FromTheRafters schrieb am Freitag, 15. Oktober 2021 um 22:06:11 UTC+2:
> After serious thinking WM wrote :
> > FromTheRafters schrieb am Donnerstag, 14. Oktober 2021 um 14:14:56 UTC+2:
> >> WM presented the following explanation :
> >
> >>> Why that? Magic of matheology? In mathematics the intersection of infinite
> >>> endegments is infinite.
> >>>
> >>> Look, if you take the infinite set of infinite endsegments and try to find
> >>> any infinite endsegment that has less than infinitely many natural numbers
> >>> in common with any infinite endsegment or any set of infinite endsegments
> >>> or all predecessors or all successors or all infinite endsegments, then
> >>> you will fail.
> >> Paraphrased:
> >>
> >> Look, if you take the -infinite- set of -infinite- endsegments and try
> >> to find any -infinite- endsegment that has less than infinitely many
> >> natural numbers in common with any -infinite- endsegment or any set of
> >> -infinite- endsegments or all predecessors or all successors or all
> >> -infinite- endsegments, then you will fail.
> >>
> >> Even cleared of obsessively superfluous infinities
> >
> > It is not superfluous. Every infinite endsegment
> There you go again. Your endsegments are defined as infinite, so the
> word "infinite" before the word "endsegment" is superfluous.

No, it shall remind you that these endsegments have indices which do not exhaust the natural numbers but only a tiny finite initial segment because almost all numbers are contents of FISONs.
>
> You do it in the hopes that it will

show you that infinitely many natural numbers cannot become indices of endsegments. Therefore the set indexes of endsegments is finite but has no fixed last element.

> between the infinite collection of endsegments as a set, and the set
> size of the individual endsegments.

The set of endsegments is actually infinite, the collection of infinite endsegments however cannot be indexed by actually infinitely many natnumbers. So it is finite. But it has no last element. Therefore it is potentially infinite.

IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMNTS EACH.

Regards, WM

On 10/15/2021 5:14 PM, WM wrote:
> FromTheRafters schrieb
> am Freitag, 15. Oktober 2021 um 22:06:11 UTC+2:

> IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS
> WITH ℵo ELEMNTS EACH.

Argumentum ad CAPS LOCK?

----
There is no last FISON.

There is no last FISON-ender.

A last end segment would begin with a last FISON-ender.
There is no last FISON-ender.
There is no last end segment.

An end segment with a last FISON-ender would have a FISON-ender
lat among all FISON-enders.
There is no FISON-ender last among all FISON-enders.
There is no end segment with a last FISON-ender.

To sum up:
There is no last end segment.
There is no end segment with a last FISON-ender.

on 10/15/2021, WM supposed :
> FromTheRafters schrieb am Freitag, 15. Oktober 2021 um 22:06:11 UTC+2:
>> After serious thinking WM wrote :
>>> FromTheRafters schrieb am Donnerstag, 14. Oktober 2021 um 14:14:56 UTC+2:
>>>> WM presented the following explanation :
>>>>> Why that? Magic of matheology? In mathematics the intersection of
>>>>> infinite endegments is infinite.
>>>>>
>>>>> Look, if you take the infinite set of infinite endsegments and try to
>>>>> find any infinite endsegment that has less than infinitely many natural
>>>>> numbers in common with any infinite endsegment or any set of infinite
>>>>> endsegments or all predecessors or all successors or all infinite
>>>>> endsegments, then you will fail.
>>>> Paraphrased:
>>>>
>>>> Look, if you take the -infinite- set of -infinite- endsegments and try
>>>> to find any -infinite- endsegment that has less than infinitely many
>>>> natural numbers in common with any -infinite- endsegment or any set of
>>>> -infinite- endsegments or all predecessors or all successors or all
>>>> -infinite- endsegments, then you will fail.
>>>>
>>>> Even cleared of obsessively superfluous infinities
>>>
>>> It is not superfluous. Every infinite endsegment
>> There you go again. Your endsegments are defined as infinite, so the
>> word "infinite" before the word "endsegment" is superfluous.
>
> No, it shall remind you that these endsegments have indices which do not
> exhaust the natural numbers but only a tiny finite initial segment because
> almost all numbers are contents of FISONs.
>>
>> You do it in the hopes that it will
>
> show you that infinitely many natural numbers cannot become indices of
> endsegments. Therefore the set indexes of endsegments is finite but has no
> fixed last element.
>
>> between the infinite collection of endsegments as a set, and the set
>> size of the individual endsegments.
>
> The set of endsegments is actually infinite, the collection of infinite
> endsegments however cannot be indexed by actually infinitely many natnumbers.
> So it is finite. But it has no last element. Therefore it is potentially
> infinite.
>
> IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMNTS EACH.

You are saying that there cannot in theory be infinitely many buses
each with infinitely many occupants.

Stop lying.

Crank Wolfgang Mueckenheim, aka WM wrotte:

> IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMNTS EACH.

Why, idiot crank Wolfgang Mueckenheim, from Hochschule Augsburg? Because
you wrote it in capital?

On 10/15/2021 4:14 PM, WM wrote:
> FromTheRafters schrieb am Freitag, 15. Oktober 2021 um 22:06:11 UTC+2:
>> After serious thinking WM wrote :
>>> FromTheRafters schrieb am Donnerstag, 14. Oktober 2021 um 14:14:56 UTC+2:
>>>> WM presented the following explanation :
>>>
>>>>> Why that? Magic of matheology? In mathematics the intersection of infinite
>>>>> endegments is infinite.
>>>>>
>>>>> Look, if you take the infinite set of infinite endsegments and try to find
>>>>> any infinite endsegment that has less than infinitely many natural numbers
>>>>> in common with any infinite endsegment or any set of infinite endsegments
>>>>> or all predecessors or all successors or all infinite endsegments, then
>>>>> you will fail.
>>>> Paraphrased:
>>>>
>>>> Look, if you take the -infinite- set of -infinite- endsegments and try
>>>> to find any -infinite- endsegment that has less than infinitely many
>>>> natural numbers in common with any -infinite- endsegment or any set of
>>>> -infinite- endsegments or all predecessors or all successors or all
>>>> -infinite- endsegments, then you will fail.
>>>>
>>>> Even cleared of obsessively superfluous infinities
>>>
>>> It is not superfluous. Every infinite endsegment
>> There you go again. Your endsegments are defined as infinite, so the
>> word "infinite" before the word "endsegment" is superfluous.
>
> No, it shall remind you that these endsegments have indices which do not exhaust the natural numbers but only a tiny finite initial segment because almost all numbers are contents of FISONs.
>>
>> You do it in the hopes that it will
>
> show you that infinitely many natural numbers cannot become indices of endsegments. Therefore the set indexes of endsegments is finite but has no fixed last element.
>
>> between the infinite collection of endsegments as a set, and the set
>> size of the individual endsegments.
>
> The set of endsegments is actually infinite, the collection of infinite endsegments however cannot be indexed by actually infinitely many natnumbers. So it is finite. But it has no last element. Therefore it is potentially infinite.
>
> IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMNTS EACH.

sure it is, and it is so.

in the x y plane, I can have infinite natural numbers for x, and infinite natural numbers for y

>
> Regards, WM
>

Python schrieb am Samstag, 16. Oktober 2021 um 03:04:16 UTC+2:
> WM wrotte:
> > IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMENTS EACH.
> Why, Wolfgang Mueckenheim, from Hochschule Augsburg? Because
> you wrote it in capital?

No, I wrote it in capitals because it is true. The reason is that there are not more than n endsegments containing the natnumbers {n, n+1, n+2, ...}.

Simple example: There are not more than four endsegments containing the natural numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.

But every infinite endsegment contains some finite number n as its index and its smallest element whereupon almost all natural numbers are following, namely ℵo. These are existing between every definable number and ω. Therefore only finitely many infinite endsegments can exist. Note that between every definable natnumber and ω there are infinitely many natnumbers
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
while between all natnumbers and ω there is nothing
|ℕ \ {1, 2, 3, ...}| = 0 .

Regards, WM

Jim Burns schrieb am Samstag, 16. Oktober 2021 um 00:24:36 UTC+2:
> On 10/15/2021 5:14 PM, WM wrote:
> > FromTheRafters schrieb
> > am Freitag, 15. Oktober 2021 um 22:06:11 UTC+2:
> > IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS
> > WITH ℵo ELEMENTS EACH.
> Argumentum ad CAPS LOCK?

Are you really unable to understand this simple argument?

There are not more than four endsegments containing the natural numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.

There are not more than n endsegments containing the natnumbers {n, n+1, n+2, ...}.

But every infinite endsegment contains some finite number n as its index and its smallest element whereupon almost all natural numbers are following, namely ℵo. As long as these are between index and ω the number of endsegments cannot be infinite.

> There is no last end segment.
There is no last endsegment definable. Dark endsegments fill the set.
> There is no end segment with a last FISON-ender.
There is no last FISON-ender definable. Potential infinity.

Note that between every definable natnumber and ω there are infinitely many dark natnumbers
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
while between all natnumbers and ω there is nothing
|ℕ \ {1, 2, 3, ...}| = 0 .

Regards, WM

FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 00:38:37 UTC+2:
> on 10/15/2021, WM supposed :

> > IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMNETS EACH.
> You are saying that there cannot in theory be infinitely many buses
> each with infinitely many occupants.

Not at all! Are you really unable to understand the difference?

Try this: there are not more than four endsegments containing the natural numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.

There are not more than n endsegments containing the natnumbers {n, n+1, n+2, ...}.

There are not more than finitely many endsegments containing infinitely many natnumbers between them and ω.

Note that between every definable natnumber and ω there are infinitely many dark natnumbers
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
while between all natnumbers and ω there is nothing
|ℕ \ {1, 2, 3, ...}| = 0 .

Regards, WM

After serious thinking WM wrote :
> FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 00:38:37 UTC+2:
>> on 10/15/2021, WM supposed :
>
>>> IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMNETS EACH.
>> You are saying that there cannot in theory be infinitely many buses
>> each with infinitely many occupants.
>
> Not at all! Are you really unable to understand the difference?

Your 'go to' stance of characterizing others as inferior doesn't work
as well here as in your classroom. Almost everyone here knows you are
just a crank.

> Try this: there are not more than four endsegments containing the natural
> numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.

So what.

> There are not more than n endsegments containing the natnumbers {n, n+1, n+2,
> ...}.

Again, so what.

> There are not more than finitely many endsegments containing infinitely many
> natnumbers between them and ω.

Conjecture.

> Note that between every definable natnumber and ω there are infinitely many
> dark natnumbers ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> while between all natnumbers and ω there is nothing
>> ℕ \ {1, 2, 3, ...}| = 0 .

Wrong, dark numbers don't exist. Insisting that they do in order to
prove that they must is typical crank behavior.

WM formulated the question :
> Python schrieb am Samstag, 16. Oktober 2021 um 03:04:16 UTC+2:
>> WM wrotte:
>>> IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMENTS EACH.
>> Why, Wolfgang Mueckenheim, from Hochschule Augsburg? Because
>> you wrote it in capital?
>
> No, I wrote it in capitals because it is true.

Ah, so that's why you normally use lowercase.

On 10/16/2021 4:42 AM, WM wrote:
> Jim Burns schrieb am Samstag, 16. Oktober 2021 um 00:24:36 UTC+2:
>> On 10/15/2021 5:14 PM, WM wrote:
>>> FromTheRafters schrieb
>>> am Freitag, 15. Oktober 2021 um 22:06:11 UTC+2:
>>> IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS
>>> WITH ℵo ELEMENTS EACH.
>> Argumentum ad CAPS LOCK?
>
> Are you really unable to understand this simple argument?
>
> There are not more than four endsegments containing the natural numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
>
> There are not more than n endsegments containing the natnumbers {n, n+1, n+2, ...}.
>
> But every infinite endsegment contains some finite number n as its index and its smallest element whereupon almost all natural numbers are following, namely ℵo. As long as these are between index and ω the number of endsegments cannot be infinite.

Each Endsegment is infinite by definition. |E(k)| = ℵo for each k

the intersection of all Endsegments is empty.
Assume a k is in the intersection,
by your above reasoning k is not in E(k+1)
therefore k cannot be in the intersection of all endsegments
therefore there are no numbers in the intersection, it is empty

On 10/16/2021 6:02 AM, FromTheRafters wrote:
> WM formulated the question :
>> Python schrieb am Samstag, 16. Oktober 2021 um 03:04:16 UTC+2:
>>> WM wrotte:
>>>> IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMENTS EACH.
>>> Why, Wolfgang Mueckenheim, from Hochschule Augsburg? Because you wrote it in capital?
>>
>> No, I wrote it in capitals because it is true.
>
> Ah, so that's why you normally use lowercase.

"IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMENTS EACH"

defiantly should all be in lower case too.

WM has Infinity Dysfunction, ID.

FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 13:00:37 UTC+2:
> After serious thinking WM wrote :
> > FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 00:38:37 UTC+2:
> >> on 10/15/2021, WM supposed :
> >
> >>> IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMNETS EACH.
> >> You are saying that there cannot in theory be infinitely many buses
> >> each with infinitely many occupants.
> >
> > Not at all! Are you really unable to understand the difference?
> Your 'go to' stance of characterizing others as inferior doesn't work
> as well here as in your classroom. Almost everyone here

is stupid and inferior to my students. Therefore you cannot even understand the simplest arguments like this

> > there are not more than four endsegments containing the natural
> > numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
> So what.

You haven't got a clue, have you? Your answer shows yout inferiority. I gues that 95 % of my students would say aha, the German expression when understanding somethimg.

> > There are not more than n endsegments containing the natnumbers {n, n+1, n+2,
> > ...}.
> Again, so what.

Again, the confession of not understanding.

> > There are not more than finitely many endsegments containing infinitely many
> > natnumbers between them and ω.
> Conjecture.

Very clear to every mediocre mathematician. Try to understand the example with 4.

> > Note that between every definable natnumber and ω there are infinitely many
> > dark natnumbers ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > while between all natnumbers and ω there is nothing
> >> |ℕ \ {1, 2, 3, ...}| = 0 .
> Wrong, dark numbers don't exist.

Fact is: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo and |ℕ \ {1, 2, 3, ...}| = 0. Whether you agree and understand or disagree and not understand is irrelevant.

Retgards, WM

On Saturday, October 16, 2021 at 12:21:12 PM UTC-4, WM wrote:
> FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 13:00:37 UTC+2:

> > > Note that between every definable natnumber and ω there are infinitely many
> > > dark natnumbers ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > > while between all natnumbers and ω there is nothing
> > >> |ℕ \ {1, 2, 3, ...}| = 0 .
> > Wrong, dark numbers don't exist.
> Fact is: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo and |ℕ \ {1, 2, 3, ...}| = 0. Whether you agree and understand or disagree and not understand is irrelevant.
>

But wait! You recently wrote:

"ℕ_def = {k ∈ ℕ : |∩{E(1), E(2), ..., E(k)}|= ℵo}"
--sci.math "ZFC knows potential infinity," 2021-08-11

Und so, N_def = N.

It seems you have made no progress whatsoever in your never ending quest (40+ years now?) to demonstrate any inconsistencies in ZFC. Must be frustrating as hell. Isn't it time, at your advanced age, to salvage a bit of dignity and quietly retire from sci.math? Honest, we won't tell anyone!

Dan

On 10/16/2021 5:42 AM, WM wrote:
> Jim Burns schrieb
> am Samstag, 16. Oktober 2021 um 00:24:36 UTC+2:

>> There is no last end segment.
>
> There is no last endsegment definable.
> Dark endsegments fill the set.
>
>> There is no end segment with a last FISON-ender.
>
> There is no last FISON-ender definable.

There is no last FISON-ender undefinable,
because there is no FISON-ender undefinable.

| Assume FISON {0,...,k} exists, and k is undefinable.
| Let NaN = { i | k >= i >= undefinable j in {0,...,k} }
| | {0,...,k}\NaN, NaN is a split, with crossing-pair j,j+1
| such that j is definable and j+1 is undefinable.
| | However, j+1 is definable as the successor of j.
| Contradiction.

Therefore,
there is no FISON-ender undefinable.

> Potential infinity.

If k is a FISON-ender, then k is in the set of FISON-enders.
If k is in the set of FISON-enders, then k is a FISON-ender.

> Note that between every definable natnumber and ω
> there are infinitely many dark natnumbers

Between each FISON-ender and anything which follows all
FISON-enders, there are infinitely-many FISON-enders.

| Assume otherwise, that G follows all FISON-enders and
| for each split of {k,...,G}, there is a crossing-pair j,j+1.
| | The split {k,...,G}\{G}, {G} has a crossing-pair j,j+1.
| G = j+1 is the successor of j.
| Each successor has its successor.
| G has its successor G+1.
| | {0,...,G}∪{G+1} is a FISON. It begins, ends, and
| has a crossing-pair j,j+1 for each split.
| G+1 is a FISON-ender.
| G does not follow all FISON-enders.
| Contradiction.

Therefore,
between each FISON-ender and anything which follows all
FISON-enders, there are infinitely-many FISON-enders.

> Note that between every definable natnumber and ω
> there are infinitely many dark natnumbers

The dark natnumbers which end a FISON are in the set of
FISON-enders.
The dark natnumbers which do NOT end a FISON are NOT
in the set of FISON-enders.

The intersection of all end segments of the FISON-enders
is empty.

> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> while between all natnumbers and ω there is nothing
> |ℕ \ {1, 2, 3, ...}| = 0 .

For each natural number, we define omega to be a follower.
For each natural number, its successor is not omega.

WM laid this down on his screen :
> FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 13:00:37 UTC+2:
>> After serious thinking WM wrote :
>>> FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 00:38:37 UTC+2:
>>>> on 10/15/2021, WM supposed :
>>>
>>>>> IT IS NOT POSSIBLE THAT THERE ARE ℵo ENDSEGMENTS WITH ℵo ELEMNETS EACH.
>>>> You are saying that there cannot in theory be infinitely many buses
>>>> each with infinitely many occupants.
>>>
>>> Not at all! Are you really unable to understand the difference?
>> Your 'go to' stance of characterizing others as inferior doesn't work
>> as well here as in your classroom. Almost everyone here
>
> is stupid and inferior to my students. Therefore you cannot even understand
> the simplest arguments like this
>
>>> there are not more than four endsegments containing the natural
>>> numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
>> So what.
>
> You haven't got a clue, have you? Your answer shows yout inferiority. I gues
> that 95 % of my students would say aha, the German expression when
> understanding somethimg.
>
>>> There are not more than n endsegments containing the natnumbers {n, n+1,
>>> n+2, ...}.
>> Again, so what.
>
> Again, the confession of not understanding.
>
>>> There are not more than finitely many endsegments containing infinitely
>>> many natnumbers between them and ω.
>> Conjecture.
>
> Very clear to every mediocre mathematician. Try to understand the example
> with 4.
>
>>> Note that between every definable natnumber and ω there are infinitely many
>>> dark natnumbers ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>> while between all natnumbers and ω there is nothing
>>>>> ℕ \ {1, 2, 3, ...}| = 0 .
>> Wrong, dark numbers don't exist.
>
> Fact is: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo and |ℕ \ {1, 2, 3, ...}| =
> 0. Whether you agree and understand or disagree and not understand is
> irrelevant.

I understand *and* disagree. I am not alone in this.

Crank Wolfgang Mueckenheim, aka WM wrote:
> FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 13:00:37 UTC+2:
....
> You haven't got a clue, have you? Your answer shows yout inferiority.
> I gues that 95 % of my students would say aha, the German expression
> when understanding somethimg.

In this specific case then it means: "Mueckenheim is a fool and a crank,
there is no point in arguing, let's repeat this fallacies then we'll
get good grades at the exam".

You are shame to German Academy and Humanity, Crank Wolfgang Mueckenheim
from Hochschule Augsburg.

FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 20:50:14 UTC+2:
> WM laid this down on his screen :

> >>> there are not more than four endsegments containing the natural
> >>> numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.

> I understand *and* disagree. I am not alone in this.

Really? Who else beliefs that there are more than four endsegments?

Regards, WM

Dan Christensen schrieb am Samstag, 16. Oktober 2021 um 19:37:03 UTC+2:
> On Saturday, October 16, 2021 at 12:21:12 PM UTC-4, WM wrote:
> > FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 13:00:37 UTC+2:
>
> > > > Note that between every definable natnumber and ω there are infinitely many
> > > > dark natnumbers ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > > > while between all natnumbers and ω there is nothing
> > > >> |ℕ \ {1, 2, 3, ...}| = 0 .
> > > Wrong, dark numbers don't exist.
> > Fact is: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo and |ℕ \ {1, 2, 3, ...}| = 0. Whether you agree and understand or disagree and not understand is irrelevant.
> >
> But wait! You recently wrote:
>
> "ℕ_def = {k ∈ ℕ : |∩{E(1), E(2), ..., E(k)}|= ℵo}"
> --sci.math "ZFC knows potential infinity," 2021-08-11
>
> Und so, N_def = N.

No, that is a erroneous conclusion. Perhaps you will understad (A), perhaps even (B), probably not (C).

(A) There are not more than four endsegments containing the naturalnumbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.

(B) There are not more than n endsegments containing the natnumbers {n, n+1, n+2, ...}.

(C) There are not more than finitely many infinite endsegments.

Regards, WM

Jim Burns schrieb am Samstag, 16. Oktober 2021 um 20:47:16 UTC+2:

> The intersection of all end segments of the FISON-enders
> is empty.

No. Perhaps you will understad (A), perhaps even (B), probably not (C).

(A) There are not more than four endsegments containing the natural numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
(B) There are not more than n endsegments containing the natnumbers {n, n+1, n+2, ...}.
(C) There are not more than finitely many infinite endsegments. Finitely many infinite endsegments have an infinite intersection.

Regards, WM

> > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > while between all natnumbers and ω there is nothing
> > |ℕ \ {1, 2, 3, ...}| = 0 .
> For each natural number, we define omega to be a follower.
> For each natural number, its successor is not omega.

But if we only subtract all natural numbers, then omega is remaining without any predecessor.

Regards, WM

Python schrieb am Samstag, 16. Oktober 2021 um 23:44:53 UTC+2:
> WM wrote:

> > I gues that 95 % of my students would say aha, the German expression
> > when understanding something.
> In this specific case then it means:

If
{1, 2, 3, 4, 5, ...} <--> {1/1, 2/1, 3/1, 4/1, 5/1, ...}
is a bijection (i.e. surjective and injective) and M is an infinite set, then
{1, 2, 3, 4, 5, ...} <--> {1/1, 2/1, 3/1, 4/1, 5/1, ..., M}
cannot be a bijection. Of course the frauds and the stupids of matheology try to disparage this clear fact. It would be alarming if they didn't.

Regards, WM

WM wrote :
> Jim Burns schrieb am Samstag, 16. Oktober 2021 um 20:47:16 UTC+2:
>
>> The intersection of all end segments of the FISON-enders
>> is empty.
>
> No. Perhaps you will understad (A), perhaps even (B), probably not (C).
>
> (A) There are not more than four endsegments containing the natural numbers
> {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.

Which means that the intersection of *all* of them is empty already --
no need to filter out any more FISONs to show this. How can you not see
this?

> (B) There are not more than n endsegments containing the natnumbers {n, n+1,
> n+2, ...}.

Again, so what? Showing how larger cardinality FISONs make the
intersection of *all* of them "more empty" doesn't help your argument.

> (C) There are not more than finitely many infinite endsegments.

A declaration with nothing to back it up. Each natural number has such
a following, no exceptions.

> Finitely many infinite endsegments have an infinite intersection.

Still wrong to assume that there are only finitely many of them.

WM formulated on Sunday :
> FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 20:50:14 UTC+2:
>> WM laid this down on his screen :
>
>>>>> there are not more than four endsegments containing the natural
>>>>> numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
>
>> I understand *and* disagree. I am not alone in this.
>
> Really? Who else beliefs that there are more than four endsegments?

Nice editing job. You know of course that I was disagreeing with your
"definable" and "dark" weasel-word numbers in the rest of your post.

You are very dishonest and should be ashamed of yourself.

FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 14:44:19 UTC+2:
> WM wrote :

> > (C) There are not more than finitely many infinite endsegments.
> A declaration with nothing to back it up. Each natural number has such
> a following, no exceptions.

Each definable natural number has infinitely many successors which are also natural numbers. So the infinite set |N of natural numbers is complete.

Each infinite endsegment has not infinitely many succeeding infinite endsegments because the infinite contents cannot be used as indexes simultaneously.

> > Finitely many infinite endsegments have an infinite intersection.
> Still wrong to assume that there are only finitely many of them.

If there are only endsegments of infinite contents it is true. Every infinite contents has some n as its smallest number. {n+1, n+2, n+3, ...} cannot be used to index further endsegments. Such an infinite set {n+1, n+2, n+3, ....} must remain as long as the endsegments are infinite.

Regards, WM

FromTheRafters schrieb am Sonntag, 17. Oktober 2021 um 14:53:42 UTC+2:
> WM formulated on Sunday :
> > FromTheRafters schrieb am Samstag, 16. Oktober 2021 um 20:50:14 UTC+2:
> >> WM laid this down on his screen :
> >
> >>>>> there are not more than four endsegments containing the natural
> >>>>> numbers {4, 5, 6, ...}, namely E(1), E(2), E(3), and E(4). E(5) fails.
> >
> >> I understand *and* disagree. I am not alone in this.
> >
> > Really? Who else beliefs that there are more than four endsegments?
> Nice editing job. You know of course that I was disagreeing with your
> "definable" and "dark" weasel-word numbers in the rest of your post.

No, how should I? If you understand that above only four can exist, then you could also understand that only n can exist containing {n, n+1, n+2, ...}.. Then you could also understand that this is true for every n starting an infinite set. What is the difference?

Regards, WM