torsdag 30 september 2021 kl. 20:56:13 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Dienstag, 28. September 2021 um 06:56:14 UTC+2:
> > måndag 27 september 2021 kl. 22:14:52 UTC+2 skrev WM:
>
> > > > Why would they need to be disjoint?
> > > Otherwise the intersection is not empty. Infinite and not disjoint gives an infinite intersection.
> > >
> > Only if there are elements present in all of the sets. If all sets have different elements the intersection can be empty.
> >
> > {a,b} {b,c} {a,c}
> > Intersection of all is empty, but intersetion of any 2 is not empty, yet none is disjoint.
> Just this is impossible for endsegments. Inclusion monotony!
>
> Regards, WM

Nope, that only shows that any FINITE intersection is non-empty, not that the intersection of ALL is emtpy

torsdag 30 september 2021 kl. 20:12:13 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Dienstag, 28. September 2021 um 06:56:41 UTC+2:
> > måndag 27 september 2021 kl. 21:34:57 UTC+2 skrev WM:
>
> > > If you claim that n has a decimal representation and this is doubted, then you have to prove it by giving it.
> > >
> > Nope, I only have to prove all natural numbers have one and that is done already.
> Irrelevant since it is just doubted that n is a natural number. If a honest mathematician is asked to show n's decimal representation, then he will do so, if he can.
>
> Regards, WM

We do not need to know decimal expansions to know there is one and to do arguments about them and you do not doubt n is a natural number. If I say it is from the set of natural number it is by definition a natural number.

torsdag 30 september 2021 kl. 20:33:05 UTC+2 skrev WM:
> Greg Cunt schrieb am Donnerstag, 30. September 2021 um 20:19:56 UTC+2:
> > On Thursday, September 30, 2021 at 8:12:13 PM UTC+2, WM wrote:
> >
> > Let n be a natural number. [Or: Let n e IN.]
> > > it is just doubted that n is a natural number.
> > BY WHOM,
> Irrelevant.
> If n is a natural number, then its decimal representation can be given.
> Give it!
> > Hint: BY DEFINITION (see above) n is an element in the set of all natural numbers.
> No, you misunderstand. "n ∈ ℕ" is simply an abbreviation for the sentence: "Consider any natural number". By the way, you can also use k ∈ ℕ or m ∈ ℕ. Are they in trichotomy? k < m < n? Or m < k, n?
> > Does this set contain anything else except natural numbers?
> No, therefore it does not contain k, m, n.
>
> Regards, WM

are you insane or just mentally retarded?

torsdag 30 september 2021 kl. 20:23:17 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Dienstag, 28. September 2021 um 06:54:42 UTC+2:
> > måndag 27 september 2021 kl. 22:04:04 UTC+2 skrev WM:
>
> > The union of all finite subsets of N is N itself.
> Look: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>
> In order omega, two consecutive infinite sets are impossible. ℵo subsequent natural numbers are always remaining. They cannot be addressed. They are dark.
>
> Regards, WM

Empty assertions that is unprovable.

There is nothing you call "dark", you imagine it but cannot prove that they are distinct from natural numbers

All natural numbers have infinitely many natural numbers after, all natural numbers are in a FISON, all natural numbers are excluded in at least 1 endsegment.

On Friday, 1 October 2021 at 02:06:23 UTC-3, zelos...@gmail.com wrote:
> torsdag 30 september 2021 kl. 20:33:05 UTC+2 skrev WM:
> > Greg Cunt schrieb am Donnerstag, 30. September 2021 um 20:19:56 UTC+2:
> > > On Thursday, September 30, 2021 at 8:12:13 PM UTC+2, WM wrote:
> > >
> > > Let n be a natural number. [Or: Let n e IN.]
> > > > it is just doubted that n is a natural number.
> > > BY WHOM,
> > Irrelevant.
> > If n is a natural number, then its decimal representation can be given.
> > Give it!
> > > Hint: BY DEFINITION (see above) n is an element in the set of all natural numbers.
> > No, you misunderstand. "n ∈ ℕ" is simply an abbreviation for the sentence: "Consider any natural number". By the way, you can also use k ∈ ℕ or m ∈ ℕ. Are they in trichotomy? k < m < n? Or m < k, n?
> > > Does this set contain anything else except natural numbers?
> > No, therefore it does not contain k, m, n.
> >
> > Regards, WM
> are you insane or just mentally retarded?

He clearly is demented, although it is not always clear whether this is Alzheimer's or the "mad scientist" variety (or both).

On 9/30/2021 1:27 PM, WM wrote:
> FromTheRafters schrieb am Dienstag, 28. September 2021 um 00:18:24 UTC+2:
>
>> You conveniently ignore the fact that even your ill fated stepwise
>> process essentially 'removes' each and every element from the
>> endsegments one-by-one leaving no initial element to feed into the
>> successor function and thus collapsing the inductive structure.
>
> No.
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

Wrong, there is no such thing as ℕ_def IAW your own daffynition.

correction:
∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

and so what ? that only says there are numbers after n in |ℕ

so trivial...

> ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo

Wrong there is no such thing as ℕ_def IAW your own daffynition.

removed "_def"

∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo

which is;

∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k)

|E(k)| = ℵo

so what? so trivial

On 9/30/2021 1:23 PM, WM wrote:
> zelos...@gmail.com schrieb am Dienstag, 28. September 2021 um 06:54:42 UTC+2:
>> måndag 27 september 2021 kl. 22:04:04 UTC+2 skrev WM:
>
>> The union of all finite subsets of N is N itself.
>
> Look: ∀n ∈ ℕ*_def*: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>
> In order omega, two consecutive infinite sets are impossible.

Wrong, google for it.

> ℵo subsequent natural numbers are always remaining.

trivial, you are just learning this ?

>They cannot be addressed.

wrong, you address each in your faulty equation above

>They are dark.

no, you in the dark about math.

>
> Regards, WM
>

On 9/27/2021 2:41 PM, WM wrote:
> Jim Burns schrieb am Sonntag, 26. September 2021 um 19:44:50 UTC+2:
>
>> I claim that I can refer to a natural number.
>> I can make claims about it (which refer to it)
>> which we know are true, even if we no one knows
>> which natural number I refer to.
>
> In that case use n.
>>
>> ( Other claims, we know are false. Still others could be
>> ( either true of false without more information.
>>
>> Starting from a claim for each natural number, we can derive
>> more claims true for each natural number.
>
> If n stands for each natural number it has no unique prime decomposition.

unique prime decomposition is red herring, a diversion.

> But every natural number has.

so you agree that every natural number has a fractional expression as well, which Cantor Enumeration lists

>
> Regards, WM
>

Gus Gassmann schrieb am Donnerstag, 30. September 2021 um 21:10:40 UTC+2:
> On Thursday, 30 September 2021 at 15:12:13 UTC-3, WM wrote:
> [...]
> > Irrelevant since it is just doubted that n is a natural number. If a honest mathematician is asked to show n's decimal representation, then he will do so, if he can.
> I also presume that you can't give the decimal representation of Graham's number. Nonetheless, it is trivial to prove that by induction it has one.

It could be givn in principle in ideal mathematics. That of n cannot be given because n is not a number.

Regards, WM

zelos...@gmail.com schrieb am Freitag, 1. Oktober 2021 um 07:06:23 UTC+2:
> torsdag 30 september 2021 kl. 20:33:05 UTC+2 skrev WM:
> > Greg Cunt schrieb am Donnerstag, 30. September 2021 um 20:19:56 UTC+2:
> > > On Thursday, September 30, 2021 at 8:12:13 PM UTC+2, WM wrote:
> > >
> > > Let n be a natural number. [Or: Let n e IN.]
> > > > it is just doubted that n is a natural number.
> > > BY WHOM,
> > Irrelevant.
> > If n is a natural number, then its decimal representation can be given.
> > Give it!
> > > Hint: BY DEFINITION (see above) n is an element in the set of all natural numbers.
> > No, you misunderstand. "n ∈ ℕ" is simply an abbreviation for the sentence: "Consider any natural number". By the way, you can also use k ∈ ℕ or m ∈ ℕ. Are they in trichotomy? k < m < n? Or m < k, n?
> > > Does this set contain anything else except natural numbers?
> > No, therefore it does not contain k, m, n.
> >
> > Regards, WM
> are you insane or just mentally retarded?

zelos...@gmail.com schrieb am Freitag, 1. Oktober 2021 um 07:04:41 UTC+2:
> torsdag 30 september 2021 kl. 20:12:13 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Dienstag, 28. September 2021 um 06:56:41 UTC+2:
> > > måndag 27 september 2021 kl. 21:34:57 UTC+2 skrev WM:
> >
> > > > If you claim that n has a decimal representation and this is doubted, then you have to prove it by giving it.
> > > >
> > > Nope, I only have to prove all natural numbers have one and that is done already.
> > Irrelevant since it is just doubted that n is a natural number. If a honest mathematician is asked to show n's decimal representation, then he will do so, if he can.

> We do not need to know decimal expansions to know there is one and to do arguments about them and you do not doubt n is a natural number. If I say it is from the set of natural number it is by definition a natural number.

It is not your competence to define what natural numbers are.
Please mark the place of n on the following scale:

....1...10...100...1000...10000...

Regards, WM

zelos...@gmail.com schrieb am Freitag, 1. Oktober 2021 um 07:06:23 UTC+2:
> torsdag 30 september 2021 kl. 20:33:05 UTC+2 skrev WM:
> > Greg Cunt schrieb am Donnerstag, 30. September 2021 um 20:19:56 UTC+2:
> > > On Thursday, September 30, 2021 at 8:12:13 PM UTC+2, WM wrote:
> > >
> > > Let n be a natural number. [Or: Let n e IN.]
> > > > it is just doubted that n is a natural number.
> > > BY WHOM,
> > Irrelevant.
> > If n is a natural number, then its decimal representation can be given.
> > Give it!
> > > Hint: BY DEFINITION (see above) n is an element in the set of all natural numbers.
> > No, you misunderstand. "n ∈ ℕ" is simply an abbreviation for the sentence: "Consider any natural number". By the way, you can also use k ∈ ℕ or m ∈ ℕ. Are they in trichotomy? k < m < n? Or m < k, n?
> > > Does this set contain anything else except natural numbers?
> > No, therefore it does not contain k, m, n.
> >
> are you insane or just mentally retarded?

Do I claim that n is a number?

Regards, WM

zelos...@gmail.com schrieb am Freitag, 1. Oktober 2021 um 07:03:30 UTC+2:
> torsdag 30 september 2021 kl. 20:56:13 UTC+2 skrev WM:
> > Just this is impossible for endsegments. Inclusion monotony!
> >
> Nope, that only shows that any FINITE intersection is non-empty, not that the intersection of ALL is emtpy

Is one endsegment among all that is not the last of a finite intersection?

Regards, WM

zelos...@gmail.com schrieb am Freitag, 1. Oktober 2021 um 07:02:54 UTC+2:
> torsdag 30 september 2021 kl. 20:54:04 UTC+2 skrev WM:
> > Gus Gassmann schrieb am Montag, 27. September 2021 um 22:52:52 UTC+2:
> > > On Monday, 27 September 2021 at 17:14:52 UTC-3, WM wrote:
ction because there is always at least 1 intersection not containing the element. So that is why it is empty. Why is this so fucking difficult for you?

Same holds for the endegments themselves. Every natural number is missing because there is at least one endsegment not containing it. What are the infintely many remaining in all endegments?

Regards, WM

Gus Gassmann schrieb am Donnerstag, 30. September 2021 um 21:17:46 UTC+2:
> On Thursday, 30 September 2021 at 15:50:44 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Montag, 27. September 2021 um 22:45:39 UTC+2:
> > > On Monday, 27 September 2021 at 17:12:55 UTC-3, WM wrote:
> >
> > > > The intersection over infinite endsegments is infinite.
> > > Often stated, never proved
> >
> > Proved by inclusion monotony.
> The only thing you prove by your insistent invocation of inclusion monotony is that you have no clue how it works with infinite intersections

Inclusion monotony works for all infinite endsegments, even for finite ones.
> > > Intersections over infinitely many end segments are empty:
> >
> > Why are the endsegments not empty?
> Because *PROVABLY* for every n in |N there is at least one set over which you intersect that does not contain it. This is really elementary stuff.'

Why are the endsegments *not* empty?

> > Why does their infinite contents differ from contents of their predecessors?
> Because *NO* natural number n can be contained in the end segment E(n+1).

In an infinite endsegment there are enough numbers remaining to yield an infinite intersection.

Regards, WM

Gus Gassmann schrieb am Donnerstag, 30. September 2021 um 21:04:33 UTC+2:
> On Thursday, 30 September 2021 at 15:27:33 UTC-3, WM wrote:
>
> > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> > ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo
> > See the universal quantifier?
>
> Yes. So?

So the collection of all FISONs is a potentially infinite set because it does not reach the first actually infinite cardinal number.

Regards, WM

FromTheRafters schrieb am Donnerstag, 30. September 2021 um 21:04:54 UTC+2:

> >> You conveniently ignore the fact that even your ill fated stepwise
> >> process essentially 'removes' each and every element from the
> >> endsegments

Why then is there no empty endsegment?

Regards, WM

On Friday, 1 October 2021 at 14:47:37 UTC-3, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 30. September 2021 um 21:10:40 UTC+2:
> > On Thursday, 30 September 2021 at 15:12:13 UTC-3, WM wrote:
> > [...]
> > > Irrelevant since it is just doubted that n is a natural number. If a honest mathematician is asked to show n's decimal representation, then he will do so, if he can.
> > I also presume that you can't give the decimal representation of Graham's number. Nonetheless, it is trivial to prove that by induction it has one.
> It could be givn in principle in ideal mathematics. That of n cannot be given because n is not a number.

Please endulge in self-gratification.

zelos...@gmail.com schrieb am Freitag, 1. Oktober 2021 um 14:21:00 UTC+2:

> All natural numbers have infinitely many natural numbers after,

So they are not an actuallyinfinite set. Note that two consecutive infinite sets of card aleph_0 are impossible in |N.

Regards, WM

On Friday, 1 October 2021 at 14:51:36 UTC-3, WM wrote:
> zelos...@gmail.com schrieb am Freitag, 1. Oktober 2021 um 07:06:23 UTC+2:
> > are you insane or just mentally retarded?
> Do I claim that n is a number?

I am not sure whether zelos would consider "Yes" to indicate the former and "No" the latter or vice versa, but I have said repeatedly that you are mad as a hatter. You are also deceitful and abusive to your students.

On Friday, 1 October 2021 at 14:58:10 UTC-3, WM wrote:
> zelos...@gmail.com schrieb am Freitag, 1. Oktober 2021 um 07:03:30 UTC+2:
> > torsdag 30 september 2021 kl. 20:56:13 UTC+2 skrev WM:
>
> > > Just this is impossible for endsegments. Inclusion monotony!
> > >
> > Nope, that only shows that any FINITE intersection is non-empty, not that the intersection of ALL is emtpy
> Is one endsegment among all that is not the last of a finite intersection?

You are delusional if you think the talk here is about *finite* intersections. I guess you are now truly down to your last marble.

On Friday, 1 October 2021 at 15:18:05 UTC-3, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 30. September 2021 um 21:04:33 UTC+2:
> > On Thursday, 30 September 2021 at 15:27:33 UTC-3, WM wrote:
> >
> > > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> > > ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo
> > > See the universal quantifier?
> >
> > Yes. So?
> So the collection of all FISONs is a potentially infinite set because it does not reach the first actually infinite cardinal number.

A set is not infinite because it "reaches" anything, you moron. A set is infinite because its cardinality is aleph_0. You are terminally stupid. I'll let that one slide because you already lost a marble today and are now down to your last one.

On Friday, 1 October 2021 at 15:22:53 UTC-3, WM wrote:
> zelos...@gmail.com schrieb am Freitag, 1. Oktober 2021 um 14:21:00 UTC+2:
>
> > All natural numbers have infinitely many natural numbers after,
> So they are not an actuallyinfinite set. Note that two consecutive infinite sets of card aleph_0 are impossible in |N.

Boohoo. And still not even the hint of a scintilla of evidence that your brain is still functioning.

Gus Gassmann schrieb am Freitag, 1. Oktober 2021 um 20:32:34 UTC+2:
> On Friday, 1 October 2021 at 15:18:05 UTC-3, WM wrote:
> > Gus Gassmann schrieb am Donnerstag, 30. September 2021 um 21:04:33 UTC+2:
> > > On Thursday, 30 September 2021 at 15:27:33 UTC-3, WM wrote:
> > >
> > > > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> > > > ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵo
> > > > See the universal quantifier?
> > >
> > > Yes. So?
> > So the collection of all FISONs is a potentially infinite set because it does not reach the first actually infinite cardinal number.
> A set is not infinite because it "reaches" anything

The smallest actually infinite set reaches cardinality ℵo. A potentially infinite set will never reach it.

> A set is infinite because its cardinality is aleph_0.

Yes, but two consecutive cardinalities ℵo in order omega are impossible.

Regards, WM

WM presented the following explanation :
> FromTheRafters schrieb am Donnerstag, 30. September 2021 um 21:04:54 UTC+2:
>
>>>> You conveniently ignore the fact that even your ill fated stepwise
>>>> process essentially 'removes' each and every element from the
>>>> endsegments
>
> Why then is there no empty endsegment?

Because there is no last FISON.