No "First" init fraction

[William]

we have
∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .
note the universal quantifier. This holds for every unit fraction. Thus for every unit fraction (unrevealed or invisible or ...) there is another unit fraction closer to zero (the gap is finite.),
Thus the is no first unit fraction.

[Chris M. Thomasson]

1/1, 1/2, 1/3, ... on its way to its limit at zero.

[FredJeffries]

No.

Not 'to its limit'.

Only TOWARDS its limit.

It never arrives 'at zero'.'

[Chris M. Thomasson]

Right. Well, its on its way to its natural limit on a per iteration
basis, right?

i[0] = 1/1
i[1] = 1/2
i[2] = 1/3
i[3] = 1/4
i[4] = 1/5
....

[Chris M. Thomasson]

None of the iterates will ever equal zero, however, zero is its limit...

Fair enough?

[Archimedes Plutonium]

Moscow,Beijing▂▄▅█████████▅▄▃▂ mushroom cloud, Xi as Putin's stooge when Russia vaporizes Shanghai with its RS-28 Sarmat "Satan II", all because Xi was too dumb to realize you can never trust an insane person

Chris, which of these is Shangai ??? Is it 1/2

> i[0] = 1/1
> i[1] = 1/2
> i[2] = 1/3
> i[3] = 1/4
> i[4] = 1/5

So spamming William, a Sarmat never reaches its exact target but sends shock waves from a blast site.

> No.
>
> Not 'to its limit'.
>
> Only TOWARDS its limit.
>
> It never arrives 'at zero'.'

William, infinite spamming nutjob comparing unit fractions of Moscow compared to Beijing. William spamming nutjob is Moscow 0/1 and Beijing fraction is 1/0 ??

yan wyck in his daily spam says Beijing was nothing in 1999 and apparently a Russian Sarmat returned Beijing to "nothingness once again".


> Shanghai, Beijing, Shenzhen, Guangzhou, Chongqing, Tianjin, Chengdu, Hangzhou, Nanjing, Wuhan, Xi'An, Suzhou, Harbin, Shenyang, Qingdao, Zhengzhou, Dongguan, Foshan, Dalian, Jinan, Changchun, Hefei
>
> Shenzhen▂▄▅█████████▅▄▃▂ Xi as Putin's stooge when Russia vaporizes Shenzhen with its RS-28 Sarmat "Satan II", all because Xi was too dumb to realize you can never trust an insane person
>
>
>
> Is Pete Olcott in his Halting Problem, halting the vaporization of Wuhan by Putin's Russia???
>
>
> On Thursday, March 30, 2023 at 11:56:20 PM UTC-5, Volney wrote:
> > Botfly of Math and Blowfly of Physics "Putin's stooge"
> >"wasn't bolted down too tight in the first place"
> 
> On Friday, September 9, 2022 at 1:16:55 AM UTC-5, Michael Moroney wrote:
> > "Imp of Science"
> >"not one single marble of commonsense in my entire brain"
> 
> Moscow█۞█ blackout, knock out Moscow electric power lines█۞█







> & wrote:
> > _And as the Baby Xi grew up from the rice paddies and reeds of Outer
> > Manchuria, stolen by the Naxi and Zani Dictator Putin in Moscow, Xi
> > learned in school in chemical engineering that Taiwan was 1/28 the size
> > of Outer Manchuria, Emperor Qing's homeland, now occupied by homeless Russians drinking vodka, as Putin bombs Ukraine. And the nascent Xi orders
> > 1,000 divisions to the Outer Manchuria border to regain back the stolen
>
>
> > > Why Putin is 2X smarter than Xi as dictators// SCIENCE COUNCIL RULES EARTH, not petty dictators
> > > 2m views
> > >
> > >
> > 2> If Putin pushes nuclear buttons, he drags down China along with Russia into a nuclear ash waste pile, and this means Xi is a inferior junior partner to Putin. Putin will drag down Xi's China, never the reverse.
> > >
> > 2> So, one can look at the present situation on Earth and ask several logical questions about the 2 dictators of Putin's Russia and China's Xi.
> > >
> > > It is little wonder that both Russia and China dictators are combative towards the West. Because dictators never want to give up on power but stay in power all their life long. So they oppose the West because the West has grown up to democracy-- let the people have power, not one single idiot having power all his life time.
> > >
> > > Naturally, Putin will want to keep the Russian people suppressed and have Russia be a second rate government as a dictator. Same goes for China-- they never want to give up power so the people themselves choose their leader.
> > >
> > > But can we find differences in Putin and Xi themselves? Well in the West we call the Chinese inscrutable-- meaning -- little logical commonsense. And is this a valid description?? Yes of course, considering that Russia had stolen the lands of Outer Manchuria, some 28 times larger of a land mass than is Taiwan island. Yet there is Xi, spending so much time on wanting to invade Taiwan, when it is Outer Manchuria and Vladivostok (Haishenwai) that he should be focusing attention upon. And while Putin is distracted with Ukraine, is the time for Xi to recapture Outer Manchuria, the Qing dynasty empire, Qing's Manchurian homeland.
> > >
> > > What does Xi do instead??? He focuses on Taiwan and befriends Russia. Why, at this rate, if Russia takes Inner Manchuria, we can expect Xi and the Chinese Communist Party to become even more loving of Russia for stealing more land of China.
> > >
> > > And there is Xi, whose China has become rich with trading with the West, yet every day, Xi foaming at the mouth in hatred of the West.
> > >
> > > So yes, Putin is 2X smarter as a dictator than is Xi, as if Putin has Xi in his side pocket.
> > >
> > > Is there some scientific explanation as to why Xi is 2X dumber than Putin?? Perhaps, in that China is densely populated and the air pollution over all of China is worse than most countries. That Xi probably has 1/2 of his brain filled with CO and CO2 isomers and lead, and mercury and nitrous oxide and sulfur dioxide from just living in that air polluted hellhole of Beijing. Xi studied chemistry and should know this. Whereas Putin likely detox..s every evening with breathing in pure oxygen at his residence and takes oxygen breathing tanks to office and work. This easily can explain the light-headed reasoning that Xi and his foreign diplomats Wang Yi display, where Putin plays them like a chess game, --- checkmate in 7 moves.
> > >
> > > This explains why Xi hates the West for not stealing any Chinese lands and making China rich in trade, while loving Putin for stealing Outer Manchuria, and proposing having Russia push nuclear buttons, making both Russia and China a nuclear waste site after ICBMs wipe China off the map.
> > >
> > > Xi's brain is full of air pollution toxins from the nasty Chinese air. They still build a new coal fired plant in China every day. The air in China is the worst air in the entire world.
> > >
> > > Why Putin is 2X smarter than Xi as dictators// SCIENCE COUNCIL RULES EARTH, not petty dictators.
>

> > > > 2/1, AP tards:
> > > > > Give Ukraine drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/2, AP tards:
> > > > > Give Ukraine drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/3, AP tards:
> > > > > Every Russian missile fired into Ukraine met with a drone from Ukraine knocking out Moscow electric power lines
> > > > >
> > > > > Give Ukraine drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/4, AP tards:
> > > > > drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/9 (vacation?), AP tards:
> > > > > drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/9, AP tards (again):
> > > > > drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/10, AP tards:
> > > > > drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/11, AP tards:
> > > > > drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/12, AP tards:
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/12, AP tards again:
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/13, AP tards:
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/14, AP tards:
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/15, AP tards:
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/16, AP tards:
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/17, AP tards:
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/18, AP tards:
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/19, AP tards:
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/20, AP tards:
> > > >
> > > > > Electricity out Novosibirsk &Volgograd█۞█knock out Moscow electric power lines█۞█
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/22, AP tards:
> > > > > Moscow electric blackout█۞█knock out Moscow electric power lines█۞█
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/23, AP tards:
> > > > > Moscow electric blackout█۞█knock out Moscow electric power lines█۞█
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/24, AP tards:
> > > > > Moscow electric blackout█۞█knock out Moscow electric power lines█۞█
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/25, AP tards:
> > > > > Moscow electric blackout█۞█knock out Moscow electric power lines█۞█
> > > > > _drones █۞█knock out Moscow electric power lines█۞█ Moscow, St.Petersburg, Volgograd, Vladivostok no electricity
> > > >
> > > > 2/26, AP tards:

[Chris M. Thomasson]

On 5/28/2023 2:01 PM, Archimedes Plutonium wrote:
>
>
> Moscow,Beijing▂▄▅█████████▅▄▃▂ mushroom cloud, Xi as Putin's stooge when Russia vaporizes Shanghai with its RS-28 Sarmat "Satan II", all because Xi was too dumb to realize you can never trust an insane person
>
> Chris, which of these is Shangai ??? Is it 1/2
>> i[0] = 1/1
>> i[1] = 1/2
>> i[2] = 1/3
>> i[3] = 1/4
>> i[4] = 1/5

[...]

i[1] = 1/2

[FromTheRafters]

on 5/28/2023, Chris M. Thomasson supposed :

This sequence does not converge.

[Chris M. Thomasson]

It certainly get closer and closer to zero? Right?

[Fritz Feldhase]

On Sunday, May 28, 2023 at 11:52:36 PM UTC+2, FromTheRafters wrote:
> on 5/28/2023, Chris M. Thomasson supposed :
> >

> > 1/1, 1/2, 1/3, ... on its way to its limit [...] zero.

> >
> This sequence does not converge.

Huh?!

The sequence (1/n)_(n e N) does not converge?

0 is not its limit? (i.e. lim_(n -> oo) 1/n =/= 0 ???).

Well... Interesting (sort of).

[FromTheRafters]

Chris M. Thomasson pretended :

Yes, and you can have a sequence converging to any element of the set.
Some sequences will not converge (they might converge in the reals) to
an element of the set of unit fractions. To me, nothing up there says
that this set is not discrete.

[Fritz Feldhase]

On Monday, May 29, 2023 at 3:03:47 AM UTC+2, FromTheRafters wrote:
> Chris M. Thomasson pretended :
> > On 5/28/2023 2:50 PM, FromTheRafters wrote:
> >> on 5/28/2023, Chris M. Thomasson supposed:
> >>>

> >>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.
> >>>
> >> This sequence does not converge.
> >

> > It certainly [converges].
> >
> Yes, and <bla>

The sequence does not converge and it converges? Are you doing the Mückenheim her?

[FromTheRafters]

Fritz Feldhase used his keyboard to write :

Sure, when you misrepresent (strawman) what I said it sure looks that
way. A sequence of rationals which approaches the value of Pi does not
converge in the rationals because Pi is not a rational number.

Sure, it gets closer and closer to Pi, but Pi is not actually in the
set. Zero is not in the set of unit fractions, so how can a sequence of
unit fractions converge to it in the rationals?

[Fritz Feldhase]

On Monday, May 29, 2023 at 11:31:10 AM UTC+2, FromTheRafters wrote:

> Zero is not in the set of unit fractions, so how can a sequence of
> unit fractions converge to it in the rationals?

Holy shit! 0 is not a rational number in your book?

Trying to do the Mückenheim?

You are talking nonsense, man.

(Another hint: We usually don't restrict our consideration to the rational numbers when talking about the convergence of a sequence of real numbers. The sequence (1/1, 1/2, 1/3, ...) concerges, and its limit is 0.)

EOD

[WM]

This is contradicted by the following: NUF(0) = 0. NUF(1) = many. Therefore the unit fractions start between 0 and 1. Never more than one occupy a point. This enforces a first one. No way to circ umvent this conclusion.

Two contradicting results cannot exist in mathematics. The second result is pure logic. The first one is not enforced by logic and avoidable by dark numbers..

Regards, WM

[FredJeffries]

On Monday, May 29, 2023 at 10:28:17 AM UTC-7, WM wrote:

> NUF(0) = 0. NUF(1) = many. Therefore the unit fractions start between 0 and 1.

Thus our beloved professor reveals himself to be a devotee of the Kalam cosmological argument.

https://en.wikipedia.org/wiki/Kalam_cosmological_argument
https://en.wikipedia.org/wiki/The_Kal%C4%81m_Cosmological_Argument

[FromTheRafters]

Fritz Feldhase expressed precisely :

> On Monday, May 29, 2023 at 11:31:10 AM UTC+2, FromTheRafters wrote:
>
>> Zero is not in the set of unit fractions, so how can a sequence of
>> unit fractions converge to it in the rationals?
>
> Holy shit! 0 is not a rational number in your book?

It is not a unit fraction in my book. It is not a positive real in my
book.

[William]

On Monday, May 29, 2023 at 2:28:17 PM UTC-3, WM wrote:
> William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:
> > we have
> > ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .
> > note the universal quantifier. This holds for every unit fraction.

>...and avoidable by dark numbers..
Hence, unit fractions cannot be "dark numbers".

[William]

On Monday, May 29, 2023 at 3:14:01 PM UTC-3, FromTheRafters wrote:
> Fritz Feldhase expressed precisely :
> > On Monday, May 29, 2023 at 11:31:10 AM UTC+2, FromTheRafters wrote:
> >
> >> Zero is not in the set of unit fractions, so how can a sequence of
> >> unit fractions converge to it in the rationals?
> >
> > Holy shit! 0 is not a rational number in your book?
> It is not a unit fraction in my book. It is not a positive real in my
> book.

https://xkcd.com/169/

[Chris M. Thomasson]

On 5/29/2023 2:29 AM, FromTheRafters wrote:
> Fritz Feldhase used his keyboard to write :
>> On Monday, May 29, 2023 at 3:03:47 AM UTC+2, FromTheRafters wrote:
>>> Chris M. Thomasson pretended :
>>>> On 5/28/2023 2:50 PM, FromTheRafters wrote:
>>>>> on 5/28/2023, Chris M. Thomasson supposed:
>>>>>>
>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.
>>>>> This sequence does not converge.
>>>>
>>>> It certainly [converges].
>>>>
>>> Yes, and <bla>
>>
>> The sequence does not converge and it converges? Are you doing the
>> Mückenheim her?
>
> Sure, when you misrepresent (strawman) what I said it sure looks that
> way. A sequence of rationals which approaches the value of Pi does not
> converge in the rationals because Pi is not a rational number.

The iterates get closer and closer to zero... Their limit.

[Ben Bacarisse]

"Chris M. Thomasson" <chris.m.t...@gmail.com> writes:

>>>>>> on 5/28/2023, Chris M. Thomasson supposed:
>>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.

> The iterates get closer and closer to zero... Their limit.

You know there are books and online courses/tutorials from which you
could learn what a limit is? Getting "closer and closer to zero" does
not mean that zero is the limit. Don't get me wrong -- zero /is/ the
limit in this case, but if you were a student of mine I'd invite you to
define a sequence that gets forever "closer and closer to zero" but
which does /not/ converge to a limit of zero.

--
Ben.

[WM]

Wrong conclusion. The unit fractions start between 0 and 1. Never more than one occupy a point. This enforces a first one.

Regards, WM

[Fritz Feldhase]

On Monday, May 29, 2023 at 8:14:01 PM UTC+2, FromTheRafters wrote:
> Fritz Feldhase expressed precisely :
> > On Monday, May 29, 2023 at 11:31:10 AM UTC+2, FromTheRafters wrote:
> >
> >> Zero is not in the set of unit fractions, so how can a sequence of
> >> unit fractions converge to it in the rationals?
> >
> > Holy shit! 0 is not a rational number in your book?
> It is not a unit fraction in my book. It is not a positive real in my
> book.

??? Ok, you are an idiot, I see.

*plonk*

> > Trying to do the Mückenheim?
> >
> > You are talking nonsense, man.
> >
> > (Another hint: We usually don't restrict our consideration to the rational
> > numbers when talking about the convergence of a sequence of real numbers. The
> > sequence (1/1, 1/2, 1/3, ...) concerges, and its limit is 0.)
> >
> > EOD

i

[Chris M. Thomasson]

Humm... Perhaps, something like:

1/2 + 1/3 + 1/4 + 1/5 ... ?

The terms get closer to zero, but the sum always gets bigger?

i[0] = 1/2
i[1] = i[0] + 1/3
i[2] = i[1] + 1/4
i[3] = i[2] + 1/5

...

?

[Chris M. Thomasson]

Not between, 1/1 is a unit fraction 0/1 is not. 1/1 is the first unit
fraction, 1/2, 1/3, ..., is their journey down to getting closer and
closer to their limit at 0/1...

[Ben Bacarisse]

"Chris M. Thomasson" <chris.m.t...@gmail.com> writes:

> On 5/29/2023 1:35 PM, Ben Bacarisse wrote:
>> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:
>>
>>>>>>>> on 5/28/2023, Chris M. Thomasson supposed:
>>>>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.
>>
>>> The iterates get closer and closer to zero... Their limit.
>> You know there are books and online courses/tutorials from which you
>> could learn what a limit is? Getting "closer and closer to zero" does
>> not mean that zero is the limit. Don't get me wrong -- zero /is/ the
>> limit in this case, but if you were a student of mine I'd invite you to
>> define a sequence that gets forever "closer and closer to zero" but
>> which does /not/ converge to a limit of zero.
>>
>
> Humm... Perhaps, something like:
>
> 1/2 + 1/3 + 1/4 + 1/5 ... ?
>
> The terms get closer to zero, but the sum always gets bigger?

No. I meant literally what I said. There no games or word trickery
going on. Find a sequence whose terms get closer and closer to zero but
which does not converge to zero. It's much simpler than you think.

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Hint: for the terms of s(n) to get "closer and closer to zero" all we
need is that

|s(n+1) - 0| < |s(n) - 0|

--
Ben.

[Chris M. Thomasson]

On 5/29/2023 1:59 PM, Ben Bacarisse wrote:
> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:
>
>> On 5/29/2023 1:35 PM, Ben Bacarisse wrote:
>>> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:
>>>
>>>>>>>>> on 5/28/2023, Chris M. Thomasson supposed:
>>>>>>>>>> 1/1, 1/2, 1/3, ... on its way to its limit at zero.
>>>
>>>> The iterates get closer and closer to zero... Their limit.
>>> You know there are books and online courses/tutorials from which you
>>> could learn what a limit is? Getting "closer and closer to zero" does
>>> not mean that zero is the limit. Don't get me wrong -- zero /is/ the
>>> limit in this case, but if you were a student of mine I'd invite you to
>>> define a sequence that gets forever "closer and closer to zero" but
>>> which does /not/ converge to a limit of zero.
>>>
>>
>> Humm... Perhaps, something like:
>>
>> 1/2 + 1/3 + 1/4 + 1/5 ... ?
>>
>> The terms get closer to zero, but the sum always gets bigger?
>
> No. I meant literally what I said. There no games or word trickery
> going on. Find a sequence whose terms get closer and closer to zero but
> which does not converge to zero. It's much simpler than you think.

[...]

Humm... For some reason I am thinking about alternating the signs of the
terms?

1/1, -1/2, 1/3, -1/4, 1/5, -1/6, ?

Damn, can't be right. I need to work on some other stuff right now, but
I will get back to you. Thanks Ben! :^)

[Chris M. Thomasson]

For some reason I also thought about alternating the roots

1, -1, 1, -1

Not sure exactly why I thought of it.

[Chris M. Thomasson]

On 5/29/2023 2:04 PM, Chris M. Thomasson wrote:

They do get closer and closer to zero if we used their absolute values,
so damn. Sorry.

[Ben Bacarisse]

This is why one can't do mathematics with vague metaphors. You have
picked one meaning for getting closer and closer to zero, and you can't
see any other.

--
Ben.

[William]

And since this "fist one" is a unit fraction, anything that is true for all unit fractions must be true for this putative "fist one". In particular
it is true for every unit fraction that there is a second unit fraction between it and zero

[Chris M. Thomasson]

I can see many ways to get closer and closer to zero. Say divide by 2:

i[0] = 10
i[1] = i[0] / 2
i[2] = i[1] / 2
i[3] = i[2] / 2
i[4] = i[3] / 2
i[5] = i[4] / 2
...

It gets closer and closer to zero. But, its not what you asked for.

[Ben Bacarisse]

How about 2, 3/2, 4/3, 5/4, 6/5, 7/6, ... 1+1/n, ... Do the terms get
closer and closer to zero, forever? What is the limit?

--
Ben.

[Ben Bacarisse]

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

> William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:
>> we have
>> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .
>> note the universal quantifier. This holds for every unit fraction. Thus for every unit fraction (unrevealed or invisible or ...) there is another unit fraction closer to zero (the gap is finite.),
>> Thus the is no first unit fraction.
>
> This is contradicted by the following: NUF(0) = 0. NUF(1) =
> many. Therefore the unit fractions start between 0 and 1. Never more
> than one occupy a point. This enforces a first one. No way to circ
> umvent this conclusion.

Oh dear. Then, according to you, there is another error in your book.
You give the set of rationals B = {x > sqrt(2)} and say that it has no
smallest element. Define NR(z) as the number of elements of B less than
z. Now NR(sqrt(2)) = 0 and NR(2) = many so the elements of your set B
start between sqrt(2) and 2. Never more than one occupy a point. This
enforces a first one, yet you say that B has no first element.

--
Ben.

[WM]

My case is this: If the number of unit fractions is 0 at 0, then it must have decreased before. That is dictated by pure logic. The case of ZFC is that every unit fraction is succeeded by another one. That is Peano, obtained from the visible natnumbers, but not dictated by logic. Logic is stronger than maths. Therefore my case prevails.

Regards, WM

[WM]

Ben Bacarisse schrieb am Dienstag, 30. Mai 2023 um 03:49:02 UTC+2:
> WM <askas...@gmail.com> writes:
> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
> Unendlichen" at Hochschule Augsburg.)
> > William schrieb am Sonntag, 28. Mai 2023 um 17:44:28 UTC+2:
> >> we have
> >> ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0 .
> >> note the universal quantifier. This holds for every unit fraction. Thus for every unit fraction (unrevealed or invisible or ...) there is another unit fraction closer to zero (the gap is finite.),
> >> Thus the is no first unit fraction.
> >
> > This is contradicted by the following: NUF(0) = 0. NUF(1) =
> > many. Therefore the unit fractions start between 0 and 1. Never more
> > than one occupy a point. This enforces a first one. No way to circ
> > umvent this conclusion.
> Oh dear. Then, according to you, there is another error in your book.
> You give the set of rationals B = {x > sqrt(2)} and say that it has no
> smallest element.

In my book there are no dark numbers.

> Define NR(z) as the number of elements of B less than
> z. Now NR(sqrt(2)) = 0 and NR(2) = many so the elements of your set B
> start between sqrt(2) and 2. Never more than one occupy a point. This
> enforces a first one, yet you say that B has no first element.

Probably my book refering correct mats is better than set theory with its completed sets. There is however the problem that every definable real number sits in a vaccum.

Regards, WM