On 7/5/2021 2:39 PM, WM wrote:
> William schrieb am Montag, 5. Juli 2021 um 21:26:59 UTC+2:
>> On Monday, July 5, 2021 at 3:06:17 PM UTC-4, WM wrote:
>
>>> As long as every passed unit fraction has aleph_0 successors, the cursor cannot reach zero
>> "As long" is a period of time [0,1)
>
> No, it means "given that". No time is involved.
>
>> (Have the cursor start at time 0 and travel at a speed of 1). It is not "forever".
>
> That proves that not always unit fractions are between the curser and zero. When the cursor has arrived, then there is no unit fraction between it and zero. All are passed. But never two or more are passed at the same position.

all numbers in the interval were identified and passed, ALL numbers.

>
>> The cursor cannot reach 0 for any time t<1. The cursor can read 0 for any time t>=1.
>
> We should forget the time. Fact is: When the cursor reaches 0 no unit fraction is between itself and 0. And fact is further that the cursor never passes aleph_0 unit fractions at the same point.

wrong. the cursor is not counting numbers, like poor Achilles and his
frames

>
> Regards, WM
>

On 7/5/2021 4:28 PM, WM wrote:
> Gus Gassmann schrieb am Montag, 5. Juli 2021 um 22:25:23 UTC+2:
>> On Monday, 5 July 2021 at 16:14:49 UTC-3, WM wrote:
>>> Gus Gassmann schrieb am Montag, 5. Juli 2021 um 20:44:09 UTC+2:
>>>> On Monday, 5 July 2021 at 15:35:13 UTC-3, WM wrote:
>>>
>>> es not play a role. The cursor has only to satisfy one condition, namely to arrive at 0.
>>>>>> that makes no sense.
>>>>> I am of different opinion.
>>>> Evidently.
>>>>
>>>> Your difference of opinion is noted but is utterly irrelevant.
>>>
>>> I am of different opinion.
>>>
>>>> Anyone who thinks the cursor can only reach zero by passing only finitely many unit fractions in order to get there "because it cannot pass two or more or even infinitely many at the same point" has lost *ALL* ability to reason along with any credibility
>>> in matheology? That is easy to bear.
>>>
>>> The cursor cannot reach zero as long as unit fractions are between itself an zero. Not even one is allowed. That is a matter of fact. Note that for ever n 0 < 1/n < cursor position. Also passing aleph_0 unit fractions simultaneously is impossible. Note that 1/n < 1/m or vice versa for different unit fractions.
>> As I said: Your ability to reason is now officially zero.
>
> At zero rematkable hings happen.
>
>> As such, it doesn't interest me any longer, and neither does this discussion.
>
> You suspect that you will no longer be able to adhere to the nonsense of set theory? Note that never two unit fractions reside at the same position, let alone aleph_0.
>
> Regards, WM
>

your conclusions are usally wrong, we try to show you the correct way,
but you do not, or don't want to understand, to protect your imagined
dark numbers.

On Monday, July 5, 2021 at 5:28:17 PM UTC-4, WM wrote:

> You suspect that you will no longer be able to adhere to the nonsense of set theory? Note that never two unit fractions reside at the same position, let alone aleph_0.

Pure gibberish, Mucke! You tried and failed to to demonstrate any inconsistencies with your FISON's, end-segments, "dark" numbers, etc. Now, you have failed trying to introduce physics into the mix with events in space-time. What next, Mucke? Quotes from the bible?

******************************************
More absurd quotes from Wolfgang Muckenheim (WM):

“In my system, two different numbers can have the same value.”
-- sci.math, 2014/10/16

“1+2 and 2+1 are different numbers.”
-- sci.math, 2014/10/20

“1/9 has no decimal representation.”
-- sci.math, 2015/09/22

"0.999... is not 1."
-- sci.logic 2015/11/25

“Axioms are rubbish!”
-- sci.math, 2014/11/19

“Formal definitions have lead to worthless crap like undefinable numbers.”
-- sci.math 2017/02/05

“No set is countable, not even |N.”
-- sci.logic, 2015/08/05

“Countable is an inconsistent notion.”
-- sci.math, 2015/12/05

Slipping ever more deeply into madness...

“There is no actually infinite set |N.”
-- sci.math, 2015/10/26

“|N is not covered by the set of natural numbers.”
-- sci.math, 2015/10/26

“The set of all rationals can be shown not to exist.”
--sci.math, 2015/11/28

“Everything is in the list of everything and therefore everything belongs to a not uncountable set.”
-- sci.math, 2015/11/30

"'Not equal' and 'equal can mean the same.”
-- sci.math, 2016/06/09

“The set of numbers will get empty after all have numbers been used..”
-- sci.math, 2016/08/24

“I need no set theory.”
-- sci.math, 2016/09/01

A special word of caution to students: Do not attempt to use WM's “system” (MuckeMath) in any course work in any high school, college or university on the planet. You will fail miserably. MuckeMath is certainly no shortcut to success in mathematics.

Using WM's “axioms” for the natural numbers, he cannot even prove that 1=/=2. His goofy little system is truly a dead-end.

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> Gus Gassmann schrieb am Montag, 5. Juli 2021 um 22:25:23 UTC+2:
> > On Monday, 5 July 2021 at 16:14:49 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Montag, 5. Juli 2021 um 20:44:09 UTC+2:
> > > > On Monday, 5 July 2021 at 15:35:13 UTC-3, WM wrote:
> > >
> > > es not play a role. The cursor has only to satisfy one condition, namely to arrive at 0.
> > > > > > that makes no sense.
> > > > > I am of different opinion.
> > > > Evidently.
> > > >
> > > > Your difference of opinion is noted but is utterly irrelevant.
> > >
> > > I am of different opinion.
> > >
> > > > Anyone who thinks the cursor can only reach zero by passing only finitely many unit fractions in order to get there "because it cannot pass two or more or even infinitely many at the same point" has lost *ALL* ability to reason along with any credibility
> > > in matheology? That is easy to bear.
> > >
> > > The cursor cannot reach zero as long as unit fractions are between itself an zero. Not even one is allowed. That is a matter of fact. Note that for ever n 0 < 1/n < cursor position. Also passing aleph_0 unit fractions simultaneously is impossible. Note that 1/n < 1/m or vice versa for different unit fractions.
> > As I said: Your ability to reason is now officially zero.
> At zero rematkable hings happen.
> > As such, it doesn't interest me any longer, and neither does this discussion.
> You suspect that you will no longer be able to adhere to the nonsense of set theory? Note that never two unit fractions reside at the same position, let alone aleph_0.

No, believe it or not, it pains me to watch your slow descent into madness and dementia. Who, other than you, have ever said or thought that different numbers could occupy the same space? Or that they would necessarily have to if there are infinitely many of them in the unit interval? This belief is utter rot, and it is not motivated by anything you have previously said or written. Ergo, it is evidence of a *new* disconnect in your brain. You *are* losing your marbles, slowly, but for everyone to see. It is not a pretty sight.

0123 let there be light numbers 0123
more formally in C++:
// let there be light numbers 0123
typedef unsigned long num; const num one = (num)true; const num zero = one >> one; const num two = one + one + (one >> one); const num three = two + one;

Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
Note that never two unit fractions reside at the same position, let alone aleph_0.

> Who, other than you, have ever said or thought that different numbers could occupy the same space?

Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1]. The cursor must get rid of them between this open interval and 0.

> Or that they would necessarily have to if there are infinitely many of them in the unit interval?

They would have to sit between 0 and the interval (0, 1].
Of course this is nonsense. Therefore the cursor passes a last unit fraction.

Regards, WM

Dan Christensen schrieb am Dienstag, 6. Juli 2021 um 06:11:22 UTC+2:
> On Monday, July 5, 2021 at 5:28:17 PM UTC-4, WM wrote:
>
> > You suspect that you will no longer be able to adhere to the nonsense of set theory? Note that never two unit fractions reside at the same position, let alone aleph_0.
> Pure gibberish

Fact.

> Now, you have failed trying to introduce physics into the mix with events in space-time.

No, that is only mathematics: The cursor moving from 1 to 0 passes all unit fractions existing there (not existing whatevers are irrelevant). It passes never two or more at the same place. Therefore it must have passed a last unit fraction when arriving at zero. This unit fraction cannot be known, like its aleph_0 predecessors. They are dark, they cannot be named, and they cannot be put in any order to be used for mappings. Otherwise we had a contradition. Only the few unit fractions belonging to finite initial segments are visible.

Regards, WM

Greg Cunt schrieb am Montag, 5. Juli 2021 um 21:48:46 UTC+2:
> On Monday, July 5, 2021 at 9:39:29 PM UTC+2, WM wrote:

> > We should forget the time.
> No, we shouldn't,
>
> Hint: "The cursor traverses the path in, say, a second." (Wolfgang Mückenheim)

That is an illustration but it is not necessary.

Regards, WM

William schrieb am Montag, 5. Juli 2021 um 21:51:35 UTC+2:
> : When the cursor reaches 0 no unit fraction is between itself and 0. And fact is further that the cursor never passes aleph_0 unit fractions at the same point.
>
> Correct. So what?

It excludes the claim that for all cursor positions > 0 there are aleph_0 unit fractions between itself and zero. Then the cursor would have to pass aleph_0 unit fractions between (0, 1] and 0.

Regards, WM

On Tuesday, July 6, 2021 at 5:59:53 PM UTC+2, WM wrote:

> cursor would have to pass aleph_0 unit fractions between (0, 1] and 0.

There is no "between", dumbo.

Either x e (0, 1], or x = 0 for all x e [0, 1].

But as long as x e (0, 1] there are infinitely many unit fractions between x and 0, right. :-)

Jim Burns schrieb am Montag, 5. Juli 2021 um 22:00:13 UTC+2:
> On 7/4/2021 9:08 AM, WM wrote:

> > None remains, not even the infinitely many following upon
> > every accessible unit fraction.
>
> Any point x before 0 and after each fraction 1/k corresponds to
> a point 1/x after each (finite) natural k. Archimedes would say
> that no such 1/x exist in the line, but that each point must be
> able to be encompassed by finitely-many unit-interval extensions
> of the line.
>
> No 1/x, no x, no dark points.

Assume that for every point 1/x in the interval (0, 1] there are aleph_0 unit fractions between cursor and 0. In 0 there all are lost. That means all get lost between (0, 1] and 0. That is impossible. Each unit fraction occupies its own place. They cannot get lost between (0, 1] and 0. The get lost one by one - until the last one is lost. But we cannot prove that it is the last one.
>
> > They are not accessible. They are dark.

> But you don't say anything about what your dark points are like.

They are unit fractions with natural denominators. Each natural has two direct neighbours, probably many predecessors and successors, but no connection by a FISON to zero.

> The only use I see for dark points is that they give you (WM)
> something to say to impertinent students who expect you to
> know something about the material you're lecturing on.

I have no impertinent students.
>
> ----
> A better description of these points is
> (i)
> definable points passed in every movement
> ("definable" might only be "finite-length decimals"),

Also many irrationals are defined, like pi, e, L. They all are surrounded by dark points. But in oder to keep things easy I have only used dark points before zero.

Regards, WM

Greg Cunt schrieb am Montag, 5. Juli 2021 um 22:06:01 UTC+2:

> That's why we use IR. In IR there are only so called "real numbers" (no dark matter)!

Every definable point is surrounded by a cloud of dark points.

Regards, WM

FromTheRafters schrieb am Montag, 5. Juli 2021 um 23:29:54 UTC+2:
> WM formulated on Monday :

> > We should forget the time.
> Then what does 'arrive' and 'passing' even mean? Time is implicit in
> your adopting a step by step method.

Only mode of speaking. "Passing" means being at the same point.

Regards, WM

On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
> Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> > On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> Note that never two unit fractions reside at the same position, let alone aleph_0.
> > Who, other than you, have ever said or thought that different numbers could occupy the same space?
> Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].

In what universe is that the same as "two numbers occupy the same space", you *fucking imbecile*? Are you really so confused, or are you deliberately lying? At any rate, your falsehood is ridiculously transparent. I suspect your are so far gone that you can't even fathom that any longer.

> > Or that they would necessarily have to if there are infinitely many of them in the unit interval?
> They would have to sit between 0 and the interval (0, 1].

Again, you are a *fucking imbecile*! All unit fractions, and indeed every numbers strictly between 0 and 1 is separated from 0 by a positive difference. How difficult can that possibly be to comprehend?

Greg Cunt schrieb am Dienstag, 6. Juli 2021 um 00:14:51 UTC+2:
> >>There are (always) infinitely many unit fractions between 0 and the position of the cursor x(t) for all t e [0, 1).

> > The cursor does not pass two or more at the same position or time.
> Did anyone claim that?

Yes, you did. See the first line above. If for all positions in (0, 1] aleph_0 unit fractions are between the cursor and zero, but none are there at zero, then aleph_0 unit fractions must have been passed between (0, 1] and 0. That is impossible.
>
> Hint: For any x, x e IR, x > 0, there are uncountably many real numbers between x and 0. For x = 0 there is no number "between" x and 0. (Between here means ... < r < ...)

That is the same nonsense, aggravated only by the use of uncountability, the greatest nonsense ever concocted in mathematics.

Regards, WM

Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 18:19:04 UTC+2:
> On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> > > On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> > Note that never two unit fractions reside at the same position, let alone aleph_0.
> > > Who, other than you, have ever said or thought that different numbers could occupy the same space?
> > Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].
> In what universe is that the same as "two numbers occupy the same space"

In our universe.

> > > Or that they would necessarily have to if there are infinitely many of them in the unit interval?
> > They would have to sit between 0 and the interval (0, 1].
> All unit fractions, and indeed every numbers strictly between 0 and 1 is separated from 0 by a positive difference. How difficult can that possibly be to comprehend?

Therefore they cannot get lost all together between (0, 1] and 0.

Regards, WM

On Tuesday, July 6, 2021 at 11:59:53 AM UTC-4, WM wrote:
> William schrieb am Montag, 5. Juli 2021 um 21:51:35 UTC+2:
> > : When the cursor reaches 0 no unit fraction is between itself and 0. And fact is further that the cursor never passes aleph_0 unit fractions at the same point.
> >
> > Correct. So what?
> It excludes the claim that for all cursor positions > 0 there are aleph_0 unit fractions between itself and zero

No it doesn't.

>Then the cursor would have to pass aleph_0 elements of U_F between (0, 1] and 0.

Nope, it is sufficient that the cursor passes aleph_0 elements of U_F in any neighborhood of 0. Of course you get a contradiction at the last neighborhood, Guess What?

--
William Hughes

On Tuesday, July 6, 2021 at 11:44:32 AM UTC-4, WM wrote:
> Dan Christensen schrieb am Dienstag, 6. Juli 2021 um 06:11:22 UTC+2:
> > On Monday, July 5, 2021 at 5:28:17 PM UTC-4, WM wrote:
> >
> > > You suspect that you will no longer be able to adhere to the nonsense of set theory? Note that never two unit fractions reside at the same position, let alone aleph_0.
> > Pure gibberish
> Fact.
> > Now, you have failed trying to introduce physics into the mix with events in space-time.

> No, that is only mathematics: The cursor moving from 1 to 0 passes all unit fractions existing there (not existing whatevers are irrelevant).

Your "cursor" is what they would call a point particle in physics. Its arrival at a point x at time t would an event in space-time. But you should know this. Why are playing dumb? Isn't physics your real field? Unless you are NOT playing at it.

> It passes never two or more at the same place. Therefore it must have passed a last unit fraction when arriving at zero. This unit fraction cannot be known, like its aleph_0 predecessors.

Contrary to what they thought 3000 years ago in ancient Greece and you continue to believe, an infinite number events can occur in finite length of time. Your point particle (cursor) passes through infinitely many points every time is moves. Deal with it, Mucke.

> They are dark, they cannot be named, and they cannot be put in any order to be used for mappings.

Pure gibberish.

> Otherwise we had a contradition. Only the few unit fractions belonging to finite initial segments are visible.
>

We have been waiting years (decades?) for your proof of any contradiction in set theory. Each of your attempts, including this one, have failed miserably. When will you learn, Mucke?

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On Tuesday, July 6, 2021 at 12:28:06 PM UTC-4, WM wrote:
> Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 18:19:04 UTC+2:
> > On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> > > > On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> > > Note that never two unit fractions reside at the same position, let alone aleph_0.
> > > > Who, other than you, have ever said or thought that different numbers could occupy the same space?
> > > Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].
> > In what universe is that the same as "two numbers occupy the same space"
> In our universe.
> > > > Or that they would necessarily have to if there are infinitely many of them in the unit interval?
> > > They would have to sit between 0 and the interval (0, 1].

Nope, for every x element of U_F there must be a set S(x) elements of U_F in the interval [0,x] and this set has cardinality aleph_0. Only if the result of applying the operator P_1(S) to the sequence S=(S(x_1), S(x_2),S(x_3)...) is non empty can you conclude that there must be elements of U_F between 0 and (0,1]
However, you cannot determine the result of applying the operator to the sequence by using a stepwise method.

--
William Hughes

William schrieb am Dienstag, 6. Juli 2021 um 18:51:51 UTC+2:
> On Tuesday, July 6, 2021 at 12:28:06 PM UTC-4, WM wrote:
> > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 18:19:04 UTC+2:
> > > On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
> > > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> > > > > On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> > > > Note that never two unit fractions reside at the same position, let alone aleph_0.
> > > > > Who, other than you, have ever said or thought that different numbers could occupy the same space?
> > > > Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].
> > > In what universe is that the same as "two numbers occupy the same space"
> > In our universe.
> > > > > Or that they would necessarily have to if there are infinitely many of them in the unit interval?
> > > > They would have to sit between 0 and the interval (0, 1].
> Nope, for every x element of U_F there must be a set S(x) elements of U_F in the interval [0,x] and this set has cardinality aleph_0.

But the set cannot have cardinality aleph_0 for every point of (0, 1]. Otherwise aleph_0 unit fractions have to disappear in a single point. Therefore the cardinality must go down earlier.

> Only if the result of applying the operator P_1(S) to the sequence S=(S(x_1), S(x_2),S(x_3)...) is non empty can you conclude that there must be elements of U_F between 0 and (0,1]

> Here we have no operators but facts.

> However, you cannot determine the result of applying the operator to the sequence by using a stepwise method.

It cannot be determined at all and is of no interest.

Regards, WM

Dan Christensen schrieb am Dienstag, 6. Juli 2021 um 18:36:44 UTC+2:
> On Tuesday, July 6, 2021 at 11:44:32 AM UTC-4, WM wrote:

> > It passes never two or more at the same place. Therefore it must have passed a last unit fraction when arriving at zero. This unit fraction cannot be known, like its aleph_0 predecessors.
> Contrary to what they thought 3000 years ago in ancient Greece and you continue to believe, an infinite number events can occur in finite length of time. Your point particle (cursor) passes through infinitely many points every time is moves.

Irrelevant. Here only unit fractions are of interest. There are infinitely many. They are passed either simultaneously or one by one. In the latter case there is a last one passed.

Regards, WM

William schrieb am Dienstag, 6. Juli 2021 um 18:32:01 UTC+2:
> On Tuesday, July 6, 2021 at 11:59:53 AM UTC-4, WM wrote:
> > William schrieb am Montag, 5. Juli 2021 um 21:51:35 UTC+2:
> > > : When the cursor reaches 0 no unit fraction is between itself and 0. And fact is further that the cursor never passes aleph_0 unit fractions at the same point.
> > >
> > > Correct. So what?
> > It excludes the claim that for all cursor positions > 0 there are aleph_0 unit fractions between itself and zero
> No it doesn't.
>
> >Then the cursor would have to pass aleph_0 elements of U_F between (0, 1] and 0.
>
> Nope, it is sufficient that the cursor passes aleph_0 elements of U_F in any neighborhood of 0. Of course you get a contradiction at the last neighborhood, Guess What?

There is a last position, it is zero and it exists. There are infinitely many unit fractions. They have been passed either simultaneously or one by one. That is independent of assumed neighbourhoods or operators. In the latter case there is a last one passed.

Regards, WM

On 7/6/2021 12:16 PM, WM wrote:
> Dan Christensen schrieb am Dienstag, 6. Juli 2021 um 18:36:44 UTC+2:
>> On Tuesday, July 6, 2021 at 11:44:32 AM UTC-4, WM wrote:
>
>>> It passes never two or more at the same place. Therefore it must have passed a last unit fraction when arriving at zero. This unit fraction cannot be known, like its aleph_0 predecessors.
>> Contrary to what they thought 3000 years ago in ancient Greece and you continue to believe, an infinite number events can occur in finite length of time. Your point particle (cursor) passes through infinitely many points every time is moves.
>
> Irrelevant. Here only unit fractions are of interest.

which is meaningless as the unit fractions are subset of real numbers in
the interval.

There are infinitely many. They are passed either simultaneously or one
by one.

wrong. None are passed simultaneously.

>In the latter case there is a last one passed.

Fail. there are no "last" unit fractions

>
> Regards, WM
>

On Tuesday, July 6, 2021 at 1:14:10 PM UTC-4, WM wrote:
> William schrieb am Dienstag, 6. Juli 2021 um 18:51:51 UTC+2:
> > On Tuesday, July 6, 2021 at 12:28:06 PM UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 18:19:04 UTC+2:
> > > > On Tuesday, 6 July 2021 at 12:35:32 UTC-3, WM wrote:
> > > > > Gus Gassmann schrieb am Dienstag, 6. Juli 2021 um 12:30:54 UTC+2:
> > > > > > On Monday, 5 July 2021 at 18:28:17 UTC-3, WM wrote:
> > > > > Note that never two unit fractions reside at the same position, let alone aleph_0.
> > > > > > Who, other than you, have ever said or thought that different numbers could occupy the same space?
> > > > > Many people have said or believed that for every point > 0 there are aleph_0 unit fractions between zero and the cursor, that means all over (0, 1].
> > > > In what universe is that the same as "two numbers occupy the same space"
> > > In our universe.
> > > > > > Or that they would necessarily have to if there are infinitely many of them in the unit interval?
> > > > > They would have to sit between 0 and the interval (0, 1].
> > Nope, for every x element of U_F there must be a set S(x) elements of U_F in the interval [0,x] and this set has cardinality aleph_0.
> But the set cannot have cardinality aleph_0 for every point of (0, 1].

Each set, (the sets s(x) are different for different x) has cardinality aleph_0

>Otherwise aleph_0 unit fractions have to disappear in a single point.

Nope. Whatever makes you think that? The sets are all different, even though they have the same cardinality, so here "Infinite intersection" (something you determineby app lying an operator to a sequence) can be the empty set.

--
William Hughes

William schrieb am Dienstag, 6. Juli 2021 um 19:28:07 UTC+2:
> On Tuesday, July 6, 2021 at 1:14:10 PM UTC-4, WM wrote:

> > But the set cannot have cardinality aleph_0 for every point of (0, 1].
> Each set, (the sets s(x) are different for different x) has cardinality aleph_0

The cursor reaches zero. If for every x the cardinality is aleph_0, then the cursore gets rid of infinitely many unit fraction at one point between (0, 1] and 0. Contradiction.

> >Otherwise aleph_0 unit fractions have to disappear in a single point.
> Nope. Whatever makes you think that?

Simple logic. The set shrinks before 0 because aleph_0 unit fractions cannot sit at one point.

> The sets are all different, even though they have the same cardinality, so here "Infinite intersection" (something you determineby app lying an operator to a sequence) can be the empty set.

My example shows that this is a foolish claim. It can be believed in the problem with endsegments of |N by some not too bright matheologians, but it is excluded in the present problem because only one unit fraction can be passed as the last one unless more are passed simultaneously. Note that is not a stepwise problem but a general question of mathematics: Can many unit fractions sit in one point or not? In order to main atin set thepry you must claim this. I am sure unbiased students will not believe it, neither your attempted tricks with operators designes according to your taste.

Regards, WM