WM laid this down on his screen :
> William schrieb am Montag, 13. Februar 2023 um 20:39:09 UTC+1:
>> There is no element of |R for which there is no larger element of |N_P. Thus
>> anything for which there is no larger element of |N_P is not an element of
>> |R (A misleading name will not change this).
>
> Let this statement remain as it stands for a while. Let us first discuss the
> following: Between all points x > 0 and 0 there are ℵo unit fractions
> remaining from the ℵo unit fractions in (0, 1]. Between x =< 0 and 0 there
> are no unit fractions. Do you agree so far?
> That means there are ℵo unit fractions (and the intervals separating them)
> between 0 and (0, 1]. But if so, these unit fractions are not accessible by
> geometric means. They are dark.

Does "accessible by geometric means" mean the same as in all your other
weasel word attempts.

On Tuesday, February 14, 2023 at 6:22:28 AM UTC-4, WM wrote:
> William schrieb am Montag, 13. Februar 2023 um 20:39:09 UTC+1:
> > There is no element of |R for which there is no larger element of |N_P. Thus anything for which there is no larger element of |N_P is not an element of |R (A misleading name will not change this).
> Let this statement remain as it stands for a while.

Note that this means that "dark unit fractions" are not unit fractions. A unit fraction is an element of U_P.

>Let us first discuss the following:
> Between all points x > 0 and 0 there are ℵo [elements of U_P] remaining from the ℵo [elements of U_P] in (0, 1].

There is a set, S(x), of cardinality ℵo . When x changes S(x) may change. There is no element of U_P in S(x) for every element, x, of |R.

> Between x =< 0 and 0 there are no [elements of U_P].
Correct

> Do you agree so far?

> That means there are ℵo [elements of U_P] (and the intervals separating them) between 0 and (0, 1].

Nope. Does not follow and is not true.

JVR schrieb am Dienstag, 14. Februar 2023 um 16:49:31 UTC+1:

> Of course. Dark numbers are the greatest invention of the decade;

Not since Hippasos 2500 years ago? But with similar personal insults and suppressions!

Regards, WM

On Tuesday, February 14, 2023 at 9:05:14 PM UTC+1, WM wrote:
> JVR schrieb am Dienstag, 14. Februar 2023 um 16:49:31 UTC+1:
> >
> > Of course. Dark numbers are the greatest invention of the decade;
> >
> Not since Hippasos 2500 years ago? But with similar personal insults and suppressions!

Yeah, Mückenheim, future generations will refer to you as "Hippasos 2"..

William schrieb am Dienstag, 14. Februar 2023 um 16:34:15 UTC+1:
> On Tuesday, February 14, 2023 at 6:22:28 AM UTC-4, WM wrote:

> > Between all points x > 0 and 0 there are ℵo [elements of U_P] remaining from the ℵo [elements of U_P] in (0, 1].
>
> There is a set, S(x), of cardinality ℵo . When x changes S(x) may change. There is no element of U_P in S(x) for every element, x, of |R.
>
> > Between x =< 0 and 0 there are no [elements of U_P].
> Correct
> > Do you agree so far?
> > That means there are ℵo [elements of U_P] (and the intervals separating them) between 0 and (0, 1].
>
> Nope. Does not follow and is not true.

Agreed. That is nonsense. Nothing fits netwee 0 and (0, 1].
But we know and can prove in ZF the theorem that at the left-hand side of every x > 0 there are ℵo unit fractions [elements of U_P] including their internal distances. Hence every x > 0 has a distance from zero which is at least given by these ℵo unit fractions including their internal distances. This is more than a point.

Therefore all x > 0 for which this theorem is true have a distance of more than one point from zero. This distance is dark.

Regards, WM

On Tuesday, February 14, 2023 at 4:11:38 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 14. Februar 2023 um 16:34:15 UTC+1:
> > On Tuesday, February 14, 2023 at 6:22:28 AM UTC-4, WM wrote:
>
> > > Between all points x > 0 and 0 there are ℵo [elements of U_P] remaining from the ℵo [elements of U_P] in (0, 1].
> >
> > There is a set, S(x), of cardinality ℵo . When x changes S(x) may change. There is no element of U_P in S(x) for every element, x, of |R.
> >
> > > Between x =< 0 and 0 there are no [elements of U_P].
> > Correct
> > > Do you agree so far?
> > > there are ℵo [elements of U_P] (and the intervals separating them) between 0 and (0, 1].
> >
> > Nope. Does not follow and is not true.
> Agreed.

Yes, without "dark unit fractions your statement That is nonsense. Nothing fits netwee 0 and (0, 1].
> But we know and can prove in ZF the theorem that at the left-hand side of every x > 0 there are ℵo unit fractions [elements of U_P] including their internal distances. Hence every x > 0 has a distance from zero which is at least given by these ℵo unit fractions including their internal distances. This is more than a point.
>
> Therefore all x > 0 for which this theorem is true have a distance of more than one point from zero. This distance is dark.
>
> Regards, WM

On Tuesday, February 14, 2023 at 9:11:38 PM UTC+1, WM wrote:

> Nothing fits between 0 and (0, 1].

Indeed!

> But we know and can prove in ZF the theorem that at the left-hand side of every x > 0 there are ℵo unit fractions [elements of U_P] including their internal distances. Hence every x > 0
> has a distance from zero which is at least given by these ℵo unit fractions including their internal distances. This is more than a point.

Indeed: ℵo unit fractions are more than a point!

How would you know?
> all x > 0 [...] have a distance of more than one point from zero.

Right, again!!!

On Tuesday, February 14, 2023 at 4:11:38 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 14. Februar 2023 um 16:34:15 UTC+1:
> > On Tuesday, February 14, 2023 at 6:22:28 AM UTC-4, WM wrote:
>
> > > Between all points x > 0 and 0 there are ℵo [elements of U_P] remaining from the ℵo [elements of U_P] in (0, 1].
> >
> > There is a set, S(x), of cardinality ℵo . When x changes S(x) may change. There is no element of U_P in S(x) for every element, x, of |R.
> >
> > > Between x =< 0 and 0 there are no [elements of U_P].
> > Correct
> > > Do you agree so far?
> > > That means there are ℵo [elements of U_P] (and the intervals separating them) between 0 and (0, 1].
> >
> > Nope. *Does not follow* and is not true
> Agreed.

So you agree that your statement "That means" is false

Fritz Feldhase schrieb am Dienstag, 14. Februar 2023 um 21:28:26 UTC+1:
> On Tuesday, February 14, 2023 at 9:11:38 PM UTC+1, WM wrote:
>
> > Nothing fits between 0 and (0, 1].
>
> Indeed!
> > But we know and can prove in ZF the theorem that at the left-hand side of every x > 0 there are ℵo unit fractions [elements of U_P] including their internal distances. Hence every x > 0
> > has a distance from zero which is at least given by these ℵo unit fractions including their internal distances. This is more than a point.
> Indeed: ℵo unit fractions are more than a point!

And they (or so many) sit at the left-hand side of every x > 0 for ehich the theorem holds. Or should by quantifier magic some x > 0 go with less of them?
>
> How would you know?
>
> > all x > 0 [...] have a distance of more than one point from zero.
>
> Right, again!!!

This "more" consists of ℵo unit fractions and the finite distances between them. A lot of dark points isolating all x > 0 and all definable unit fractions from zero.

Regards, WM

William schrieb am Dienstag, 14. Februar 2023 um 21:34:02 UTC+1:
> On Tuesday, February 14, 2023 at 4:11:38 PM UTC-4, WM wrote:
> > William schrieb am Dienstag, 14. Februar 2023 um 16:34:15 UTC+1:
> > > On Tuesday, February 14, 2023 at 6:22:28 AM UTC-4, WM wrote:
> >
> > > > Between all points x > 0 and 0 there are ℵo [elements of U_P] remaining from the ℵo [elements of U_P] in (0, 1].
> > >
> > > There is a set, S(x), of cardinality ℵo . When x changes S(x) may change. There is no element of U_P in S(x) for every element, x, of |R..
> > >
> > > > Between x =< 0 and 0 there are no [elements of U_P].
> > > Correct
> > > > Do you agree so far?
> > > > That means there are ℵo [elements of U_P] (and the intervals separating them) between 0 and (0, 1].
> > >
> > > Nope. *Does not follow* and is not true
> > Agreed.
>
> So you agree that your statement "That means" is false.

Of course. This statement is nonsense because nothing fits between 0 and (0, 1]. (See my "correct" above.) It was only rhetoric in order to provoke you to state that these ℵo [elements of U_P] (and the intervals separating them) are in (0, 1] at the left-hand side of every x > 0 which satisfies the theorem of ZF. All these x > 0 are at the right-hand side of the dark mess consisting of these ℵo unit fractions and the intervals separating them, a considerable set of points, isolating them from zero.

Regards, WM

On Tuesday, February 14, 2023 at 5:40:41 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 14. Februar 2023 um 21:34:02 UTC+1:
> > On Tuesday, February 14, 2023 at 4:11:38 PM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 14. Februar 2023 um 16:34:15 UTC+1:
> > > > On Tuesday, February 14, 2023 at 6:22:28 AM UTC-4, WM wrote:
> > >
> > > > > Between all points x > 0 and 0 there are ℵo [elements of U_P] remaining from the ℵo [elements of U_P] in (0, 1].
> > > >
> > > > There is a set, S(x), of cardinality ℵo . When x changes S(x) may change. There is no element of U_P in S(x) for every element, x, of |R.
> > > >
> > > > > Between x =< 0 and 0 there are no [elements of U_P].
> > > > Correct
> > > > > Do you agree so far?
> > > > > That means there are ℵo [elements of U_P] (and the intervals separating them) between 0 and (0, 1].
> > > >
> > > > Nope. *Does not follow* and is not true
> > > Agreed.
> >
> > So you agree that your statement "That means" is false.
>
> Of course.

So, you agree that the fact that for each x>0, there are an infinite number of unit fractions between 0 and (0,1] does not imply that there is an infinite number of unit factions between 0 and the set of real numbers (0,1]. So you agree there is no contradiction in saying that for each x>0, there are an infinite number of unit fractions between 0 and (0,1] and saying that there is no unit faction between 0 and (0,1]

Note that no "dark unit faction" is a unit faction

On Tuesday, February 14, 2023 at 5:40:41 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 14. Februar 2023 um 21:34:02 UTC+1:
> > On Tuesday, February 14, 2023 at 4:11:38 PM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 14. Februar 2023 um 16:34:15 UTC+1:
> > > > On Tuesday, February 14, 2023 at 6:22:28 AM UTC-4, WM wrote:
> > >
> > > > > Between all points x > 0 and 0 there are ℵo [elements of U_P] remaining from the ℵo [elements of U_P] in (0, 1].
> > > >
> > > > There is a set, S(x), of cardinality ℵo . When x changes S(x) may change. There is no element of U_P in S(x) for every element, x, of |R.
> > > >
> > > > > Between x =< 0 and 0 there are no [elements of U_P].
> > > > Correct
> > > > > Do you agree so far?
> > > > > That means there are ℵo [elements of U_P] (and the intervals separating them) between 0 and (0, 1].
> > > >
> > > > Nope. *Does not follow* and is not true
> > > Agreed.
> >
> > So you agree that your statement "That means" is false.
>
> Of course.

So, you agree that the fact that for each x>0, there are an infinite number of unit fractions between 0 and x does not imply that there is an infinite number of unit factions between 0 and the set of real numbers (0,1]. So you agree there is no contradiction in saying that for each x>0, there are an infinite number of unit fractions between 0 and x and saying that there is no unit fraction between 0 and (0,1]

Note that no "dark unit faction" is a real number. So there are no "dark unit fractions" in the set of real numbers (0.1] (indeed the is no "dark" anything in the set of real numbers (0,1]

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

> Ben Bacarisse schrieb am Dienstag, 14. Februar 2023 um 03:20:42 UTC+1:

>> Do you still think that this
>>
>> (c) ({u | u ∈ U and u <= x} is empty for x <= 0) /\
>> ({u | u ∈ U and u <= x} is not finite for x > 0)
>>
>> is wrong, or is this a fact we agree to move on from?
>
> If both parts of (c) were true, then...

Hmm... You seem afraid to answer. (c) is either a true statement or it
is not. You appear to want to dispute it, but you won't answer the
direct question. Is (c) wrong (in your opinion) or not?

--
Ben.

William schrieb am Dienstag, 14. Februar 2023 um 23:35:32 UTC+1:

> So, you agree that the fact that for each x>0, there are an infinite number of unit fractions between 0 and x does not imply that there is an infinite number of unit factions between 0 and the set of real numbers (0,1].

For each definable x > 0, there are an infinite number of unit fractions between 0 and x.

> So you agree there is no contradiction in saying that for each x>0, there are an infinite number of unit fractions between 0 and x and saying that there is no unit fraction between 0 and (0,1]

There is a contradiction.

(1) According to a theorem of ZF there are ℵo unit fractions between zero and every point x > 0 on the real axis. That means every positive x has a distance from zero which contains ℵo unit fractions and their internal distances. The complete interval (0, 1] however has no distance from zero, i.e., between zero and (0, 1] there is nothing.

(2) Therefore all positive points x which are subject to the theorem (that there is a distance from zero which contains ℵo unit fractions and their internal distances), have a distance from zero which contains ℵo unit fractions and their internal distances.

(3) Since the interval (0, 1] has not even one point distance from zero, it contains real points x > 0 which are not obeying said theorem.

(4) These points are dark, i.e., they cannot be chosen as individuals but only collectively. Every point that can be chosen is subject to said theorem..

(5) Note that there is no quantifier exchange! I did not claim that there is a distance (which contains ℵo unit fractions and their internal distances) between zero and all positive points x > 0. That would be a quantifier exchange. It would be wrong because there is nothing between zero and (0, 1]. I only claimed that the distance (to zero which contains ℵo unit fractions and their internal distances) lies between zero and all positive points x > 0 which are subject to said theorem of ZF.

Regards, WM

Ben Bacarisse schrieb am Mittwoch, 15. Februar 2023 um 04:30:57 UTC+1:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
> > Ben Bacarisse schrieb am Dienstag, 14. Februar 2023 um 03:20:42 UTC+1:
>
> >> Do you still think that this
> >>
> >> (c) ({u | u ∈ U and u <= x} is empty for x <= 0) /\
> >> ({u | u ∈ U and u <= x} is not finite for x > 0)
> >>
> >> is wrong, or is this a fact we agree to move on from?
> >
> > If both parts of (c) were true, then...
>
> Hmm... You seem afraid to answer. (c) is either a true statement or it
> is not. You appear to want to dispute it, but you won't answer the
> direct question. Is (c) wrong (in your opinion) or not?

Statement (c) is blatantly wrong. Here you can learn why:

(1) According to a theorem of ZF there are ℵo unit fractions between zero and every point x > 0 on the real axis. That means every positive x has a distance from zero which contains ℵo unit fractions and their internal distances. The complete interval (0, 1] however has no distance from zero, i.e., between zero and (0, 1] there is nothing.

(2) Therefore all positive points x which are subject to the theorem (that there is a distance from zero which contains ℵo unit fractions and their internal distances), have a distance from zero which contains ℵo unit fractions and their internal distances.

(3) Since the interval (0, 1] has not even one point distance from zero, it contains real points x > 0 which are not obeying said theorem.

(4) These points are dark, i.e., they cannot be chosen as individuals but only collectively. Every point that can be chosen is subject to said theorem..

(5) Note that there is no quantifier exchange! I did not claim that there is a distance (which contains ℵo unit fractions and their internal distances) between zero and all positive points x > 0. That would be a quantifier exchange. It would be wrong because there is nothing between zero and (0, 1]. I only claimed that the distance (to zero which contains ℵo unit fractions and their internal distances) lies between zero and all positive points x > 0 which are subject to said theorem of ZF.

Regards, WM

On Wednesday, February 15, 2023 at 1:35:57 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 14. Februar 2023 um 23:35:32 UTC+1:
>

> For each definable x > 0, there are an infinite number of unit fractions between 0 and x.

And since each real numbers is finite, and hence has the property it can be written down. we can lose the "definable".

For each real number x>0 there are an infinite number of unit fractions [elements of U_P] between 0 and x.

Specifically, for every real number x, there is a set S(x) of elements of U_P with the cardinality of S(x) aleph_0.

(Note that S(x) can change if x changes, and there is no unit fraction {Element of U_P] that is in S(x) for every x)

On Wednesday, February 15, 2023 at 6:39:01 PM UTC+1, WM wrote:
> Ben Bacarisse schrieb am Mittwoch, 15. Februar 2023 um 04:30:57 UTC+1:
> >
> > Do you still think that this
> >
> > (c) ({u | u ∈ U and u <= x} is empty for x <= 0) /\ ({u | u ∈ U and u <= x} is not finite for x > 0)
> >
> > is wrong[?]
>
> Statement (c) is blatantly wrong.

Ok. Which part(s) of it:

(a) {u | u ∈ U and u <= x} is empty for x <= 0

and/or

(b) {u | u ∈ U and u <= x} is not finite for x > 0 ,

where U = {1/n : n e IN} and x e IR.

?

On Wednesday, February 15, 2023 at 1:35:57 PM UTC-4, WM wrote:

> (3) Since the interval (0, 1] has not even one point distance from zero, it contains real points x > 0 which are not obeying said theorem.

Nope. the fact that the set of real numbers (0,1] has some property ("has not even one point distant from zero") , does not mean that an element of the set has this property.

William schrieb am Mittwoch, 15. Februar 2023 um 20:00:26 UTC+1:
> On Wednesday, February 15, 2023 at 1:35:57 PM UTC-4, WM wrote:
> > (3) Since the interval (0, 1] has not even one point distance from zero, it contains real points x > 0 which are not obeying said theorem.
> Nope. the fact that the set of real numbers (0,1] has some property ("has not even one point distant from zero") , does not mean that an element of the set has this property.

A set has other properties than its elements. The set is a single object whereas its elements are many. A set can be infinite although all its elements may be finite.

But that is not a reason for outrageously foolish claims.

The set (0, 1] reaches precisely as far as its finite initial segments [1/n, ..., 1/1]. The set covers no point without at least one of its finite initial segments covering this point. Only in matheology this could be forgotten.

Regards, WM