On 10/16/2021 11:21 AM, WM wrote:
> 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

you get angry ?? you are the one that ALWAYS posts wrong stuff. Every single Post is Wrong.

No proofs, No Algebra, No variables,... you are a kook.

>
>>> 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.

How many of them believe in your dark numbers after your class is finished?

On 10/17/2021 2:49 PM, WM wrote:
> Dan Christensen schrieb am Sonntag, 17. Oktober 2021 um 21:13:23 UTC+2:
>> On Sunday, October 17, 2021 at 6:23:28 AM UTC-4, WM wrote:
>
>>>>> 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 an erroneous conclusion.
>>
>> E(n) = {n, n+1, n+2, ... } for all n in N.
>>
>> |E(n)|= ℵo for all n in N
>
> That appears so to the superficial thinker.
> (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, ...}.

n is not a number, you said.

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

wrong

>
> Regards, WM
>

söndag 17 oktober 2021 kl. 20:31:22 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Sonntag, 17. Oktober 2021 um 15:14:44 UTC+2:
> > söndag 17 oktober 2021 kl. 15:00:10 UTC+2 skrev 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?
> > Yes, it is weasel words
> No, every infinite endsegment of ℕ starts at some finite natural number.
>
> Regards, WM

Yes but all you do is use weasle words to not do proper mathematical proofs

söndag 17 oktober 2021 kl. 20:51:47 UTC+2 skrev WM:
> Gus Gassmann schrieb am Sonntag, 17. Oktober 2021 um 17:47:34 UTC+2:
> > On Sunday, 17 October 2021 at 09:57:03 UTC-3, WM wrote:
> > [...]
> > > Each definable natural number has infinitely many successors which are also natural numbers. So the infinite set |N of natural numbers is complete.
> > Correct.
> > > Each infinite endsegment has not infinitely many succeeding infinite endsegments because the infinite contents cannot be used as indexes simultaneously.
> > This nonsense seems to stem again from your utter inability to respect the order of quantifiers.
> This simple fact cannot be remedied by quantifier tricks. Infinite endsegments have an infinite intersection and there are not aleph_0 infinite endsegments. There are not aleph_0 natural numbers having alweh_0 successors.
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>
> Regards, WM
incorrect, ALL natural numbers have aleph_0 successors you imbecile.

zelos...@gmail.com schrieb am Montag, 18. Oktober 2021 um 08:28:25 UTC+2:

> ALL natural numbers have aleph_0 successors

Here is the refutation: |ℕ \ {1, 2, 3, ...}| = 0 .

Regards, WM

Gus Gassmann schrieb am Sonntag, 17. Oktober 2021 um 22:18:16 UTC+2:
> On Sunday, 17 October 2021 at 15:51:47 UTC-3, WM wrote:

> > This simple fact cannot be remedied by quantifier tricks. Infinite endsegments have an infinite intersection and there are not aleph_0 infinite endsegments. There are not aleph_0 natural numbers having aleph_0 successors.
> Of course there are! *EVERY* one of the aleph_0 natural numbers has aleph_0 successors.

That would require two consecutive sets in omega. Impossible.

> > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> Finite intersections of end segments (as you also wrote!) have cardinality aleph_0.

Yes.

> To extend that to infinite intersections you are switching quantifiers.

No, I simply apply mathematics. It says truly

|ℕ \ {1, 2, 3, ...}| = 0 .

Regards, WM

On Monday, 18 October 2021 at 06:23:05 UTC-3, WM wrote:
> zelos...@gmail.com schrieb am Montag, 18. Oktober 2021 um 08:28:25 UTC+2:
>
> > ALL natural numbers have aleph_0 successors
> Here is the refutation: |ℕ \ {1, 2, 3, ...}| = 0 .

You're not even trying any more. To call this statement a "refutation" and think that one can get away with it shows a level of insanity that is truly astounding!

On Monday, 18 October 2021 at 06:31:25 UTC-3, WM wrote:
> Gus Gassmann schrieb am Sonntag, 17. Oktober 2021 um 22:18:16 UTC+2:
> > On Sunday, 17 October 2021 at 15:51:47 UTC-3, WM wrote:
> > > This simple fact cannot be remedied by quantifier tricks. Infinite endsegments have an infinite intersection and there are not aleph_0 infinite endsegments. There are not aleph_0 natural numbers having aleph_0 successors.

Yes, there are. Every end segment E(n) has the form {n+0, n+1, n+2, n+3, n+4, ...}, which even in your warped mind I think you would accept as being in bijection with the set {{0, 1, 2, 3, 4, ...}. It therefore has cardinality aleph_0.

> > Of course there are! *EVERY* one of the aleph_0 natural numbers has aleph_0 successors.
> That would require two consecutive sets in omega. Impossible.

Bullshit. It requires -- on your part -- an illegal quantifier switch, but that is par for the course.

> > > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> > Finite intersections of end segments (as you also wrote!) have cardinality aleph_0.
> Yes.
> > To extend that to infinite intersections you are switching quantifiers.
> No, I simply apply mathematics. It says truly
> |ℕ \ {1, 2, 3, ...}| = 0 .

You apply bullshit rules that you cannot elucidate properly, you mumble, you equivocate, you toss in the ever-present quantifier switch, but it is crystal-clear that there is one thing you do not do: mathematics. Your mind is fried; I don't think you were ever able to do serious mathematics, and these days it is clearly impossible for you to formulate one single coherent thought.

Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 12:34:21 UTC+2:
> On Monday, 18 October 2021 at 06:31:25 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Sonntag, 17. Oktober 2021 um 22:18:16 UTC+2:

> > > > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> > > Finite intersections of end segments (as you also wrote!) have cardinality aleph_0.
> > Yes.
> > > To extend that to infinite intersections you are switching quantifiers.
> > No, I simply apply mathematics. It says truly
> > |ℕ \ {1, 2, 3, ...}| = 0 .
> You apply bullshit rules that you cannot elucidate properly

What is wrong with |ℕ \ {1, 2, 3, ...}| = 0 ?
What is wrong with ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo ?
What is to elucidate here?

you toss in the ever-present quantifier switch

What is wrong above?

Regards, WM

WM was thinking very hard :
> Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 12:34:21 UTC+2:
>> On Monday, 18 October 2021 at 06:31:25 UTC-3, WM wrote:
>>> Gus Gassmann schrieb am Sonntag, 17. Oktober 2021 um 22:18:16 UTC+2:
>
>>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>>>> Finite intersections of end segments (as you also wrote!) have cardinality
>>>> aleph_0. Yes. To extend that to infinite intersections you are switching
>>>> quantifiers. No, I simply apply mathematics. It says truly ℕ \ {1, 2, 3,
>>>> ...}| = 0 .
>> You apply bullshit rules that you cannot elucidate properly
>
> What is wrong with |ℕ \ {1, 2, 3, ...}| = 0 ?

Since you 'subtract' the complement or endsegment of the zero element,
the zero gets 'left behind' as you've taken a proper subset the
cardinality of the new set is one, not zero.

There is no zero in the set of naturals you adhere to. The cardinality
of the emptyset is undefined in your arena. Why not use non-negative
integers like the rest of the world does?

On 10/17/2021 5:27 AM, 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.
> (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
>

duh. Generalize. There are k endsegments that have k as an element. Done.

On 10/18/2021 5:28 AM, Gus Gassmann wrote:
> On Monday, 18 October 2021 at 06:23:05 UTC-3, WM wrote:
>> zelos...@gmail.com schrieb am Montag, 18. Oktober 2021 um 08:28:25 UTC+2:
>>
>>> ALL natural numbers have aleph_0 successors
>> Here is the refutation: |ℕ \ {1, 2, 3, ...}| = 0 .
>
> You're not even trying any more. To call this statement a "refutation" and think that one can get away with it shows a level of insanity that is truly astounding!
>

gad, WMs extra LAMO lately, perhaps he should take his marble and stay home

On Sunday, October 17, 2021 at 12:27:40 PM UTC+2, WM wrote:

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

I guess this should read (after all {n, n+1, n+2, ...} is a set, not "natnumbers"):

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

Well, actually, there are no natural numbers in {n, n+1, n+2, ...}.

Remember, you claimed that "n isn't a natural number", als well as "n+1 is not a natural number too."

I guess that will hold for all elements in {n, n+1, n+2, ...}. Hence there are no natural numbers in {n, n+1, n+2, ...}.

On Monday, 18 October 2021 at 10:48:45 UTC-3, WM wrote:
[...]
> What is wrong with |ℕ \ {1, 2, 3, ...}| = 0 ?
> What is wrong with ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo ?
> What is to elucidate here?

[>] > you toss in the ever-present quantifier switch

That was my remark. Please respect level of indentation! (Fixed; this time)

> What is wrong above?
Everything. You have no clue, as usual. First, ℕ = ℕ_def. Second, you switch quantifiers, as usual. And third, you don't know, or understand, or care, that your argument is hare-brained.

Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 17:10:55 UTC+2:
> On Monday, 18 October 2021 at 10:48:45 UTC-3, WM wrote:
> [...]
> > What is wrong with |ℕ \ {1, 2, 3, ...}| = 0 ?
> > What is wrong with ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo ?
> > What is to elucidate here?
> > What is wrong above?
> Everything. First, ℕ = ℕ_def.
Obviously not. But even if so, this would not be a mistake.

> Second, you switch quantifiers
What do you mean by this? Please be more specific.

> And third, you don't know, or understand, or care, that your argument is hare-brained.
No mathematical argument.

Regards, WM

On Monday, 18 October 2021 at 12:20:29 UTC-3, WM wrote:
> Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 17:10:55 UTC+2:
> > On Monday, 18 October 2021 at 10:48:45 UTC-3, WM wrote:
> > [...]
> > > What is wrong with |ℕ \ {1, 2, 3, ...}| = 0 ?
> > > What is wrong with ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo ?
> > > What is to elucidate here?
> > > What is wrong above?
> > Everything. First, ℕ = ℕ_def.
> Obviously not. But even if so, this would not be a mistake.
>
> > Second, you switch quantifiers
> What do you mean by this? Please be more specific.

Look. If you cannot see the difference between "for every n in find the set difference N \ {1, 2, 3, ..., n}" and "find the set difference N \ *ALL* sets {1, 2, 3, ..., n}" then you are truly beyond hope. And if you do know the difference and are trying to yank my chain, then piss off. Either way, there is no point to discuss this with you any further. And yes, your burbling *is* hare-brained.

Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 17:41:08 UTC+2:

> If you cannot see the difference between "for every n in find the set difference N \ {1, 2, 3, ..., n}" and "find the set difference N \ *ALL* sets {1, 2, 3, ..., n}" then you are truly beyond hope.

I see the difference. It is very clear. If all FISONs {1, 2, 3, ..., n} exhaust ℕ, then there must be the set ℕ among them, because all FISONs differ from each other, and all FISONs smaller than ℕ will leave a remainder and therefore will not exhaust ℕ. Only a brainless fool could believe that all FISONs smaller than ℕ will exhaust ℕ. In particular all finite FISONs will be applied completely in vain.

But since there is no FISON ℕ,
ℕ \ *ALL* sets {1, 2, 3, ..., n}
is not empty and not the same as
ℕ \ {1, 2, 3, ...} = { }

Regards, WM

Greg Cunt schrieb am Montag, 18. Oktober 2021 um 17:00:46 UTC+2:
> On Sunday, October 17, 2021 at 12:27:40 PM UTC+2, WM wrote:
>
> > (B) There are not more than n endsegments containing the natnumbers {n, n+1, n+2, ...}.
> I guess this should read (after all {n, n+1, n+2, ...} is a set, not "natnumbers"):
>
> | (B) There are not more than n endsegments containing the natnumbers *in* {n, n+1, n+2, ...}.

Yes.
>
> Well, actually, there are no natural numbers in {n, n+1, n+2, ...}.
>
> Remember, you claimed that "n isn't a natural number", als well as "n+1 is not a natural number too."

Thank you for your attention. But I told you already that this phrase "the natural number n has property X" is merely an abbreviation for "every natural number which can be inserted in place of n has property X". For instance if you insert 4711 in place of n in E(n), then you must also insert 4711 in place of n in {n, n+1, n+2, ...}
>
Regards, WM

Dan Christensen schrieb am Sonntag, 17. Oktober 2021 um 22:43:25 UTC+2:

> E(1) = {1, 2, 3, 4, ... }
> E(2) = {2, 3, 4, 5, ... }
> E(3) = {3, 4, 5, 6, ... }
> ...
> E(n) = {n, n+1, n+2, ... }
> ...
>
> Contrary to your outrageous claim here, each of these infinitely many end-segments has in it infinitely many elements.

There are only potentially infinitely many endsegments. If the set of infinite endsegments had the cardinality aleph_0, then all natnumbers were used as indices and none would remain as contents.

Regards, WM

On Monday, 18 October 2021 at 12:59:36 UTC-3, WM wrote:
> Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 17:41:08 UTC+2:
>
> > If you cannot see the difference between "for every n in find the set difference N \ {1, 2, 3, ..., n}" and "find the set difference N \ *ALL* sets {1, 2, 3, ..., n}" then you are truly beyond hope.
> I see the difference.

You have no clue. I have removed your utter nonsense. You cannot fathom that there can be infinitely many finite objects, and as long as you hide behind that mental defect, you will *NEVER* get anywhere. In particular, your insistence that N =/= union (n in N: {n}) is breathtaking in its stupidity.

Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 18:26:38 UTC+2:
> On Monday, 18 October 2021 at 12:59:36 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 17:41:08 UTC+2:
> >
> > > If you cannot see the difference between "for every n in find the set difference N \ {1, 2, 3, ..., n}" and "find the set difference N \ *ALL* sets {1, 2, 3, ..., n}" then you are truly beyond hope.
> > I see the difference.
> You cannot fathom that there can be infinitely many finite objects

Why should I accept nonsense? I can prove that there are not infinitely many FISONs because of the simple pigeon hole principle which holds for all sets the elements of which must be distinguished - irrespective of whether the sets are finite ot infinite.

> In particular, your insistence that N =/= union (n in N: {n}) is breathtaking in its stupidity.

I did not say so. So the stupidity is not mine. Of course ℕ = {1, 2, 3, ...} is the union of the singletons of natural numbers. But most of them are dark. Therefore ℕ is not the union of FISONs. On the contrary, for all FISONs we have ℕ \ {1, 2, 3, ..., n} is an infinite endsegment.

Regards, WM

On Monday, 18 October 2021 at 13:38:53 UTC-3, WM wrote:
> Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 18:26:38 UTC+2:
> > On Monday, 18 October 2021 at 12:59:36 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 17:41:08 UTC+2:
> > >
> > > > If you cannot see the difference between "for every n in find the set difference N \ {1, 2, 3, ..., n}" and "find the set difference N \ *ALL* sets {1, 2, 3, ..., n}" then you are truly beyond hope.
> > > I see the difference.
> > You cannot fathom that there can be infinitely many finite objects
> Why should I accept nonsense? I can prove that there are not infinitely many FISONs because of the simple pigeon hole principle which holds for all sets the elements of which must be distinguished - irrespective of whether the sets are finite ot infinite.

There you go again with your idiocies. The pigeonhole principle does *NOT* apply to infinite sets. You thinking it does only shows that you have no clue. EOD.

WM formulated the question :
> Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 18:26:38 UTC+2:
>> On Monday, 18 October 2021 at 12:59:36 UTC-3, WM wrote:
>>> Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 17:41:08 UTC+2:
>>>
>>>> If you cannot see the difference between "for every n in find the set
>>>> difference N \ {1, 2, 3, ..., n}" and "find the set difference N \ *ALL*
>>>> sets {1, 2, 3, ..., n}" then you are truly beyond hope.
>>> I see the difference.
>> You cannot fathom that there can be infinitely many finite objects
>
> Why should I accept nonsense? I can prove that there are not infinitely many
> FISONs because of the simple pigeon hole principle which holds for all sets
> the elements of which must be distinguished - irrespective of whether the
> sets are finite ot infinite.

A sign outside the Hilbert says "No Double Occupancy" so you're out of
luck there.

On Monday, October 18, 2021 at 6:08:01 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Montag, 18. Oktober 2021 um 17:00:46 UTC+2:
> >
> > Remember, you claimed that "n isn't a natural number", als well as "n+1 is not a natural number too."
> >
> I told you already <bla bla>

Look, you silly asshole, you explicitly claimed: "n isn't a natural number" and "n+1 is not a natural number too."

Hence the claim

"There are natnumbers in {n, n+1}"

implies a contradiction.

You see, IF "n isn't a natural number" and "n+1 is not a natural number too" THEN there are no natural numbers in {n, n+1}.

EOD.

Fuck off, Mückenheim!

Gus Gassmann schrieb am Montag, 18. Oktober 2021 um 19:01:36 UTC+2:

> There you go again with your idiocies. The pigeonhole principle does *NOT* apply to infinite sets.

It applies to all FISONs which shall be distinguished, because otherwise it is not possible to distinguish them. Only by different number of natnumbers or symbols like
o oo
ooo
oooo
....
it is possible to distinguish FISONs. If it does not apply, then there is no distinction possible. But even in infinite sets it is necessary to distinguish elements if any useful result shall be obtained. What fool has taught you, without considering this simple fact, that Dirichlet's principle does not apply?

Regards, WM