William schrieb am Sonntag, 5. Februar 2023 um 23:26:07 UTC+1:
> On Sunday, February 5, 2023 at 5:21:32 PM UTC-4, WM wrote:
> unit faction.
>
> >... there is nothing between all unit fractions and zero.
>
> Correct, there is no element of |R between the set of unit fractions ("all unit fractions") and 0,

NUF((0, 1]) = 0

> There are an infinite number of "infinitesimal unit factions" between 0 and the set of elements of |R, (0,1].

No. There is no point of the real measur betwee 0 and (0, 1]. This proves that there is nothing. No infinitesimal unit factions either.

Only ∀n ∈ ℕ_def: NUF(1/n) = ℵo.

This shows that many dark unit fractions are between every definable unit fraction and zero.

Regards, WM

Jim Burns schrieb am Montag, 6. Februar 2023 um 01:59:54 UTC+1:

> Now I'm thinking of
> infinitely-many finite fractions.
>
> Each of those fractions is stationary
> with respect to zero.
>
They all are present in the interval (0, 1], no additional unit fraction can be created. If you remove any point of the edge of the interval you remove at least one unit fraction. They fill it to the brim. But ∀n ∈ ℕ_def: NUF(1/n) = ℵo. The definable unit fractions don't fill it to the brim. What is the difference made of?

Regards, WM

Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 09:54:15 UTC+1:

> If x is a unit fraction, then [by definition] there is a natural number n such that x = 1/n. Then the unit fraction 1/(n+1) is between x and zero: 0 < 1(n+1) < 1/n = x.

That is true for definable unit fractions only.
∀n ∈ ℕ_def: NUF(1/n) = ℵo.
But NUF ((0, 1]) = 0.
If you remove any point of the edge of the interval you remove at least one unit fraction. They fill it to the brim. But ∀n ∈ ℕ_def: NUF(1/n) = ℵo. The definable unit fractions don't fill it to the brim. What is the difference made of?

Regards, WM

On Monday, February 6, 2023 at 11:35:50 AM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 09:54:15 UTC+1:
> >
> > If x is a unit fraction, then [by definition] there is a natural number n such that x = 1/n. Then the unit fraction 1/(n+1) is between x and zero: 0 < 1(n+1) < 1/n = x.
>>
> That is true for

each and every x in (0, 1].

Learn some logic learn some math, Mückenheim.

Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 12:02:14 UTC+1:
> On Monday, February 6, 2023 at 11:35:50 AM UTC+1, WM wrote:
> > Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 09:54:15 UTC+1:
> > >
> > > If x is a unit fraction, then [by definition] there is a natural number n such that x = 1/n. Then the unit fraction 1/(n+1) is between x and zero: 0 < 1(n+1) < 1/n = x.
> >>
> > That is true for
> each and every x in (0, 1].

No. It is not true for each and every x because (0, 1] is nothing but all these x. And between those closest to zero and zero there is nothing.
>
> Learn some logic

Here we see by your example that "'Mathematical logic' has completely deformed the thinking of mathematicians and of philosophers". [V.48] [L. Wittgenstein: "Remarks on the foundations of mathematics", Wiley-Blackwell (1991)]

Regards, WM

Fritz Feldhase <franz.fritschee.ff@gmail.com> writes:

> On Monday, February 6, 2023 at 11:35:50 AM UTC+1, WM wrote:
>> Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 09:54:15 UTC+1:
>> >
>> > If x is a unit fraction, then [by definition] there is a natural
>> > number n such that x = 1/n. Then the unit fraction 1/(n+1) is
>> > between x and zero: 0 < 1(n+1) < 1/n = x.
>>>
>> That is true for
>
> each and every x in (0, 1].
>
> Learn some logic learn some math, Mückenheim.

I am never 100% sure that WM is not just "trying it on" as the saying
goes. He seems always to be searching for mysteries and for the most
mysterious ways to word them. I'm sure 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.

The trick is to find a way to make it mysterious and "distance" is
promising, especially when we add a dash of the usual every/all magic:
"Look", he tells his unfortunate students, "there is infinite distance
between every 1/n and 0, but no distance between all 1/n and zero!".
(And to Cantor et. al. he says "Look on my Works, ye Mighty, and
despair!".)

--
Ben.

On Monday, February 6, 2023 at 12:57:27 PM UTC+1, Ben Bacarisse wrote:
> Fritz Feldhase <franz.fri...@gmail.com> writes:
>
> > On Monday, February 6, 2023 at 11:35:50 AM UTC+1, WM wrote:
> >> Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 09:54:15 UTC+1:
> >> >
> >> > If x is a unit fraction, then [by definition] there is a natural
> >> > number n such that x = 1/n. Then the unit fraction 1/(n+1) is
> >> > between x and zero: 0 < 1(n+1) < 1/n = x.
> >>>
> >> That is true for
> >
> > each and every x in (0, 1].
> >
> I am never 100% sure that WM is not just "trying it on" as the saying
> goes. He seems always to be searching for mysteries and for the most
> mysterious ways to word them. I'm sure 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.

If you say so. WM's reply:

| "No. It is not true for each and every x because (0, 1] is nothing but all these x. And between those closest to zero and zero there is nothing."

Complete nonsense, as usual.

Hard to believe that this man taught Analysis at Hochschule Augsburg.

On Monday, February 6, 2023 at 12:22:54 PM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 12:02:14 UTC+1:
> > On Monday, February 6, 2023 at 11:35:50 AM UTC+1, WM wrote:
> > > Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 09:54:15 UTC+1:
> > > >
> > > > If x is a unit fraction, then [by definition] there is a natural number n such that x = 1/n. Then the unit fraction 1/(n+1) is between x and zero: 0 < 1(n+1) < 1/n = x.
> > > >
> > > That is true for
> > >
> > each and every x in (0, 1].
> >
> No. <nonsense>

Yes.

On Monday, February 6, 2023 at 12:57:27 PM UTC+1, Ben Bacarisse wrote:
> Fritz Feldhase <franz.fri...@gmail.com> writes:
>
> > On Monday, February 6, 2023 at 11:35:50 AM UTC+1, WM wrote:
> >> Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 09:54:15 UTC+1:
> >> >
> >> > If x is a unit fraction, then [by definition] there is a natural
> >> > number n such that x = 1/n. Then the unit fraction 1/(n+1) is
> >> > between x and zero: 0 < 1(n+1) < 1/n = x.
> >>>
> >> That is true for
> >
> > each and every x in (0, 1].
> >
> > Learn some logic learn some math, Mückenheim.
> I am never 100% sure that WM is not just "trying it on" as the saying
> goes. He seems always to be searching for mysteries and for the most
> mysterious ways to word them. I'm sure 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.
>
> The trick is to find a way to make it mysterious and "distance" is
> promising, especially when we add a dash of the usual every/all magic:
> "Look", he tells his unfortunate students, "there is infinite distance
> between every 1/n and 0, but no distance between all 1/n and zero!".
> (And to Cantor et. al. he says "Look on my Works, ye Mighty, and
> despair!".)
>
> --
> Ben.

I think Mücke actually believes that by babbling about 1/n 'being far' from 0
he is saying something that couldn't be said, mutatis mutandis, in a natural manner
about n 'being far' from infinity.

Primarily he is fooling himself. Meanwhile, he is demonstrating that he doesn't
understand basic concepts such as limit, metric and the set functions known as 'measure'.

I am quite sure that you assume that he understands a lot more than he actually does.
It's obvious, for example, that his book consists almost entirely of stuff he copied
blindly.

On Monday, February 6, 2023 at 6:23:40 AM UTC-4, WM wrote:
> William schrieb am Sonntag, 5. Februar 2023 um 23:26:07 UTC+1:
> > On Sunday, February 5, 2023 at 5:21:32 PM UTC-4, WM wrote:
> > unit faction.
> >
> > >... there is nothing between all unit fractions and zero.
> >
> > Correct, there is no element of |R between the set of unit fractions ("all unit fractions") and 0,
> NUF((0, 1]) = 0
> > There are an infinite number of "infinitesimal unit factions" between 0 and the set of elements of |R, (0,1].
> No. There is no point of the real measur betwee 0 and (0, 1].

An "infinitesimal unit fraction" is not a unit fraction. It does not correspond to a point on the real number line.

Ben Bacarisse schrieb am Montag, 6. Februar 2023 um 12:57:27 UTC+1:
> I'm sure 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.

Of course there is no problem to understand what you are doing. But you have problems to understand what I am doing unless you are trying very hard, although it is very simple:

All the unit fractions fill the half open interval (0, 1] to the brim. That means you cannot cut any point at its edges without cutting a unit fraction too. Note that I need not and do talk about the set of unit fractions here. So you have no chance to apply your matheologial nonsense about sets filling differences like that between all FISONs and ℕ.

Can you understand this?

That means there is no unit fraction between all unit fractions and zero.
Similarly there is no natural number between all natural numbers and omega.

But there are ℵo unit fractions between every chosen unit fraction and zero.
Similarly there are ℵo natural numbers between every chosen natural number and omega,

Can you understand this when you are trying very, very hard?

Regards, WM

JVR schrieb am Montag, 6. Februar 2023 um 15:51:47 UTC+1:

> I think

I doubt that. But here you may try to prove the contrary.

All the unit fractions 1/n with n ∈ ℕ fill the half open interval (0, 1] to the brim. That means you cannot cut any point at its edges without cutting a unit fraction too.
That means there is no unit fraction between all unit fractions, which are points on the real line, and zero.

But there are ℵo unit fractions between every chosen unit fraction and zero. That means you cannot choose every unit fraction.

Regards, WM

William schrieb am Montag, 6. Februar 2023 um 15:53:22 UTC+1:
> On Monday, February 6, 2023 at 6:23:40 AM UTC-4, WM wrote:
>
> > There is no point of the real measur between 0 and (0, 1].
> An "infinitesimal unit fraction" is not a unit fraction. It does not correspond to a point on the real number line.

All the *real* unit fractions 1/n with n ∈ ℕ fill the half open interval (0, 1] to the brim. That means you cannot cut any point at its edges without cutting a real unit fraction too.
That means there is no unit fraction between all real unit fractions, which are points on the real line, and zero.

But there are ℵo unit fractions between every chosen unit fraction and zero.

Regards, WM

Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 13:12:40 UTC+1:
(0, 1] is nothing but all these x. And between those closest to zero and zero there is nothing."
> Complete nonsense, as usual.
>
All the unit fractions fill the half open interval (0, 1] to the brim. That means you cannot cut any point at its edges without cutting a unit fraction too. Note that I need not and do talk about the set of unit fractions here. So you have no chance to apply your matheologial nonsense about sets filling differences like that between all FISONs and ℕ.

Between all unit fractions and zero there is nothing. But between all unit fractions which can be chosen, there is more. Hence not all unit fractions can be chosen.

Regards, WM

On Monday, February 6, 2023 at 11:05:57 AM UTC-4, WM wrote:
> William schrieb am Montag, 6. Februar 2023 um 15:53:22 UTC+1:

> > An "infinitesimal unit fraction" is not a unit fraction. It does not correspond to a point on the real number line.
> All the *real* unit fractions 1/n with n ∈ ℕ fill the half open interval (0, 1] to the brim.
>That means you cannot cut any point at its edges without cutting a real unit fraction too.

Correct. So what? An "infinitesimal unit fraction" not a unit fraction.

Fritz Feldhase <franz.fritschee.ff@gmail.com> writes:

> On Monday, February 6, 2023 at 12:57:27 PM UTC+1, Ben Bacarisse wrote:
>> Fritz Feldhase <franz.fri...@gmail.com> writes:
>>
>> > On Monday, February 6, 2023 at 11:35:50 AM UTC+1, WM wrote:
>> >> Fritz Feldhase schrieb am Montag, 6. Februar 2023 um 09:54:15 UTC+1:
>> >> >
>> >> > If x is a unit fraction, then [by definition] there is a natural
>> >> > number n such that x = 1/n. Then the unit fraction 1/(n+1) is
>> >> > between x and zero: 0 < 1(n+1) < 1/n = x.
>> >>>
>> >> That is true for
>> >
>> > each and every x in (0, 1].
>> >
>> I am never 100% sure that WM is not just "trying it on" as the saying
>> goes. He seems always to be searching for mysteries and for the most
>> mysterious ways to word them. I'm sure 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.
>
> If you say so. WM's reply:
>
> | "No. It is not true for each and every x because (0, 1] is nothing
> | but all these x. And between those closest to zero and zero there is
> | nothing."

Sure he says that. But you can't take that "no" to mean he disagrees
(seriously, you can't!). If pressed he will say that it's "matheology"
that says "no". In any tight corner, he'll agree with an irrefutable
fact saying it true in "potential infinity" but will claims its /us/ who
disagree (with ourselves!) due to something vaguely stated that he holds
to be contradictory.

It would take 10 or 20 cycles, pressing him again and again to
explicitly deny that there is an x with 0 < x < 1/n for each n in N+.
If you kept at it he would agree, with a blazé "of course that's true in
potential infinity, but mathelolgians say there is nothing between 0 and
all (0,1]".

> Complete nonsense, as usual.

Yes, but safe, flexible noncommittal nonsense that can be repudiated at
any time.

> Hard to believe that this man taught Analysis at Hochschule Augsburg.

Apart from the errors, his textbook is entirely conventional
mathematics, so I can (just about) believe it. What gives me pause is
that he does not really know what his textbook says. He denies that any
non-constant set sequence has a limit, yet his book defines one (and
gives an example). He denies that his definition of point-wise
convergence (surely a run-of-the-mill concept in analysis) applies to
discrete functions like indicator functions. I don't think he really
know what he's written in the book.

--
Ben.

William schrieb am Montag, 6. Februar 2023 um 16:11:00 UTC+1:
> On Monday, February 6, 2023 at 11:05:57 AM UTC-4, WM wrote:
> > William schrieb am Montag, 6. Februar 2023 um 15:53:22 UTC+1:
>
> > > An "infinitesimal unit fraction" is not a unit fraction. It does not correspond to a point on the real number line.
> > All the *real* unit fractions 1/n with n ∈ ℕ fill the half open interval (0, 1] to the brim.
> >That means you cannot cut any point at its edges without cutting a real unit fraction too.
> Correct. So what? An "infinitesimal unit fraction" not a unit fraction.

Which unit fractions filling the interval are infinitesimal?

Regards, WM

On Monday, February 6, 2023 at 11:26:25 AM UTC-4, WM wrote:
> William schrieb am Montag, 6. Februar 2023 um 16:11:00 UTC+1:
> > On Monday, February 6, 2023 at 11:05:57 AM UTC-4, WM wrote:
> > > William schrieb am Montag, 6. Februar 2023 um 15:53:22 UTC+1:
> >
> > > > An "infinitesimal unit fraction" is not a unit fraction. It does not correspond to a point on the real number line.
> > > All the *real* unit fractions 1/n with n ∈ ℕ fill the half open interval (0, 1] to the brim.
> > >That means you cannot cut any point at its edges without cutting a real unit fraction too.
> > Correct. So what? An "infinitesimal unit fraction" not a unit fraction.
> Which unit fractions filling the interval are infinitesimal?
There is no unit fraction filling the interval (0,1] that is infinitesimal. So what? The things you call "infinitesimal unit fractions" are not unit fractions.

William schrieb am Montag, 6. Februar 2023 um 16:38:13 UTC+1:
> On Monday, February 6, 2023 at 11:26:25 AM UTC-4, WM wrote:
> > William schrieb am Montag, 6. Februar 2023 um 16:11:00 UTC+1:
> > > On Monday, February 6, 2023 at 11:05:57 AM UTC-4, WM wrote:
> > > > William schrieb am Montag, 6. Februar 2023 um 15:53:22 UTC+1:
> > >
> > > > > An "infinitesimal unit fraction" is not a unit fraction. It does not correspond to a point on the real number line.
> > > > All the *real* unit fractions 1/n with n ∈ ℕ fill the half open interval (0, 1] to the brim.
> > > >That means you cannot cut any point at its edges without cutting a real unit fraction too.
> > > Correct. So what? An "infinitesimal unit fraction" not a unit fraction.
> > Which unit fractions filling the interval are infinitesimal?
> There is no unit fraction filling the interval (0,1] that is infinitesimal. So what?

Nevertheless some are so close to 0 that nothing fits in between.

> The things you call "infinitesimal unit fractions" are not unit fractions..

I did not call them so. That was your idea. I only use unit fractions 1/n with n ∈ ℕ.

Regards, WM

On Monday, February 6, 2023 at 11:53:48 AM UTC-4, WM wrote:
>I only use unit fractions 1/n with n ∈ ℕ.

Every unit fraction is an element of |R The reciprocals of the things you call "dark natural numbers" are infinitesimal and thus are not elements of |R. So the things you call "dark natural numbers" are not natural numbers..

On Monday, February 6, 2023 at 4:05:54 PM UTC+1, WM wrote:
> JVR schrieb am Montag, 6. Februar 2023 um 15:51:47 UTC+1:
>
> > I think
>
> I doubt that. But here you may try to prove the contrary.
>
> All the unit fractions 1/n with n ∈ ℕ fill the half open interval (0, 1] to the brim.

You are right. I admit it. I have never heard that half-open intervals have a
brim. I assume that's a recent discovery. Please tell us more.

> That means you cannot cut any point at its edges without cutting a unit fraction too.

You are right. I admit it. I didn't know that points have edges and that you can cut them.
Are there perforations for prescribed cuts, or can you decide yourself where
to cut? So presumably you can make 2 points out of one. Can you then continue
the process? Can you make 2 out of 4?

And what happens when you accidentally cut a unit fraction while cutting a point?

> That means there is no unit fraction between all unit fractions, which are points on the real line, and zero.

I admit it. I can't make head or tail of that sentence. Are you quite sure you are saying what
you intended to say?

> But there are ℵo unit fractions between every chosen unit fraction and zero. That means you cannot choose every unit fraction.

I admit it. That's a mystery to me. Which unit fraction is the first one you can't choose?

On Monday, February 6, 2023 at 4:53:48 PM UTC+1, WM wrote:
> William schrieb am Montag, 6. Februar 2023 um 16:38:13 UTC+1:
> > On Monday, February 6, 2023 at 11:26:25 AM UTC-4, WM wrote:
> > > William schrieb am Montag, 6. Februar 2023 um 16:11:00 UTC+1:
> > > > On Monday, February 6, 2023 at 11:05:57 AM UTC-4, WM wrote:
> > > > > William schrieb am Montag, 6. Februar 2023 um 15:53:22 UTC+1:
> > > >
> > > > > > An "infinitesimal unit fraction" is not a unit fraction. It does not correspond to a point on the real number line.
> > > > > All the *real* unit fractions 1/n with n ∈ ℕ fill the half open interval (0, 1] to the brim.
> > > > >That means you cannot cut any point at its edges without cutting a real unit fraction too.
> > > > Correct. So what? An "infinitesimal unit fraction" not a unit fraction.
> > > Which unit fractions filling the interval are infinitesimal?
> > There is no unit fraction filling the interval (0,1] that is infinitesimal. So what?
> Nevertheless some are so close to 0 that nothing fits in between.

You are right. That's inconceivable. They can't possibly all fit. That would mean
that some unit fractions don't have a brim and can't be cut, since they would be too thin.

Remember Achilles and why he couldn't catch the turtle.

William schrieb am Montag, 6. Februar 2023 um 17:22:39 UTC+1:
> On Monday, February 6, 2023 at 11:53:48 AM UTC-4, WM wrote:
> >I only use unit fractions 1/n with n ∈ ℕ.
> Every unit fraction is an element of |R

Right!

> The reciprocals of the things you call "dark natural numbers" are infinitesimal and thus are not elements of |R.

Wrong.

All real unit fractions are in (0, 1]. Their infimum is 0. In every epsilon-neighbourhood of 0 there are almost all unit fractions. They are there as real numbers and as points on the real line.

> So the things you call "dark natural numbers" are not natural numbers.

They are natural numbers like all fractions are fractions. The darkness occurs only because you cannot choose a unit fractions with less than ℵo successors. But there are real unit fractions with less successors, because the slightest cut will remove them.

Regards, WM

JVR schrieb am Montag, 6. Februar 2023 um 18:13:27 UTC+1:
> On Monday, February 6, 2023 at 4:05:54 PM UTC+1, WM wrote:

> > All the unit fractions 1/n with n ∈ ℕ fill the half open interval (0, 1] to the brim.
> You are right. I admit it. I have never heard that half-open intervals have a
> brim. I assume that's a recent discovery. Please tell us more.

The brim is the supremum or infinimum. Here it is the infinimum 0.

> > That means you cannot cut any point at its edges without cutting a unit fraction too.
> You are right. I admit it. I didn't know that points have edges

Edges are points.
> and that you can cut them.

WE cannot cut a point from the left edge of (0, 1], but the slightes cut will remove all dark unit fractions.

> Are there perforations for prescribed cuts, or can you decide yourself where
> to cut?

We can decide it. It is called epsilon neighbourhood and contains all dark unit fractions and a lot of visible unit fractions.
>
> And what happens when you accidentally cut a unit fraction while cutting a point?

Irrelevant. You can cut at a unit fraction or between two.

> > That means there is no unit fraction between all unit fractions, which are points on the real line, and zero.
> I admit it. I can't make head or tail of that sentence.

All unit fractions are in (0, 1]. No smaller interval contains all. And between 0 and (0, 1] there is no unit fraction.

> > But there are ℵo unit fractions between every chosen unit fraction and zero. That means you cannot choose every unit fraction.
> I admit it. That's a mystery to me. Which unit fraction is the first one you can't choose?

There is none definable. For every unit fraction there is a smaller one that you can choose. Nevertheless always ℵo unit fractions remain dark.. They cannot be distinguished but can only be chosen collectively, for instance by (0, eps)

Regards, WM

On Monday, February 6, 2023 at 2:23:43 PM UTC-4, WM wrote:
> William schrieb am Montag, 6. Februar 2023 um 17:22:39 UTC+1:
> > the things you call "dark natural numbers" are not natural numbers.
> They are natural numbers

Nope, the reciprocal of a natural number is a real number. The reciprocal or the things you call "dark natural numbers" are smaller than any element of U_P and thus infinitesimal.