On Thursday, 23 September 2021 at 16:42:24 UTC-3, WM wrote:
> [...] All natural numbers together are not followed by any natural number.

Typical Muckenheim gobbledygook. You couldn't give a coherent description of "[a]ll natural numbers together" if your life depended on it. You are now far too dense, stupid, dishonest and demented to do any kind of mathematics.

On Thursday, 23 September 2021 at 16:59:32 UTC-3, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 23. September 2021 um 15:36:27 UTC+2:
> > "Infinite" means that there is *no end* to the successor operation.
> Nevertheless there is an end to the natural numbers, namely omega. Like the zero ends the sequence of unit fractions.

Excellent observation! But just as there is no smallest reciprocal larger than 0 there is no largest natural number smaller than omega. (And vice versa.) There is a reason why omega is called a limit ordinal, you know... (Not that you could ever come close to understanding that reason.)

> > Every natural number has an end segment that contains alpha_0 direct and indirect successors.
> There are definable natural numbers each of which is followed by infinitely many natural numbers. All natural numbers together are not followed by any natural number. Can you see a difference between "definable natural numbers" and "all natural numbers"? What causes this difference?
Inability to express yourself unambiguously, quantifier dyslexia and progressive dementia, to name three obvious afflictions of yours.

On 9/23/2021 3:38 PM, WM wrote:

> Simplified it is this:
> There are definable natural numbers
> each of which is followed by infinitely many natural numbers.

forall j, exists k, k > j

> All natural numbers are not followed by any natural number.

not exists j, forall k, k =< j

> Can you see a difference between "definable natural numbers"
> and "all natural numbers"?

That's not a difference. They say the same thing.

(k > j) xor
(k =< j)

forall k, k =< j xor
exists k, not(k =< j)

forall k, k =< j xor
exists k, k > j

forall j, exists k, k > j xor
exists j, not(exists k, k > j)

forall j, exists k, k > j xor
exists j, forall k, k =< j

forall j, exists k, k > j iff
not exists j, forall k, k =< j

On 9/23/2021 2:56 PM, WM wrote:
> FromTheRafters schrieb am Donnerstag, 23. September 2021 um 18:09:53 UTC+2:
>
>> "Followed by" makes even less sense here; why would natural numbers
>> "follow" an infinite set of endsegments?
>>> namely an infinite
>>> subset of the contents of all endsegments.
>> ???
>
> Every index n of endsegment E(n) is followed by the infinite contents of the endsegment n+1, n+2, n+3, ... .

? Just state each endsegment is an infinite set.

> This is true for every endsegment and hence for all endsegments: There are infinitely many natural numbers required which are larger than all indices of endsegments.

false. Just state each endsegment is an infinite set.

>
> Regards, WM
>

On 9/23/2021 4:08 PM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 23. September 2021 um 19:32:17 UTC+2:
>> On 9/23/2021 9:23 AM, WM wrote:
>>> Jim Burns schrieb
>>> am Mittwoch, 22. September 2021 um 21:41:54 UTC+2:

>>>> If you change the definition of "natural number" to include
>>>> "must have its UPF known", then you are talking about
>>>> something else, something other than natural numbers.
>>>
>>> My definition:
>>> Any natnumbers must have its UPF known in principle or knowable.
>>
>> What have you defined?
>> Is it a thing that you intended to define?
>
> No it is a thing that every mathematician should have learnt
> in the first semester and always remember and know.

You don't mean "definition".
You mean "theorem".

Definition marks out the boundaries of the topic.
Changing the definition changes the topic.

A theorem extends what we know about the topic
without changing the topic.
This is what makes mathematics so impressive.
All these theorems, vast knowledge, about _the same things_
as were _defined_ very simply.

Adding to or deleting from the _definition_ breaks that.
The point is that the theorems are about _the same things_

We can define a right triangle.
It's a plane figure with three sides and a right angle.

<scribble scribble scribble>
It happens that
"It's a plane figure with three sides and a right angle"
tells us enough that, if we know that, we _also_ know that
the square of its longest side is equal to the sum of
the squares of the two remaining sides.

*What we do not want to do* is to _re-define_ our topic
to right triangles for which the square of its longest side
is equal to the sum of the squares of the two remaining sides.

Why would we bother with a claim like that?
What would it tell us about ABC which is only defined to be
a right triangle?

It is a _theorem_ not a _definition_ that
a natural number >= 2 has a unique prime factorization.

WM wrote on 9/23/2021 :
> Serg io schrieb am Donnerstag, 23. September 2021 um 16:23:39 UTC+2:
>> On 9/23/2021 7:56 AM, WM wrote:
>
>>> Why does this contents not appear in the intersection?
>> there is no element common to all endsegments.
>
> If two non-empty sets, A and B, have an empty intersection, then

They are disjoint sets.

> A has no element of B and B has no element of A.

Wow! Both at the same time? Brilliant!

> Then A and B are not inclusion monotonic.

I prefer my disjoint sets to be so.

WM has brought this to us :
> Gus Gassmann schrieb am Donnerstag, 23. September 2021 um 15:36:27 UTC+2:
>> "Infinite" means that there is *no end* to the successor operation.
>
> Nevertheless there is an end to the natural numbers, namely omega. Like the
> zero ends the sequence of unit fractions.

Stop lying.

>> Every natural number has an end segment that contains alpha_0 direct and
>> indirect successors.
>
> There are definable natural numbers each of which is followed by infinitely
> many natural numbers.

Each natural number is both definable and is "followed" by infinitely
more definable natural nunbers.

> All natural numbers together are not followed by any
> natural number. Can you see a difference between "definable natural numbers"
> and "all natural numbers"? What causes this difference?

Your unfounded implication that the 'defined' ara a proper subset of
the 'just plain old' naturals. they are not, you just have a half-baked
notion that you want them to be.

After serious thinking WM wrote :
> Jim Burns schrieb am Donnerstag, 23. September 2021 um 19:32:17 UTC+2:
>> On 9/23/2021 9:23 AM, WM wrote:
>>> Jim Burns schrieb
>>> am Mittwoch, 22. September 2021 um 21:41:54 UTC+2:
>>>> If you change the definition of "natural number" to include
>>>> "must have its UPF known", then you are talking about
>>>> something else, something other than natural numbers.
>>>
>>> My definition:
>>> Any natnumbers must have its UPF known in principle or knowable.
>> What have you defined?
>> Is it a thing that you intended to define?
>
> No it is a thing that every mathematician should have learnt in the first
> semester and always remember and know.
>
>> I know that you (WM) don't know because you *changed the topic*
>> from one of 0,1,2,3,4,5,...
>> to some thing with a UPF that we can know.
>
> I did not change. These things 1,2,3,4,5,... are just those which have unique
> prime decompositions.

Four is the only composite number listed explicitly, how do you
decompose that which is not composite?

On 9/23/2021 2:38 PM, WM wrote:
> zelos...@gmail.com schrieb am Donnerstag, 23. September 2021 um 15:10:49 UTC+2:
>> torsdag 23 september 2021 kl. 14:58:52 UTC+2 skrev WM:
>>>
>>> Irrelevant sind E(n) has infinitely many successors. Consider the whole set of endsegments. What remains as the contents of endsegments?
>>>
>> What are you talking about?
>
> Simplified it is this: There are definable natural numbers each of which is followed by infinitely many natural numbers. All natural numbers are not followed by any natural number. Can you see a difference between "definable natural numbers" and "all natural numbers"?
>
> Regards, WM
>

let A = natural numbers

substitute;

"Simplified it is this: There are definable A each of which is followed by infinitely many A. All A are not followed by any A. Can you see a difference
between "definable A" and "all A"?"

didn't help at all...

On 9/23/2021 2:49 PM, WM wrote:
> Serg io schrieb am Donnerstag, 23. September 2021 um 16:23:39 UTC+2:
>> On 9/23/2021 7:56 AM, WM wrote:
>
>>> Why does this contents not appear in the intersection?
>> there is no element common to all endsegments.
>
> If two non-empty sets, A and B, have an empty intersection, then A has no element of B and B has no element of A. Then A and B are not inclusion monotonic.

A and B are distraction.

There is no element common to all endsegments.

>>>
>>> The contents of the endsegments succeeds their indices.
>> Wrong.
>
> By definition: E(n) = (n, n+1, n+2, ...)

still wrong, that is equal, not succeeds

>
> Regards, WM
>

On Thursday, September 23, 2021 at 2:57:01 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Mittwoch, 22. September 2021 um 21:06:25 UTC+2:

Look, you silly asshole:

> > If E is an endsegment, then min(E) is its index and it contains all natural numbers >= min(E) (and nothing else).
> >
> > This holds for each and every endsegment, i.e. for ALL endsegments.
> >
> No.

Yes.

Proof: Let be E is an endsegment. Then there is a unique index i e IN such that E = E(i). Let n e IN be this index. Hence E = E(n), i.e. E = {m e IN : m >= n}. Since min({m e IN : m >= n}) = n we get that min(E) = n. Hence min(E) is E's index and E contains all natural numbers >= min(E) (and nothing else). Since E was an arbitrary endsegment, this holds for each and every endsegment, i.e. for ALL endsegments. qed

On Thursday, September 23, 2021 at 9:49:25 PM UTC+2, WM wrote:

> By definition: E(n) = (n, n+1, n+2, ...)

Nope. E(n) = {n, n+1, n+2, ...}

That's a BIG difference in the context set theory, idiot.

On Thursday, September 23, 2021 at 10:08:15 PM UTC+2, WM wrote:

Let n e IN \ {1}.

> These things 1,2,3,4,5,... are just those which have unique prime decompositions.

Certainly 1 is not one of "these things". Btw. these things are the natural numbers > 1. n is one of them.

On Thursday, September 23, 2021 at 10:08:15 PM UTC+2, WM wrote:

Let n e IN \ {1}.

> These things 1,2,3,4,5,... are just those which have [a] unique prime [factorization].

Certainly 1 is not one of "these things". Btw. these things are the natural numbers > 1. n is one of them.

tisdag 21 september 2021 kl. 15:23:16 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Dienstag, 21. September 2021 um 14:02:08 UTC+2:
> > fredag 17 september 2021 kl. 21:44:43 UTC+2 skrev WM:
> > > Greg Cunt schrieb am Donnerstag, 16. September 2021 um 18:04:33 UTC+2:
> > >
> > > > No, n is a natural number, not a "place holder".
> > > What is its prime decomposition? How many factors has it?
> > >
> > It has as many as it does.
> It has none that anybody could find. It is matheology like all your ZF-nonsense.
>
> Regards, WM
It has as many as it does. That is how many it has.

Stop your idiocy for fuck sake.

torsdag 23 september 2021 kl. 15:23:55 UTC+2 skrev WM:
> Jim Burns schrieb am Mittwoch, 22. September 2021 um 21:41:54 UTC+2:
> > On 9/22/2021 3:14 PM, WM wrote:
> > > Jim Burns schrieb
> > > am Dienstag, 21. September 2021 um 19:42:24 UTC+2:
> > >> On 9/21/2021 6:57 AM, WM wrote:
> >
> > >>> Every natural number has a unique prime decomposition.
> > >>> n has none.
> > >>
> > >> We know some things about n. We don't know other things.
> > >
> > > Every natural number less than 1 billion has a
> > > known prime decomposition. Is n larger?
> > It's possible that n is larger.
> > Whether larger or smaller, it has a UPF.
> Whether larger or smaller, the UPF of a natural number is known in principle. A UPF of n is not known in principle.
> > >> We know n has a unique prime factorization.
> > >> We don't know what its UPF is.
> > >
> > > For a natural number you would know what it is.
> > I know other things about n, but not that.
> Therefore n is something else but not a natural number.
> > > You don't.
> > > Therefore n is not a natural number.
> > n can be counted to in principle.
> No, not even in principle. You are extremely deranged.
> > If you change the definition of "natural number" to include
> > "must have its UPF known", then you are talking about
> > something else, something other than natural numbers.
> My definition: Any natnumbers must have its UPF known in principle or knowable.
>
> Regards, WM

if I say n is a natural number, it is a natural number. Why are these basic things so fucking difficult for you!?

torsdag 23 september 2021 kl. 21:38:20 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 23. September 2021 um 15:10:49 UTC+2:
> > torsdag 23 september 2021 kl. 14:58:52 UTC+2 skrev WM:
> > >
> > > Irrelevant sind E(n) has infinitely many successors. Consider the whole set of endsegments. What remains as the contents of endsegments?
> > >
> > What are you talking about?
> Simplified it is this: There are definable natural numbers each of which is followed by infinitely many natural numbers. All natural numbers are not followed by any natural number. Can you see a difference between "definable natural numbers" and "all natural numbers"?
>
> Regards, WM

>There are definable natural numbers

Which by your definition is all natural numbers.

>each of which is followed by infinitely many natural numbers

Again, that is all natural numbers.

>All natural numbers are not followed by any natural number.

Incorrect, again, all natural numbers are followed by infinitely many more natural numbers.

>Can you see a difference between "definable natural numbers" and "all natural numbers"?

No because the "proeprty" you state that "definable" means apply to all natural numbers.

>All natural numbers together are not followed by any natural number.

You are talking about the set of natural numbers when you mean "All natural numbers together", which of course has no successor because the successor function is not defined for the set N, it is only defind on N

torsdag 23 september 2021 kl. 15:18:46 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 23. September 2021 um 06:43:45 UTC+2:
>
> > Very possible because that is the definition of natural numbers. There is always a successor to each and every one of them. Therefore each one is followed by infinitely many natural numbers
> The set of all of them is not followed by natural numbers. Therefore it must contain more than each one that is followed by infinitely many numbers. Or why do the numbers cease to have successors?
>
> Regards, WM
of course the set N is not followed by anything, it is a meaningless thing to ask because the successor function cannot take N in it as it is not defined for it!

The natural numbers IN the set N however ALWAYS have successors, there is no one where it stops you idiot!

zelos...@gmail.com schrieb am Freitag, 24. September 2021 um 07:08:11 UTC+2:
> torsdag 23 september 2021 kl. 15:23:55 UTC+2 skrev WM:
> > My definition: Any natnumbers must have its UPF known in principle or knowable.
> >
> if I say n is a natural number, it is a natural number.

It is not difficult to prove you wrong. You claim that every natural number has a decimal representation. Name the decimal representation of n. Fail.

Regards, WM

After serious thinking WM wrote :
> zelos...@gmail.com schrieb am Freitag, 24. September 2021 um 07:08:11 UTC+2:
>> torsdag 23 september 2021 kl. 15:23:55 UTC+2 skrev WM:
>
>>> My definition: Any natnumbers must have its UPF known in principle or
>>> knowable.
>>>
>> if I say n is a natural number, it is a natural number.
>
> It is not difficult to prove you wrong. You claim that every natural number
> has a decimal representation. Name the decimal representation of n. Fail.

n.000...

On Friday, September 24, 2021 at 3:48:09 PM UTC+2, WM wrote:
> zelos...@gmail.com schrieb am Freitag, 24. September 2021 um 07:08:11 UTC+2:
> >
> > if I say n is a natural number,

by stating "Let n be a natural number."

> > it is a natural number.
> >
> It is not difficult to prove you wrong. You claim that every natural number has a decimal representation. Name the decimal representation of n. Fail..

Holy shit! There's a big difference in math between proving the existence of some mathematical object and being able to "specify" ("name" it). MEINE FRESSE, SIND SIE BLÖDE!

For example, there's the Schnirelmann constant, which certainly is a natural number. But it's not KNOWN if it is 3, 4 or 5 dumbo.

See: https://primes.utm.edu/glossary/xpage/SchnirelmannsConstant.html

On 9/24/2021 8:48 AM, WM wrote:
> zelos...@gmail.com schrieb am Freitag, 24. September 2021 um 07:08:11 UTC+2:
>> torsdag 23 september 2021 kl. 15:23:55 UTC+2 skrev WM:
>
>>> My definition: Any natnumbers must have its UPF known in principle or knowable.
>>>
>> if I say n is a natural number, it is a natural number.
>
> It is not difficult to prove you wrong. You claim that every natural number has a decimal representation. Name the decimal representation of n. Fail.
>
> Regards, WM
>

(observers and lurkers: we now intellectually descend below the age of 2 years old, when most of us have already been taught to count on on our fingers.
WM denies fingers as natural numbers, as they have no decimal representation. IAW WM, you must physically mark each finger with its number by pen. OR
else it is NOT an number, and your digit counting FAILS. Some call this baby math, or counting rocks math, or counting sheeps, or troll math. )

Jim Burns schrieb am Donnerstag, 23. September 2021 um 23:04:42 UTC+2:

> *What we do not want to do* is to _re-define_ our topic

My definition is the valid one.

> It is a _theorem_ not a _definition_ that
> a natural number >= 2 has a unique prime factorization.

That is what we use for the definition of natural numbers if some fools claim that the expression "natural number" is a natural number.

Regards, WM

FromTheRafters schrieb am Freitag, 24. September 2021 um 00:13:58 UTC+2:
> After serious thinking WM wrote :

> > I did not change. These things 1,2,3,4,5,... are just those which have unique
> > prime decompositions.
> Four is the only composite number listed explicitly, how do you
> decompose that which is not composite?

How do you raise a number to the power of 1?

Regards, WM

Greg Cunt schrieb am Freitag, 24. September 2021 um 05:27:09 UTC+2:
> On Thursday, September 23, 2021 at 10:08:15 PM UTC+2, WM wrote:
>
> Let n e IN \ {1}.
>
> > These things 1,2,3,4,5,... are just those which have [a] unique prime [factorization].
>
> Certainly 1 is not one of "these things". Btw. these things are the natural numbers > 1. n is one of them.

They have prime factorizations that can be known. n has not. Do you claim that "natural number" is a natural number too?

Regards, WM