novaBBS - sci.math - Re: The testimony of unit fractions

Re: The testimony of unit fractions

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https://www.novabbs.com/tech/article-flat.php?id=126432&group=sci.math#126432

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From: pyt...@invalid.org (Python)
Newsgroups: sci.math
Subject: Re: The testimony of unit fractions
Date: Sun, 5 Feb 2023 02:36:33 +0100
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by: Python - Sun, 5 Feb 2023 01:36 UTC

Le 04/02/2023 à 19:21, WM a écrit :
> William schrieb am Samstag, 4. Februar 2023 um 19:14:17 UTC+1:
>> On Saturday, February 4, 2023 at 2:05:13 PM UTC-4, WM wrote:
>>> William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
>>>> On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
>>>
>>>>> Well, this is an interesting topic. Are you interested to discuss the consequences?
>>>> A natural number is an element of |R. The reciprocal of a natural number is an element of |R and thus not an infinitesimal. The reciprocal of a "dark natural number" is an infinitesimal. Thus a "dark natural number" is not a natural number. An "infinitesimal unit faction" is not a unit fraction.
>>> But between all real unit fractions and zero there are dark unit fractions.
>> So what? "Dark unit fractions" are still not unit fractions.
>>>>> Are there ℵo infinitesimal unit fractions?
>>>> Indeed, and not one "infinitesimal unit fraction" is a unit fraction.
>>> The infinitesimal unit fractions are residing between the real unit fractions and zero.
>>>>
>>>>> What about their reciprocals?
>>>>
>>>> They are "dark natural numbers". Not one of them is a natural number.
>>> They are residing between the Peano numbers and ω.
>> So what? They, "dark natural numbers", are still not natural numbers.
>
> I agree that they are not Peano-natural-numbers. But they are there. Why not investigate them?

Sure investigate, capture, torture, abuse, molest, put them
in konzentration kamps Dr Wolfgang Mückenheim, from Hochschule
Augsburg.

Re: The testimony of unit fractions

<b8554618-1e5b-cdfd-f261-2a23df9f2d4b@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: The testimony of unit fractions
Date: Sat, 4 Feb 2023 20:38:51 -0500
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by: Jim Burns - Sun, 5 Feb 2023 01:38 UTC

On 2/4/2023 5:43 PM, WM wrote:
> Jim Burns schrieb am Samstag,
> 4. Februar 2023 um 20:42:01 UTC+1:

>> what you are talking about is d
>> with nothing between it and 0
>> which is d < each finite 1/n
>
> No problem.
> Nobody can use it and its infinitely many
> colleagues.

| Sooner or later, all our games turn into
| Calvinball.
| -- Calvin
https://calvinandhobbes.fandom.com/wiki/Calvinball

We can make a not-possibly-false claim
about d that nothing is between 0 and d

Because
d exists, point d < each finite 1/n
D exists, set D bounded non-empty holding reals

Because
D exists, set D bounded non-empty holding reals
f exists, function f continuous and jumping

Because
f exists, function f continuous and jumping
you (WM) are not talking about "line"

> No problem.
> Nobody can use it and its infinitely many
> colleagues.

Yes.
Since your "line" allows continuous functions
to jump, and our "line" doesn't allow that,
nothing you say can be a problem for us.

Re: The testimony of unit fractions

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From: james.g....@att.net (Jim Burns)
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Subject: Re: The testimony of unit fractions
Date: Sat, 4 Feb 2023 20:50:15 -0500
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by: Jim Burns - Sun, 5 Feb 2023 01:50 UTC

On 2/4/2023 8:38 PM, Jim Burns wrote:
> On 2/4/2023 5:43 PM, WM wrote:
>> Jim Burns schrieb am Samstag,
>> 4. Februar 2023 um 20:42:01 UTC+1:
>
>>> what you are talking about is d
>>> with nothing between it and 0
>>> which is d < each finite 1/n
>>
>> No problem.
>> Nobody can use it and its infinitely many
>> colleagues.
>
> | Sooner or later, all our games turn into
> | Calvinball.
> |
> -- Calvin
> https://calvinandhobbes.fandom.com/wiki/Calvinball
>
> We can make a not-possibly-false claim
> about d that nothing is between 0 and d
>
> Because
> d exists, point d < each finite 1/n
> D exists, set D bounded non-empty holding reals

bounded non-empty holding reals
and not having a LUB

> Because
> D exists, set D bounded non-empty holding reals
> f exists, function f continuous and jumping
>
> Because
> f exists, function f continuous and jumping
> you (WM) are not talking about "line"
>
>> No problem.
>> Nobody can use it and its infinitely many
>> colleagues.
>
> Yes.
> Since your "line" allows continuous functions
> to jump, and our "line" doesn't allow that,
> nothing you say can be a problem for us.
>
>

Re: The testimony of unit fractions

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Subject: Re: The testimony of unit fractions
From: wpihug...@gmail.com (William)
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by: William - Sun, 5 Feb 2023 02:16 UTC

On Saturday, February 4, 2023 at 7:56:52 PM UTC-4, WM wrote:
> William schrieb am Sonntag, 5. Februar 2023 um 00:26:42 UTC+1:
> > On Saturday, February 4, 2023 at 6:37:31 PM UTC-4, WM wrote:
> > > William schrieb am Samstag, 4. Februar 2023 um 19:29:52 UTC+1:
> > > > On Saturday, February 4, 2023 at 2:21:16 PM UTC-4, WM wrote:
> > > > > William schrieb am Samstag, 4. Februar 2023 um 19:14:17 UTC+1:
> > > > > > On Saturday, February 4, 2023 at 2:05:13 PM UTC-4, WM wrote:
> > > > > > > William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
> > > > > > > > On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
> > > > > > >
> > > > > > > > > Well, this is an interesting topic. Are you interested to discuss the consequences?
> > > > > > > > A natural number is an element of |R. The reciprocal of a natural number is an element of |R and thus not an infinitesimal. The reciprocal of a "dark natural number" is an infinitesimal. Thus a "dark natural number" is not a natural number. An "infinitesimal unit faction" is not a unit fraction.
> > > > > > > But between all real unit fractions and zero there are dark unit fractions.
> > > > > > So what? "Dark unit fractions" are still not unit fractions.
> > > > > > > > > Are there ℵo infinitesimal unit fractions?
> > > > > > > > Indeed, and not one "infinitesimal unit fraction" is a unit fraction.
> > > > > > > The infinitesimal unit fractions are residing between the real unit fractions and zero.
> > > > > > > >
> > > > > > > > > What about their reciprocals?
> > > > > > > >
> > > > > > > > They are "dark natural numbers". Not one of them is a natural number.
> > > > > > > They are residing between the Peano numbers and ω.
> > > > > > So what? They, "dark natural numbers", are still not natural numbers.
> > > > > I agree that they are not Peano-natural-numbers.
> > > > They are not any type of natural numbers. "Dark natural numbers" are not natural numbers.
> > > They are, as you call them, dark natural numbers. Don't be a racist. You cannot exclude the dark unit fractions from the real axis.
> > >
> > The real number line contains only points corresponding to elements of |R.
> Where is the unit fraction next to zero?

There is no unit fraction "next to" zero. A unit fraction (unlike an "infinitesimal unit fraction" which is not a unit fraction.) is an element of |R and cannot be infinitesimal.

Re: The testimony of unit fractions

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by: FromTheRafters - Sun, 5 Feb 2023 08:59 UTC

WM pretended :
> William schrieb am Sonntag, 5. Februar 2023 um 00:26:42 UTC+1:
>> On Saturday, February 4, 2023 at 6:37:31 PM UTC-4, WM wrote:
>>> William schrieb am Samstag, 4. Februar 2023 um 19:29:52 UTC+1:
>>>> On Saturday, February 4, 2023 at 2:21:16 PM UTC-4, WM wrote:
>>>>> William schrieb am Samstag, 4. Februar 2023 um 19:14:17 UTC+1:
>>>>>> On Saturday, February 4, 2023 at 2:05:13 PM UTC-4, WM wrote:
>>>>>>> William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
>>>>>>>> On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
>>>>>>>>> Well, this is an interesting topic. Are you interested to discuss the
>>>>>>>>> consequences?
>>>>>>>> A natural number is an element of |R. The reciprocal of a natural
>>>>>>>> number is an element of |R and thus not an infinitesimal. The
>>>>>>>> reciprocal of a "dark natural number" is an infinitesimal. Thus a
>>>>>>>> "dark natural number" is not a natural number. An "infinitesimal unit
>>>>>>>> faction" is not a unit fraction.
>>>>>>> But between all real unit fractions and zero there are dark unit
>>>>>>> fractions. So what? "Dark unit fractions" are still not unit
>>>>>>> fractions.
>>>>>>>>> Are there ℵo infinitesimal unit fractions?
>>>>>>>> Indeed, and not one "infinitesimal unit fraction" is a unit fraction.
>>>>>>>> The infinitesimal unit fractions are residing between the real unit
>>>>>>>> fractions and zero.
>>>>>>>>
>>>>>>>>> What about their reciprocals?
>>>>>>>>
>>>>>>>> They are "dark natural numbers". Not one of them is a natural number.
>>>>>>> They are residing between the Peano numbers and ω.
>>>>>> So what? They, "dark natural numbers", are still not natural numbers.
>>>>> I agree that they are not Peano-natural-numbers.
>>>> They are not any type of natural numbers. "Dark natural numbers" are not
>>>> natural numbers.
>>> They are, as you call them, dark natural numbers. Don't be a racist. You
>>> cannot exclude the dark unit fractions from the real axis.
>>>
>> The real number line contains only points corresponding to elements of |R.
>
> Where is the unit fraction next to zero?

Explain what 'next to' means in the reals.

Re: The testimony of unit fractions

<trnvaf$2eqgc$6@dont-email.me>

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https://www.novabbs.com/tech/article-flat.php?id=126458&group=sci.math#126458

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: The testimony of unit fractions
Date: Sun, 5 Feb 2023 02:10:23 -0800
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by: Chris M. Thomasson - Sun, 5 Feb 2023 10:10 UTC

On 2/5/2023 12:59 AM, FromTheRafters wrote:
> WM pretended :
>> William schrieb am Sonntag, 5. Februar 2023 um 00:26:42 UTC+1:
>>> On Saturday, February 4, 2023 at 6:37:31 PM UTC-4, WM wrote:
>>>> William schrieb am Samstag, 4. Februar 2023 um 19:29:52 UTC+1:
>>>>> On Saturday, February 4, 2023 at 2:21:16 PM UTC-4, WM wrote:
>>>>>> William schrieb am Samstag, 4. Februar 2023 um 19:14:17 UTC+1:
>>>>>>> On Saturday, February 4, 2023 at 2:05:13 PM UTC-4, WM wrote:
>>>>>>>> William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
>>>>>>>>> On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
>>>>>>>>>> Well, this is an interesting topic. Are you interested to
>>>>>>>>>> discuss the consequences?
>>>>>>>>> A natural number is an element of |R. The reciprocal of a
>>>>>>>>> natural number is an element of |R and thus not an
>>>>>>>>> infinitesimal. The reciprocal of a "dark natural number" is an
>>>>>>>>> infinitesimal. Thus a "dark natural number" is not a natural
>>>>>>>>> number. An "infinitesimal unit faction" is not a unit fraction.
>>>>>>>> But between all real unit fractions and zero there are dark unit
>>>>>>>> fractions. So what? "Dark unit fractions" are still not unit
>>>>>>>> fractions.
>>>>>>>>>> Are there ℵo infinitesimal unit fractions?
>>>>>>>>> Indeed, and not one "infinitesimal unit fraction" is a unit
>>>>>>>>> fraction. The infinitesimal unit fractions are residing between
>>>>>>>>> the real unit fractions and zero.
>>>>>>>>>> What about their reciprocals?
>>>>>>>>>
>>>>>>>>> They are "dark natural numbers". Not one of them is a natural
>>>>>>>>> number.
>>>>>>>> They are residing between the Peano numbers and ω.
>>>>>>> So what? They, "dark natural numbers", are still not natural
>>>>>>> numbers.
>>>>>> I agree that they are not Peano-natural-numbers.
>>>>> They are not any type of natural numbers. "Dark natural numbers"
>>>>> are not natural numbers.
>>>> They are, as you call them, dark natural numbers. Don't be a racist.
>>>> You cannot exclude the dark unit fractions from the real axis.
>>> The real number line contains only points corresponding to elements
>>> of |R.
>>
>> Where is the unit fraction next to zero?
>
> Explain what 'next to' means in the reals.

The only thing I can think of is infinitely close as in:

..1
..01
..001
..0001
....

So zero wrt its limit?

Re: The testimony of unit fractions

<cebefc29-fbb1-4b0e-ac43-dd1fe4edd566n@googlegroups.com>

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Subject: Re: The testimony of unit fractions
From: askaske...@gmail.com (WM)
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by: WM - Sun, 5 Feb 2023 12:29 UTC

William schrieb am Sonntag, 5. Februar 2023 um 03:16:19 UTC+1:
> On Saturday, February 4, 2023 at 7:56:52 PM UTC-4, WM wrote:

> > > The real number line contains only points corresponding to elements of |R.
> > Where is the unit fraction next to zero?
>
> There is no unit fraction "next to" zero.

There are many dark unit fractions "next to zero", i.e. between 0 and every definable unit fraction, because the dark unit fractions cannot be distinguished.

> A unit fraction (unlike an "infinitesimal unit fraction" which is not a unit fraction.) is an element of |R and cannot be infinitesimal.

But the dark unit fractions are points on the real line. They are the infinitely many unit fractions between every definable unit fraction and 0.

Regards, WM

Re: The testimony of unit fractions

<tro7kv$2gh5c$1@dont-email.me>

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: The testimony of unit fractions
Date: Sun, 05 Feb 2023 07:32:29 -0500
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by: FromTheRafters - Sun, 5 Feb 2023 12:32 UTC

Chris M. Thomasson laid this down on his screen :
> On 2/5/2023 12:59 AM, FromTheRafters wrote:
>> WM pretended :
>>> William schrieb am Sonntag, 5. Februar 2023 um 00:26:42 UTC+1:
>>>> On Saturday, February 4, 2023 at 6:37:31 PM UTC-4, WM wrote:
>>>>> William schrieb am Samstag, 4. Februar 2023 um 19:29:52 UTC+1:
>>>>>> On Saturday, February 4, 2023 at 2:21:16 PM UTC-4, WM wrote:
>>>>>>> William schrieb am Samstag, 4. Februar 2023 um 19:14:17 UTC+1:
>>>>>>>> On Saturday, February 4, 2023 at 2:05:13 PM UTC-4, WM wrote:
>>>>>>>>> William schrieb am Samstag, 4. Februar 2023 um 15:46:27 UTC+1:
>>>>>>>>>> On Saturday, February 4, 2023 at 4:32:07 AM UTC-4, WM wrote:
>>>>>>>>>>> Well, this is an interesting topic. Are you interested to discuss
>>>>>>>>>>> the consequences?
>>>>>>>>>> A natural number is an element of |R. The reciprocal of a natural
>>>>>>>>>> number is an element of |R and thus not an infinitesimal. The
>>>>>>>>>> reciprocal of a "dark natural number" is an infinitesimal. Thus a
>>>>>>>>>> "dark natural number" is not a natural number. An "infinitesimal
>>>>>>>>>> unit faction" is not a unit fraction.
>>>>>>>>> But between all real unit fractions and zero there are dark unit
>>>>>>>>> fractions. So what? "Dark unit fractions" are still not unit
>>>>>>>>> fractions.
>>>>>>>>>>> Are there ℵo infinitesimal unit fractions?
>>>>>>>>>> Indeed, and not one "infinitesimal unit fraction" is a unit
>>>>>>>>>> fraction. The infinitesimal unit fractions are residing between the
>>>>>>>>>> real unit fractions and zero.
>>>>>>>>>>> What about their reciprocals?
>>>>>>>>>>
>>>>>>>>>> They are "dark natural numbers". Not one of them is a natural
>>>>>>>>>> number.
>>>>>>>>> They are residing between the Peano numbers and ω.
>>>>>>>> So what? They, "dark natural numbers", are still not natural numbers.
>>>>>>> I agree that they are not Peano-natural-numbers.
>>>>>> They are not any type of natural numbers. "Dark natural numbers" are
>>>>>> not natural numbers.
>>>>> They are, as you call them, dark natural numbers. Don't be a racist. You
>>>>> cannot exclude the dark unit fractions from the real axis.
>>>> The real number line contains only points corresponding to elements of
>>>> |R.
>>>
>>> Where is the unit fraction next to zero?
>>
>> Explain what 'next to' means in the reals.
>
> The only thing I can think of is infinitely close as in:
>
> .1
> .01
> .001
> .0001
> ...
>
> So zero wrt its limit?

The idea of infinitely close seems to come from the hyperreals.

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

Re: The testimony of unit fractions

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Subject: Re: The testimony of unit fractions
From: wpihug...@gmail.com (William)
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by: William - Sun, 5 Feb 2023 12:57 UTC

On Sunday, February 5, 2023 at 8:29:14 AM UTC-4, WM wrote:
> William schrieb am Sonntag, 5. Februar 2023 um 03:16:19 UTC+1:
> > On Saturday, February 4, 2023 at 7:56:52 PM UTC-4, WM wrote:
>
> > > > The real number line contains only points corresponding to elements of |R.
> > > Where is the unit fraction next to zero?
> >
> > There is no unit fraction "next to" zero.
> There are many dark unit fractions "next to zero",

So what? Despite the misleading name "dark unit fractions" are not unit fractions. The is no unit fraction "next to"

> i.e. between 0 and every definable unit fraction, because the dark unit fractions cannot be distinguished.
> > A unit fraction (unlike an "infinitesimal unit fraction" which is not a unit fraction.) is an element of |R and cannot be infinitesimal.
> But the dark unit fractions are points on the real line.

Nope the real number line only has points corresponding to elements of |R. An infinitesimal is not an element of |R,

Re: The testimony of unit fractions

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Subject: Re: The testimony of unit fractions
From: askaske...@gmail.com (WM)
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by: WM - Sun, 5 Feb 2023 15:02 UTC

Re: The testimony of unit fractions

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Subject: Re: The testimony of unit fractions
From: askaske...@gmail.com (WM)
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by: WM - Sun, 5 Feb 2023 15:11 UTC

William schrieb am Sonntag, 5. Februar 2023 um 13:57:39 UTC+1:
> On Sunday, February 5, 2023 at 8:29:14 AM UTC-4, WM wrote:
> > William schrieb am Sonntag, 5. Februar 2023 um 03:16:19 UTC+1:
> > > On Saturday, February 4, 2023 at 7:56:52 PM UTC-4, WM wrote:
> >
> > > > > The real number line contains only points corresponding to elements of |R.
> > > > Where is the unit fraction next to zero?
> > >
> > > There is no unit fraction "next to" zero.
> > There are many dark unit fractions "next to zero",
> So what? Despite the misleading name "dark unit fractions" are not unit fractions.

All unit fractions which you can find can be surpassed when you look closer.. This is an infinite process. Whether these unit fractions have been there before already or whether you have created them by looking closer, is not decided.

> The is no unit fraction "next to"

The unit fraction u is next to zero, if there is no other unit fraction between u and zero.
The unit fraction u is next to zero, if the cursor passes no is no other unit fraction on his way.
Alas such a unit fraction cannot be found, because ℵo unit fractions are not distinguishable. Therefore they all are next to zero. But they are points on the real line.

> > But the dark unit fractions are points on the real line.
> Nope the real number line only has points corresponding to elements of |R.. An infinitesimal is not an element of |R,

You are provably wrong here.
The cursor will pass more unit fractions on the real line than can be discerned.
There is no point between 0 and (0, 1]. Hence there cannot exist a unit fraction between 0 and (0, 1]. Hence the Number of Unit Fractions between zero and (0, 1] is 0.
However the Number of Unit Fractions between zero and every discernible 1/n and (0, 1] is ℵo.

Regards, WM

Re: The testimony of unit fractions

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Subject: Re: The testimony of unit fractions
From: wpihug...@gmail.com (William)
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by: William - Sun, 5 Feb 2023 16:48 UTC

On Sunday, February 5, 2023 at 11:11:48 AM UTC-4, WM wrote:
> William schrieb am Sonntag, 5. Februar 2023 um 13:57:39 UTC+1:
> > On Sunday, February 5, 2023 at 8:29:14 AM UTC-4, WM wrote:
> > > William schrieb am Sonntag, 5. Februar 2023 um 03:16:19 UTC+1:
> > > > On Saturday, February 4, 2023 at 7:56:52 PM UTC-4, WM wrote:
> > >
> > > > > > The real number line contains only points corresponding to elements of |R.
> > > > > Where is the unit fraction next to zero?
> > > >
> > > > There is no unit fraction "next to" zero.
> > > There are many dark unit fractions "next to zero",
> > So what? Despite the misleading name "dark unit fractions" are not unit fractions.
> All unit fractions which you can find can be surpassed when you look closer. This is an infinite process. Whether these unit fractions have been there before already or whether you have created them by looking closer, is not decided.

Yes, It has been decided, A unit fraction (unlike an "infinitesimal unit fraction" which is not a unit fraction) is an element of the set |R. A set does not change. An element of |R has always been an element of |R

> > The is no unit fraction "next to"
> The unit fraction u is next to zero, if there is no other unit fraction between u and zero.

Correct. And since if x is a unit fraction there a unit fraction between x and zero there is no unit fraction next to 0.

Re: The testimony of unit fractions

<trp45a$2mhfk$1@dont-email.me>

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by: Chris M. Thomasson - Sun, 5 Feb 2023 20:39 UTC

Re: The testimony of unit fractions

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by: FromTheRafters - Sun, 5 Feb 2023 21:12 UTC

Re: The testimony of unit fractions

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From: FTR...@nomail.afraid.org (FromTheRafters)
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Subject: Re: The testimony of unit fractions
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by: FromTheRafters - Sun, 5 Feb 2023 21:12 UTC

Re: The testimony of unit fractions

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Subject: Re: The testimony of unit fractions
From: askaske...@gmail.com (WM)
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by: WM - Sun, 5 Feb 2023 21:21 UTC

William schrieb am Sonntag, 5. Februar 2023 um 17:48:44 UTC+1:
> On Sunday, February 5, 2023 at 11:11:48 AM UTC-4, WM wrote:

> > All unit fractions which you can find can be surpassed when you look closer. This is an infinite process. Whether these unit fractions have been there before already or whether you have created them by looking closer, is not decided.
> Yes, It has been decided, A unit fraction (unlike an "infinitesimal unit fraction" which is not a unit fraction) is an element of the set |R. A set does not change. An element of |R has always been an element of |R.

There is a last unit fraction *noticed*. This last unit fraction depends on your attention and can be surpassed. The unit fractions surpassing it either are already present at the first cursor movement or not. Here we assume that they and all further surpassing unit fractions are already existing as dark terms. The set is not fixed because it can be expanded at will.

> > > The is no unit fraction "next to"
> > The unit fraction u is next to zero, if there is no other unit fraction between u and zero.
> Correct. And since if x is a unit fraction there a unit fraction between x and zero there is no unit fraction next to 0.

Wrong. There is nothing between (0,1] and zero, but removing anything from the edges of (0,1] will remove unit fractions too. Hence there is nothing between all unit fractions and zero.

Regards, WM

Re: The testimony of unit fractions

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Subject: Re: The testimony of unit fractions
From: askaske...@gmail.com (WM)
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by: WM - Sun, 5 Feb 2023 21:21 UTC

Re: The testimony of unit fractions

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Subject: Re: The testimony of unit fractions
From: wpihug...@gmail.com (William)
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by: William - Sun, 5 Feb 2023 22:26 UTC

Re: The testimony of unit fractions

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: The testimony of unit fractions
Date: Sun, 5 Feb 2023 16:09:05 -0800
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by: Chris M. Thomasson - Mon, 6 Feb 2023 00:09 UTC

On 2/5/2023 1:12 PM, FromTheRafters wrote:
> Chris M. Thomasson brought next idea :
>> On 2/5/2023 7:02 AM, WM wrote:
>>> FromTheRafters schrieb am Sonntag, 5. Februar 2023 um 09:59:53 UTC+1:
>>>> WM pretended :
>>>
>>>>> Where is the unit fraction next to zero?
>>>> Explain what 'next to' means in the reals.
>>>
>>> The unit fraction u is next to zero, if there is no other unit
>>> fraction between u and zero.
>>> The unit fraction u is next to zero, if the cursor passes no is no
>>> other unit fraction on his way.
>>> Alas such a unit fraction cannot be found, because ℵo unit fractions
>>> are not distinguishable. Therefore they all are next to zero.
>>
>> There are infinite fractions that get closer and closer to zero.
>
> What is an infinite fraction?

Think of:

1/1, 1/2, 1/3, 1/4, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, ...

Re: The testimony of unit fractions

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From: james.g....@att.net (Jim Burns)
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Subject: Re: The testimony of unit fractions
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by: Jim Burns - Mon, 6 Feb 2023 00:58 UTC

Re: The testimony of unit fractions

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: The testimony of unit fractions
Date: Sun, 05 Feb 2023 20:08:49 -0500
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by: FromTheRafters - Mon, 6 Feb 2023 01:08 UTC

Chris M. Thomasson used his keyboard to write :
> On 2/5/2023 1:12 PM, FromTheRafters wrote:
>> Chris M. Thomasson brought next idea :
>>> On 2/5/2023 7:02 AM, WM wrote:
>>>> FromTheRafters schrieb am Sonntag, 5. Februar 2023 um 09:59:53 UTC+1:
>>>>> WM pretended :
>>>>
>>>>>> Where is the unit fraction next to zero?
>>>>> Explain what 'next to' means in the reals.
>>>>
>>>> The unit fraction u is next to zero, if there is no other unit fraction
>>>> between u and zero.
>>>> The unit fraction u is next to zero, if the cursor passes no is no other
>>>> unit fraction on his way.
>>>> Alas such a unit fraction cannot be found, because ℵo unit fractions are
>>>> not distinguishable. Therefore they all are next to zero.
>>>
>>> There are infinite fractions that get closer and closer to zero.
>>
>> What is an infinite fraction?
>
> Think of:
>
> 1/1, 1/2, 1/3, 1/4, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, ...

Ah, *infinitely many* unit fractions.

Re: The testimony of unit fractions

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: The testimony of unit fractions
Date: Sun, 5 Feb 2023 19:44:54 -0800
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by: Chris M. Thomasson - Mon, 6 Feb 2023 03:44 UTC

Re: The testimony of unit fractions

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: The testimony of unit fractions
Date: Sun, 5 Feb 2023 19:45:25 -0800
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by: Chris M. Thomasson - Mon, 6 Feb 2023 03:45 UTC

On 2/5/2023 1:21 PM, WM wrote:
> FromTheRafters schrieb am Sonntag, 5. Februar 2023 um 22:12:44 UTC+1:
>> WM laid this down on his screen :
>>> FromTheRafters schrieb am Sonntag, 5. Februar 2023 um 09:59:53 UTC+1:
>>>> WM pretended :
>>>
>>>>> Where is the unit fraction next to zero?
>>>> Explain what 'next to' means in the reals.
>>>
>>> The unit fraction u is next to zero, if there is no other unit fraction
>>> between u and zero.
>> Since there is 'always' something else between u and zero,
>
> No. There is nothing between (0,1] and zero, but removing anything from the edges of (0,1] will remove unit fractions too. Hence there is nothing between all unit fractions and zero.

What about 1/2?

0, 1/2, 1

Re: The testimony of unit fractions

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: The testimony of unit fractions
Date: Mon, 06 Feb 2023 03:47:44 -0500
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by: FromTheRafters - Mon, 6 Feb 2023 08:47 UTC

Chris M. Thomasson explained :
> On 2/5/2023 1:21 PM, WM wrote:
>> FromTheRafters schrieb am Sonntag, 5. Februar 2023 um 22:12:44 UTC+1:
>>> WM laid this down on his screen :
>>>> FromTheRafters schrieb am Sonntag, 5. Februar 2023 um 09:59:53 UTC+1:
>>>>> WM pretended :
>>>>
>>>>>> Where is the unit fraction next to zero?
>>>>> Explain what 'next to' means in the reals.
>>>>
>>>> The unit fraction u is next to zero, if there is no other unit fraction
>>>> between u and zero.
>>> Since there is 'always' something else between u and zero,
>>
>> No. There is nothing between (0,1] and zero, but removing anything from the
>> edges of (0,1] will remove unit fractions too. Hence there is nothing
>> between all unit fractions and zero.
>
> What about 1/2?
>
> 0, 1/2, 1
>
> ?

In this case you have an object (an open interval) and a zero. Two
different objects with nothing between them.

Re: The testimony of unit fractions

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Subject: Re: The testimony of unit fractions
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Mon, 6 Feb 2023 08:54 UTC

When Dexter's on the Internet, can Hell be far behind?"

tech / sci.math / Re: The testimony of unit fractions

tech / sci.math / Re: The testimony of unit fractions

Subject	Author
The testimony of unit fractions	WM
Re: The testimony of unit fractions	FromTheRafters
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	FromTheRafters
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	FromTheRafters
Re: The testimony of unit fractions	JVR
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	Chris M. Thomasson
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	Chris M. Thomasson
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	JVR
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	FredJeffries
Re: The testimony of unit fractions	Chris M. Thomasson
Re: The testimony of unit fractions	Gus Gassmann
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Gus Gassmann
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Gus Gassmann
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	JVR
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	FromTheRafters
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Python
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	FromTheRafters
Re: The testimony of unit fractions	Chris M. Thomasson
Re: The testimony of unit fractions	FromTheRafters
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Chris M. Thomasson
Re: The testimony of unit fractions	FromTheRafters
Re: The testimony of unit fractions	Chris M. Thomasson
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	Chris M. Thomasson
Re: The testimony of unit fractions	FromTheRafters
Re: The testimony of unit fractions	FromTheRafters
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	JVR
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	William
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	WM
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	zelos...@gmail.com
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Timothy Golden
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Eram semper recta
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Timothy Golden
Re: The testimony of unit fractions	zelos...@gmail.com
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Ross A. Finlayson
Re: The testimony of unit fractions	Ross A. Finlayson
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	zelos...@gmail.com
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Fritz Feldhase
Re: The testimony of unit fractions	Jim Burns
Re: The testimony of unit fractions	FredJeffries
Re: The testimony of unit fractions	FredJeffries
Re: The testimony of unit fractions	FromTheRafters
Re: The testimony of unit fractions	Ezimene nimi Teine nimi
Re: The testimony of unit fractions	K