delta, an infinite spike at zero with area one?

Have you ever heard of quasi-invariance or Ramsey theory?

How about near-field and far-field and fast and slow waves?

Did you know the highly non-linear is essentially resonant and harmonic?

The "equivalency function" or sweep is very simple and right past standard,
also it's very interesting, and it's very interesting for probability theory that
the uniform numbers have multiple distributions, where "Standard" probability theory
knows only one, and, it's very interesting for differential analysis and singularity theory,
because its its own anti-derivative, and the spiral space-filling curve is a perfect mathematical
construct of sorts.

In fact in the _real_ old days, like pre-history, that's also the symbol of mathematics
writ large: the spiral, what's the origin.

Then, that the Archimedean spiral for example is a great third tool after compass
and edge, fulfills a lot more constructions than what you might be missing.

This simple "sweep" is a _central_ continuous domain relevant all higher mathematics.

Or, "don't you know"?

On 2/11/2023 3:16 AM, WM wrote:
> Ben Bacarisse schrieb am Samstag, 11. Februar 2023 um 03:31:24 UTC+1:
>> WM <askas...@gmail.com> writes:
>
>>>> Both proper mathematics and WMaths agree that every FISON has infinitely
>>>> many successors[1] and that a set of FISONS is not a natnumber and
>>>> therefore has no natnumber successor.
>>>
>>> Instead of the set consider all FISONs.
>> I considered and addressed what you said. We both agree on these two
>> points
>
> Not at all! The set of FISONs is a FISON in classical mathematics. This was changed only by Cantor and Bolzano.
>
> "To the idea to consider the infinite large not only in the form of the unlimited growing and the closely connected form of the convergent infinite series, introduced first in the seventeenth century," [Cantor, p. 175]
>
> "If they, like Hegel, Erdmann, and others, imagine the mathematical infinite only as a magnitude which is variable and only has no limit in its growth (like some mathematicians, as we will see soon, have assumed to explain their notion) so I agree in their reproach of this notion of a magnitude only growing into the infinite but never reaching it." [Bernard Bolzano: "Paradoxien des Unendlichen", Reclam, Leipzig (1851) p. 6ff]
>
>> so now you want to change the argument by waffling about
>> something else.
>
> You should first try to understand classical mathematics! In order to help you, I have switched to unit fractions. Even you should be able to understand from Cantor's set theory and ZF that the last noticed unit fraction 1/n has ℵo successors which remain unseen because 1/n, the last one seen, was passed before.
>
>>> It is easily seen by the unit fractions.
>> No, the unit fractions, 1/, 1/2, 1/3, ..., have "none missing" (by
>> definition) but don't "have an end".
>
> After arriving at zero there are no further unit fractions following.
[...]

What about the negative unit fractions?

WM <askasker48@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)

> Ben Bacarisse schrieb am Samstag, 11. Februar 2023 um 03:31:24 UTC+1:
>> WM <askas...@gmail.com> writes:
>
>> >> Both proper mathematics and WMaths agree that every FISON has infinitely
>> >> many successors[1] and that a set of FISONS is not a natnumber and
>> >> therefore has no natnumber successor.
>> >
>> > Instead of the set consider all FISONs.
>> I considered and addressed what you said. We both agree on these two
>> points
>
> Not at all! The set of FISONs is a FISON in classical
> mathematics. This was changed only by Cantor and Bolzano.

We've been agreeing about the two basic facts you want to use to present
yet another mystery to your students for over a week now:

| ... he knows that 0 is the GLB of U (where U is {1/n} with n in N⁺)
| so that {u | u ∈ U and u <= x} is empty for x <= 0 and not finite
| when x > 0 so I doubt he disputes any matter of fact here.

Your problem is to find some form of words that makes this discontinuity
from x > 0 (infinite unit factions) to x <= (no unit fractions) seem
like a mystery so you can bamboozle your students with it. You'll get
there eventually and you are good at the word games, but there will
still be no dispute over these facts.

>> > It is easily seen by the unit fractions.
>> No, the unit fractions, 1/, 1/2, 1/3, ..., have "none missing" (by
>> definition) but don't "have an end".
>
> After arriving at zero there are no further unit fractions
> following. Hence there was an end before.

No. Proofs don't work by just adding in "hence" before the conclusion
you want. But I suspect that if you say this with enough conviction,
you students will dutifully write it down and parrot it back to you in
the exam.

The sequence of unit fractions is bounded below. Zero is the greatest
lower bound. The sequence is, none the less, endless.

--
Ben.

On Saturday, February 11, 2023 at 12:16:15 PM UTC+1, WM wrote:

> The set of FISONs is a FISON in classical mathematics.

Nonsense.

Hint: The set of FISONs is infinite, while each and every FISON is finite. qed

Lit.: https://en.m.wikipedia.org/wiki/Classical_mathematics

WM explained on 2/11/2023 :
> Fritz Feldhase schrieb am Samstag, 11. Februar 2023 um 20:19:07 UTC+1:
>> On Saturday, February 11, 2023 at 5:51:34 PM UTC+1, WM wrote:
>>
>>> Passing the [...] unit fractions the cursor will not reach zero.
>>
>> Indeed! That's why Achill will never catch up with the turtle!
>>> There are always ℵo missing.
>> Exactly! (Except when the curser arrives at zero.)
>
> But continuity forbids that ℵo points of the real line are placed between 0
> and (0, 1].

Wrong, continuity guarantees that 2^aleph_zero points of the real line
are placed 'between' any element of Q and any other element of Q.

Also recall that 2^aleph_zero is greater than aleph_zero.

Timothy Golden was thinking very hard :
> On Saturday, February 11, 2023 at 1:35:33 PM UTC-5, FromTheRafters wrote:
>> Timothy Golden used his keyboard to write :
>>> On Saturday, February 11, 2023 at 11:11:22 AM UTC-5, Fritz Feldhase wrote:
>>>> On Saturday, February 11, 2023 at 4:34:59 PM UTC+1, Timothy Golden wrote:
>>>>
>>>>>> so 15 is a natural
>>>>>>
>>>>> I <bla>
>>>>
>>>> Look, you fucking asshole full of shit, there's a difference between a
>>>> number and its name.
>>>>
>>>> 15 is a natural number and "15" is its name.
>>>>
>>>> Actually, there's a "naming scheme" for (natural) numbers which is called
>>>> /decimal representation/.
>>>>
>>>> See: https://en.wikipedia.org/wiki/Decimal_representation
>>>
>>> And when we look at the Peano axioms, in their great simplicity, and
>>> staying with that great simplicity, we see that the value 15 is
>>> represented as: s(s(s(s(s(s(s(s(s(1))))))) or some such, and those who
>>> claim to be working within those axioms ought not then to rely upon radix
>>> ten technology, for they have already committed themselves to a radix one
>>> technology, or something extremely close by.
>> You're confused. The naturals and the integers don't need or use the
>> post radix fractional part in any of the positional notations, and they
>> are finite.
>
> Without regard to the title of the thread I'd be polluting the discussion,
> and I can see that status quo thinking does believe that I am polluting this
> thread. It seems to me that you've raised the confusion if you delete my
> claim then falsify it. Obviously I am applying some radix-centric thinking
> here which many are in denial of. I don't disagree with what you are saying,
> but still the radix analysis holds up. In radix one the radix point will not
> function as it does in the higher radix forms. It is an interesting
> conformity, but it contradicts the reliance upon a radix one analysis (Peano)
> to work unit fractions, other than 1/1, which is 0.1 in radix one notation.

You're confused. The naturals and the integers don't need or use the
post radix fractional part in any of the positional notations, and they
are finite.

William schrieb am Samstag, 11. Februar 2023 um 21:16:51 UTC+1:
> On Saturday, February 11, 2023 at 4:08:26 PM UTC-4, WM wrote:
> > William schrieb am Samstag, 11. Februar 2023 um 19:21:47 UTC+1:
> > > On Saturday, February 11, 2023 at 2:06:21 PM UTC-4, WM wrote:
> > > > William schrieb am Samstag, 11. Februar 2023 um 18:42:40 UTC+1:
> > > > > You have agreed that a "dark natural number" is larger than any element of |N_P.
> > > > Of course.
> > > > > No element of |R is larger than any element of |N_P.
> > > > Almost all elements of |R and almost all elements of |N are larger than the elements of |N_P.
> > > Nope. No element of |R is larger than each element of |N_P. (Each element of R is followed by an infinite number of elements of |N_P none of which is larger than any element of |N_P)
> > Your opinion
> You agree than a "dark natural number" is larger than any element of N_P,

Of course.

> and that any element of |R is followed by an infinite number of elements of |N_P.

No. Every visible element of |R like every visible natural number is followed by infinitely many naturla numbers (namely by potentiallyinfinitely many visible natural numbers and by actualle infinitely many dark natural numbers).

> It is not just my opinion.

It is proven wrong by the cursor.

Regards, WM

WM wrote on 2/12/2023 :
> William schrieb am Samstag, 11. Februar 2023 um 21:16:51 UTC+1:
>> On Saturday, February 11, 2023 at 4:08:26 PM UTC-4, WM wrote:
>>> William schrieb am Samstag, 11. Februar 2023 um 19:21:47 UTC+1:
>>>> On Saturday, February 11, 2023 at 2:06:21 PM UTC-4, WM wrote:
>>>>> William schrieb am Samstag, 11. Februar 2023 um 18:42:40 UTC+1:
>>>>>> You have agreed that a "dark natural number" is larger than any element
>>>>>> of |N_P. Of course. No element of |R is larger than any element of
>>>>>> |N_P.
>>>>> Almost all elements of |R and almost all elements of |N are larger than
>>>>> the elements of |N_P.
>>>> Nope. No element of |R is larger than each element of |N_P. (Each element
>>>> of R is followed by an infinite number of elements of |N_P none of which
>>>> is larger than any element of |N_P)
>>> Your opinion
>> You agree than a "dark natural number" is larger than any element of N_P,
>
> Of course.
>
>> and that any element of |R is followed by an infinite number of elements of
>> |N_P.
>
> No. Every visible element of |R like every visible natural number is followed
> by infinitely many naturla numbers (namely by potentiallyinfinitely many
> visible natural numbers and by actualle infinitely many dark natural
> numbers).
>
>> It is not just my opinion.
>
> It is proven wrong by the cursor.

Curse all you want, it's still wrong.

Fritz Feldhase schrieb am Samstag, 11. Februar 2023 um 21:27:51 UTC+1:
> On Saturday, February 11, 2023 at 9:06:03 PM UTC+1, WM wrote:
>
> > continuity forbids that ℵo points of the real line are placed between 0 and (0, 1].
> If you say so.

All unit fractions are different points on the real line. Not even one could be placed between 0 and (0, 1], let alone ℵo unit fractions.

> On the other hand, that's just fine, since there are no points "between" 0 and (0, 1]. With other words, there is no real number x such that for all y e (0, 1]: 0 < x < y.

But there are many unit fractions, in fact ℵo, between every noticed unit fraction and zero. These are ℵo different points on the real line and cannot be removed by quantifier-magic. But they cannot discerned.

Regards, WM

Fritz Feldhase schrieb am Sonntag, 12. Februar 2023 um 06:48:21 UTC+1:
> On Saturday, February 11, 2023 at 12:16:15 PM UTC+1, WM wrote:
>
> > The set of FISONs is a FISON in classical mathematics.
> Nonsense.
>
> Hint: The set of FISONs is infinite,

It is potentially infinite.
Proof by unit fractions.

Regards, WM

FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 11:17:46 UTC+1:
> WM explained on 2/11/2023 :
> > Fritz Feldhase schrieb am Samstag, 11. Februar 2023 um 20:19:07 UTC+1:
> >> On Saturday, February 11, 2023 at 5:51:34 PM UTC+1, WM wrote:
> >>
> >>> Passing the [...] unit fractions the cursor will not reach zero.
> >>
> >> Indeed! That's why Achill will never catch up with the turtle!
> >>> There are always ℵo missing.
> >> Exactly! (Except when the curser arrives at zero.)
> >
> > But continuity forbids that ℵo points of the real line are placed between 0
> > and (0, 1].
> Wrong, continuity guarantees that 2^aleph_zero points of the real line
> are placed 'between' any element of Q and any other element of Q.

Nonsense. There is an actual infinity of dark real numbers. But not between 0 and (0, 1].
>
> Also recall that 2^aleph_zero is greater than aleph_zero.

Recall that only fools will continue to accept 2^aleph_zero when they have been taught about dark numbers.

Regards, WM

Ben Bacarisse schrieb am Sonntag, 12. Februar 2023 um 02:12:13 UTC+1:
> WM <askas...@gmail.com> writes:

> >> >> Both proper mathematics and WMaths agree that every FISON has infinitely
> >> >> many successors[1] and that a set of FISONS is not a natnumber and
> >> >> therefore has no natnumber successor.
> >> >
> >> > Instead of the set consider all FISONs.
> >> I considered and addressed what you said. We both agree on these two
> >> points
> >
> > Not at all! The set of FISONs is a FISON in classical
> > mathematics. This was changed only by Cantor and Bolzano.

> | ... he knows that 0 is the GLB of U (where U is {1/n} with n in N⁺)

Of course.
> | so that {u | u ∈ U and u <= x} is empty for x <= 0 and not finite
> | when x > 0 so I doubt he disputes any matter of fact here.

Wrong. Infinitely many unit fractions do not fit between 0 and (0, 1]. But the unit fractions stretch to every point of (0, 1].
>
> Your problem is to find some form of words that makes this discontinuity
> from x > 0 (infinite unit factions) to x <= (no unit fractions) seem
> like a mystery

It is not a mystery, but it is impossible geometrically, because ℵo unit fractions occupy ℵo *different* geometrical points. A mystery is only how mathematicians could overlook this until now.

> so you can bamboozle your students with it.

Do you really claim that there are ℵo different points between 0 and (0, 1]? That will be a good teaser for my next lecture.

> You'll get ℵo
> there eventually and you are good at the word games, but there will
> still be no dispute over these facts.

I would be intersted if anybody shares your opinion. Not even YBM I think will be so fanatic.

> The sequence of unit fractions is bounded below. Zero is the greatest
> lower bound. The sequence is, none the less, endless.

Its end cannot be seen or discerned but it can be passed by the cursor collectively because none of the ℵo points remains to be passed.

Regards, WM

WM explained :
> FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 11:17:46 UTC+1:
>> WM explained on 2/11/2023 :
>>> Fritz Feldhase schrieb am Samstag, 11. Februar 2023 um 20:19:07 UTC+1:
>>>> On Saturday, February 11, 2023 at 5:51:34 PM UTC+1, WM wrote:
>>>>
>>>>> Passing the [...] unit fractions the cursor will not reach zero.
>>>>
>>>> Indeed! That's why Achill will never catch up with the turtle!
>>>>> There are always ℵo missing.
>>>> Exactly! (Except when the curser arrives at zero.)
>>>
>>> But continuity forbids that ℵo points of the real line are placed between 0
>>> and (0, 1].
>> Wrong, continuity guarantees that 2^aleph_zero points of the real line
>> are placed 'between' any element of Q and any other element of Q.
>
> Nonsense. There is an actual infinity of dark real numbers. But not between 0
> and (0, 1].
>>
>> Also recall that 2^aleph_zero is greater than aleph_zero.
>
> Recall that only fools will continue to accept 2^aleph_zero when they have
> been taught about dark numbers.

Which don't exist except in your addled brain.

On Sunday, February 12, 2023 at 11:30:14 AM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Samstag, 11. Februar 2023 um 21:27:51 UTC+1:

> All unit fractions are [...] points on the real line. Not even one could be placed between 0 and (0, 1], let alone ℵo unit fractions.

Indeed! Did ANYONE claim otherwise?

Hint:

> > [...] that's just fine, since there are no points "between" 0 and (0, 1] [on the real number line]. With other words, there is no real number x such that for all y e (0, 1]: 0 < x < y.

> But there are many unit fractions, in fact ℵo, between every [...] unit fraction and zero.

Indeed! Did ANYONE claim otherwise?

> These are ℵo [...] points on the real line [...]

Indeed!

> and cannot be removed by quantifier-magic.

Huh?! What are you babbling about?

> But they cannot discerned.

Yeah, whatever, Mückenheim!

(Seems that you are just playing idiotic word games. Please fuck off, Mückenheim!)

On Sunday, February 12, 2023 at 11:31:10 AM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 12. Februar 2023 um 06:48:21 UTC+1:
> > On Saturday, February 11, 2023 at 12:16:15 PM UTC+1, WM wrote:
> > >
> > > The set of FISONs is a FISON in classical mathematics. [WM]
> > >
> > Nonsense.

You might try to learn the (generally accepted) meaning of the word "classical mathematics", Mückenheim.

See: https://en.m.wikipedia.org/wiki/Classical_mathematics

Hint: The set of FISONs is infinite, while each and every FISON is finite [by definition]. qed

> It is

infinite. I already told you that. The proof is easy:

If the set of of FISONs Fin were finite there would be a maximum in the set {max(F) : F e Fin}, after all this would be a finite set of natural numbers. Letzt's call this maximum WM. Then the FISON {0, ..., WM+1} would be a FISON not in Fin (since WM e IN and hence WM+1 e IN and WM+1 > WM). Contradiction! Hence the set of of FISONs Fin is not finite, hence infinite. qed

Fritz Feldhase schrieb am Sonntag, 12. Februar 2023 um 12:07:08 UTC+1:
> On Sunday, February 12, 2023 at 11:30:14 AM UTC+1, WM wrote:
> > Fritz Feldhase schrieb am Samstag, 11. Februar 2023 um 21:27:51 UTC+1:
> > All unit fractions are [...] points on the real line. Not even one could be placed between 0 and (0, 1], let alone ℵo unit fractions.
>
> Indeed! Did ANYONE claim otherwise?

Many claim that ℵo unit fractions follow upon every point of (0, 1].. That means they cannot be points in (0, 1].

> Indeed! Did ANYONE claim otherwise?

Unit fractions are positive. If they cannot be in (0, 1] and between 0 and (0, 1], then they cannot be anywhere.

Regards, WM

On Sunday, February 12, 2023 at 11:48:26 AM UTC+1, WM wrote:

> ℵo unit fractions occupy ℵo *different* geometrical points.

If you say so. But where's the problem? ℵo unit fractions are ℵo points on the real number line. So what?

> A mystery is only how mathematicians could overlook this until now.

Errr? What did they "overlook"?

> Do you really claim that there are ℵo different points between 0 and (0, 1]?

Who claimed that nonsense? Can you quote such a claim, please?

MAN, YOU ARE DELUDED, no joke.