For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x. At x = 0 there is no unit fraction between 0 and x. That means between (0, 1] and 0 all ℵo points (together with the intervals between adjacent unit fraction) must have gone. That however is impossible.

Regards, WM

on 2/12/2023, WM supposed :

> For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x.

.... and possibly even more irrational numbers. So what?

> At x = 0 there is no unit fraction between 0 and x.

Again, so what?

> That means between (0, 1] and
> 0 all ℵo points (together with the intervals between adjacent unit fraction)
> must have gone. That however is impossible.

You say 'gone' like it was an action slowly happening.

FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 20:20:59 UTC+1:
> on 2/12/2023, WM supposed :
> > For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x.
> ... and possibly even more irrational numbers. So what?
> > At x = 0 there is no unit fraction between 0 and x.
> Again, so what?
> > That means between (0, 1] and
> > 0 all ℵo points (together with the intervals between adjacent unit fraction)
> > must have gone. That however is impossible.
> You say 'gone' like it was an action slowly happening.

No action. There is a function: NUF(x), the number of unit fractions between 0 and x. This function is
f(x) = ℵo for 0 < x < oo
and f(0) = 0.

That means the NUF must decrease from ℵo to 0 in no distance on the real axis. That is not possible because every unit fraction occupies its own point. In addition between every two adjacent unit fractions there is a non-empty interval.

Regards, WM

WM pretended :
> FromTheRafters schrieb am Sonntag, 12. Februar 2023 um 20:20:59 UTC+1:
>> on 2/12/2023, WM supposed :
>>> For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x. ...
>>> and possibly even more irrational numbers. So what? At x = 0 there is no
>>> unit fraction between 0 and x. Again, so what? That means between (0, 1]
>>> and 0 all ℵo points (together with the intervals between adjacent unit
>>> fraction) must have gone. That however is impossible.
>> You say 'gone' like it was an action slowly happening.
>
> No action. There is a function: NUF(x), the number of unit fractions between
> 0 and x. This function is f(x) = ℵo for 0 < x < oo
> and f(0) = 0.
>
> That means the NUF must decrease from ℵo to 0 in no distance on the real
> axis.

Does it? Is it necessarily a continuous function?

FromTheRafters schrieb am Montag, 13. Februar 2023 um 00:25:38 UTC+1:
> WM pretended :
> > There is a function: NUF(x), the number of unit fractions between
> > 0 and x. This function is f(x) = ℵo for 0 < x < oo
> > and f(0) = 0.
> >
> > That means the NUF must decrease from ℵo to 0 in no distance on the real
> > axis.
> Does it? Is it necessarily a continuous function?

It gives the number of unit fractions sitting as geometric points on the reral line at the left-hand side of x. These points cannot disperse. So they must sit between 0 and (0, oo). But they can't. So many do not fit in between. Not even a single one, let alone infinitely many including the non-empty intervals between them.

Regards, WM

WM was thinking very hard :
> FromTheRafters schrieb am Montag, 13. Februar 2023 um 00:25:38 UTC+1:
>> WM pretended :
>>> There is a function: NUF(x), the number of unit fractions between
>>> 0 and x. This function is f(x) = ℵo for 0 < x < oo
>>> and f(0) = 0.
>>>
>>> That means the NUF must decrease from ℵo to 0 in no distance on the real
>>> axis.
>> Does it? Is it necessarily a continuous function?
>
> It gives the number of unit fractions sitting as geometric points on the
> reral line at the left-hand side of x.

No it doesn't.

FromTheRafters schrieb am Montag, 13. Februar 2023 um 10:09:56 UTC+1:
> WM was thinking very hard :
> > FromTheRafters schrieb am Montag, 13. Februar 2023 um 00:25:38 UTC+1:
> >> WM pretended :
> >>> There is a function: NUF(x), the number of unit fractions between
> >>> 0 and x. This function is f(x) = ℵo for 0 < x < oo
> >>> and f(0) = 0.
> >>>
> >>> That means the NUF must decrease from ℵo to 0 in no distance on the real
> >>> axis.
> >> Does it? Is it necessarily a continuous function?
> >
> > It gives the number of unit fractions sitting as geometric points on the
> > reral line at the left-hand side of x.
> No it doesn't.

There are ℵo unit fractions at the left-hand side of every x > 1. ℵo of them (of these very unit fractions) are at the left-hand side of every x > 0. So says set theory. If this was true, then ℵo of them, together with the spaces between them, would sit between all positive x and 0., i.e. between 0 and (0, 1]. Or the positive points x > 0 do not fill the interval (0, oo) completely. Or matheologians accept this contradiction and dishonestly lie that it was no contradiction.

Regards, WM

WM explained :
> FromTheRafters schrieb am Montag, 13. Februar 2023 um 10:09:56 UTC+1:
>> WM was thinking very hard :
>>> FromTheRafters schrieb am Montag, 13. Februar 2023 um 00:25:38 UTC+1:
>>>> WM pretended :
>>>>> There is a function: NUF(x), the number of unit fractions between
>>>>> 0 and x. This function is f(x) = ℵo for 0 < x < oo
>>>>> and f(0) = 0.
>>>>>
>>>>> That means the NUF must decrease from ℵo to 0 in no distance on the real
>>>>> axis.
>>>> Does it? Is it necessarily a continuous function?
>>>
>>> It gives the number of unit fractions sitting as geometric points on the
>>> reral line at the left-hand side of x.
>> No it doesn't.
>
> There are ℵo unit fractions at the left-hand side of every x > 1. ℵo of them
> (of these very unit fractions) are at the left-hand side of every x > 0. So
> says set theory. If this was true, then ℵo of them, together with the spaces
> between them, would sit between all positive x and 0., i.e. between 0 and (0,
> 1]. Or the positive points x > 0 do not fill the interval (0, oo) completely.
> Or matheologians accept this contradiction and dishonestly lie that it was no
> contradiction.

Your points are too thick.

https://mathworld.wolfram.com/AccumulationPoint.html

FromTheRafters schrieb am Montag, 13. Februar 2023 um 19:51:42 UTC+1:
> WM explained :
> > FromTheRafters schrieb am Montag, 13. Februar 2023 um 10:09:56 UTC+1:
> >> WM was thinking very hard :
> >>> FromTheRafters schrieb am Montag, 13. Februar 2023 um 00:25:38 UTC+1:
> >>>> WM pretended :
> >>>>> There is a function: NUF(x), the number of unit fractions between
> >>>>> 0 and x. This function is f(x) = ℵo for 0 < x < oo
> >>>>> and f(0) = 0.
> >>>>>
> >>>>> That means the NUF must decrease from ℵo to 0 in no distance on the real
> >>>>> axis.
> >>>> Does it? Is it necessarily a continuous function?
> >>>
> >>> It gives the number of unit fractions sitting as geometric points on the
> >>> reral line at the left-hand side of x.
> >> No it doesn't.
> >
> > There are ℵo unit fractions at the left-hand side of every x > 1. ℵo of them
> > (of these very unit fractions) are at the left-hand side of every x > 0.. So
> > says set theory. If this was true, then ℵo of them, together with the spaces
> > between them, would sit between all positive x and 0., i.e. between 0 and (0,
> > 1]. Or the positive points x > 0 do not fill the interval (0, oo) completely.
> > Or matheologians accept this contradiction and dishonestly lie that it was no
> > contradiction.
> Your points are too thick.
>
Points are neither thick nor thin. But different points always differ, by definition.

Regards, WM

On Monday, February 13, 2023 at 8:31:25 PM UTC+1, WM wrote:

> different points always differ [...]

Yeah, an incredible insight, Mückenheim!

WM expressed precisely :
> FromTheRafters schrieb am Montag, 13. Februar 2023 um 19:51:42 UTC+1:
>> WM explained :
>>> FromTheRafters schrieb am Montag, 13. Februar 2023 um 10:09:56 UTC+1:
>>>> WM was thinking very hard :
>>>>> FromTheRafters schrieb am Montag, 13. Februar 2023 um 00:25:38 UTC+1:
>>>>>> WM pretended :
>>>>>>> There is a function: NUF(x), the number of unit fractions between
>>>>>>> 0 and x. This function is f(x) = ℵo for 0 < x < oo
>>>>>>> and f(0) = 0.
>>>>>>>
>>>>>>> That means the NUF must decrease from ℵo to 0 in no distance on the
>>>>>>> real axis.
>>>>>> Does it? Is it necessarily a continuous function?
>>>>>
>>>>> It gives the number of unit fractions sitting as geometric points on the
>>>>> reral line at the left-hand side of x.
>>>> No it doesn't.
>>>
>>> There are ℵo unit fractions at the left-hand side of every x > 1. ℵo of
>>> them (of these very unit fractions) are at the left-hand side of every x >
>>> 0. So says set theory. If this was true, then ℵo of them, together with
>>> the spaces between them, would sit between all positive x and 0., i.e.
>>> between 0 and (0, 1]. Or the positive points x > 0 do not fill the
>>> interval (0, oo) completely. Or matheologians accept this contradiction
>>> and dishonestly lie that it was no contradiction.
>> Your points are too thick.
>>
> Points are neither thick nor thin. But different points always differ, by
> definition.

Too thick and they won't fit, too thin and they won't fill.

Maybe some sort of epsilon would help?

On Sunday, February 12, 2023 at 11:08:29 AM UTC-8, WM wrote:
> For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x. At x = 0 there is no unit fraction between 0 and x. That means between (0, 1] and 0 all ℵo points (together with the intervals between adjacent unit fraction) must have gone. That however is impossible.
>
> Regards, WM

Here's what Chatgpt says about what you wrote:

The statement you provided is a variation of the old paradox known as "Hilbert's Hotel." The paradox arises when you consider the concept of infinite sets and the idea that they can be "filled up." In this case, the idea is that there are an infinite number of unit fractions between 0 and 1, and that for each of these fractions, there is another fraction between it and 0. This would lead to the conclusion that there are an infinite number of points between 0 and 1, which is a paradox.

However, the resolution to this paradox is that it is not accurate to assume that there is a one-to-one correspondence between the points in (0, 1] and the unit fractions between 0 and 1. While there are an infinite number of unit fractions between 0 and 1, this does not mean that every point in (0, 1] can be represented by one of these unit fractions. In mathematics, it is not possible to assign a unique unit fraction to every point in (0, 1], as there are simply too many points in this interval for there to be a one-to-one correspondence.

In other words, the idea that all of the points in (0, 1] can be represented by the unit fractions between 0 and 1 is an illusion created by the concept of infinity. While it may seem intuitive to think that there are an infinite number of points between 0 and 1, this idea is not supported by mathematical principles.

David Petry schrieb am Dienstag, 14. Februar 2023 um 03:35:01 UTC+1:
> On Sunday, February 12, 2023 at 11:08:29 AM UTC-8, WM wrote:
> > For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x. At x = 0 there is no unit fraction between 0 and x. That means between (0, 1] and 0 all ℵo points (together with the intervals between adjacent unit fraction) must have gone. That however is impossible.

> Here's what Chatgpt says about what you wrote:
>
> The statement you provided is a variation of the old paradox known as "Hilbert's Hotel."

No.

> The paradox arises when you consider the concept of infinite sets and the idea that they can be "filled up." In this case, the idea is that there are an infinite number of unit fractions between 0 and 1, and that for each of these fractions, there is another fraction between it and 0. This would lead to the conclusion that there are an infinite number of points between 0 and 1, which is a paradox.

That is Cantor's claim, even that they all can be put in bijection with all the fractions in (-oo, oo). But it is off topic here.
>
> However, the resolution to this paradox is that it is not accurate to assume that there is a one-to-one correspondence between the points in (0, 1] and the unit fractions between 0 and 1.

The number of points, reals, in (0, 1] is not the topic.

> While there are an infinite number of unit fractions between 0 and 1, this does not mean that every point in (0, 1] can be represented by one of these unit fractions. In mathematics, it is not possible to assign a unique unit fraction to every point in (0, 1], as there are simply too many points in this interval for there to be a one-to-one correspondence.

Again that is not the topic (and further it is wrong).
>
> In other words, the idea that all of the points in (0, 1] can be represented by the unit fractions between 0 and 1 is an illusion created by the concept of infinity. While it may seem intuitive to think that there are an infinite number of points between 0 and 1, this idea is not supported by mathematical principles.

This idea is fact.

Conclusion ChatGPT has to learn a lot.

Regards, WM

Fritz Feldhase schrieb am Montag, 13. Februar 2023 um 21:40:15 UTC+1:
> On Monday, February 13, 2023 at 8:31:25 PM UTC+1, WM wrote:
>
> > different points always differ [...]
>
> Yeah, an incredible insight, Mückenheim!

You seem to miss that insight.
Note: 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].

The alternative of this antinomy is this: not all points x > 0 are visible. Then the unit fractions fit within (0, 1]. They must only lie at the left-hand side of the visible x > 0.

Regards, WM

WM explained on 2/14/2023 :
> David Petry schrieb am Dienstag, 14. Februar 2023 um 03:35:01 UTC+1:
>> On Sunday, February 12, 2023 at 11:08:29 AM UTC-8, WM wrote:
>>> For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x. At
>>> x = 0 there is no unit fraction between 0 and x. That means between (0, 1]
>>> and 0 all ℵo points (together with the intervals between adjacent unit
>>> fraction) must have gone. That however is impossible.
>
>> Here's what Chatgpt says about what you wrote:
>>
>> The statement you provided is a variation of the old paradox known as
>> "Hilbert's Hotel."
>
> No.
>
>> The paradox arises when you consider the concept of infinite sets and the
>> idea that they can be "filled up." In this case, the idea is that there are
>> an infinite number of unit fractions between 0 and 1, and that for each of
>> these fractions, there is another fraction between it and 0. This would lead
>> to the conclusion that there are an infinite number of points between 0 and
>> 1, which is a paradox.
>
> That is Cantor's claim, even that they all can be put in bijection with all
> the fractions in (-oo, oo). But it is off topic here.
>>
>> However, the resolution to this paradox is that it is not accurate to assume
>> that there is a one-to-one correspondence between the points in (0, 1] and
>> the unit fractions between 0 and 1.
>
> The number of points, reals, in (0, 1] is not the topic.
>
>> While there are an infinite number of unit fractions between 0 and 1, this
>> does not mean that every point in (0, 1] can be represented by one of these
>> unit fractions. In mathematics, it is not possible to assign a unique unit
>> fraction to every point in (0, 1], as there are simply too many points in
>> this interval for there to be a one-to-one correspondence.
>
> Again that is not the topic (and further it is wrong).
>>
>> In other words, the idea that all of the points in (0, 1] can be represented
>> by the unit fractions between 0 and 1 is an illusion created by the concept
>> of infinity. While it may seem intuitive to think that there are an infinite
>> number of points between 0 and 1, this idea is not supported by mathematical
>> principles.
>
> This idea is fact.
>
> Conclusion ChatGPT has to learn a lot.

Agreed, but not from you hopefully.

On Tuesday, February 14, 2023 at 1:07:06 AM UTC-8, WM wrote:

> Conclusion ChatGPT has to learn a lot.

That's absolutely right.

Anyway, as an experiment, I typed in this to chatgpt:

rewrite this so a ten year old could understand it. For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x. At x = 0 there is no unit fraction between 0 and x. That means between (0, 1] and 0 all ℵo points (together with the intervals between adjacent unit fraction) must have gone. That however is impossible.

This is the response from ChatGPT:

*start*

This passage talks about how there are lots of little fractions (like 1/2, 1/4, 1/8, and so on) that can fit between 0 and any number between 0 and 1.

But when we get to 0, there aren't any of these little fractions that can fit between 0 and itself.

So if we look at all the numbers between 0 and 1, plus 0 itself, there are an endless amount of points we need to cover.

However, it's impossible to cover all of these points because there are just too many of them!

*end*

How did it do?

Maybe I'm just being silly, but you might ask ChatGPT to rewrite the things you want people in this newsgroup to understand, and ask it to write it so that a ten-year-old could understand it.

David Petry schrieb am Mittwoch, 15. Februar 2023 um 06:20:04 UTC+1:
> On Tuesday, February 14, 2023 at 1:07:06 AM UTC-8, WM wrote:
>
>
> > Conclusion ChatGPT has to learn a lot.
> That's absolutely right.
>
> Anyway, as an experiment, I typed in this to chatgpt:
>
> rewrite this so a ten year old could understand it. For every point x ∈ (0, 1] there are ℵo unit fractions between 0 and x. At x = 0 there is no unit fraction between 0 and x. That means between (0, 1] and 0 all ℵo points (together with the intervals between adjacent unit fraction) must have gone. That however is impossible.
>
> This is the response from ChatGPT:
>
> *start*
>
> This passage talks about how there are lots of little fractions (like 1/2, 1/4, 1/8, and so on) that can fit between 0 and any number between 0 and 1.
>
> But when we get to 0, there aren't any of these little fractions that can fit between 0 and itself.
>
> So if we look at all the numbers between 0 and 1, plus 0 itself, there are an endless amount of points we need to cover.
>
> However, it's impossible to cover all of these points because there are just too many of them!

Cantor did not only cover them all but used them to enumerate much more. And the cursor moving from 1 to -1 also passes all these points.
>
> *end*
>
> How did it do?
>
> Maybe I'm just being silly, but you might ask ChatGPT to rewrite the things you want people in this newsgroup to understand, and ask it to write it so that a ten-year-old could understand it.

Thank you very much. I am not (yet) in contact with ChatGPT although I am really interested in his way of "thinking".

I have modified my argument to the hopefully last version in order to refute the accusation of quantifier swapping. If you could ask him this?

(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 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