onsdag 22 september 2021 kl. 17:36:16 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Dienstag, 21. September 2021 um 14:02:41 UTC+2:
> > tisdag 21 september 2021 kl. 13:03:30 UTC+2 skrev WM:
> > > zelos...@gmail.com schrieb am Dienstag, 21. September 2021 um 07:11:54 UTC+2:
> > > > fredag 17 september 2021 kl. 22:10:38 UTC+2 skrev WM:
> > >
> > > > > All infinite endsegments have an infinite intersection with each other. This is proved by inclusion monotony! As long as an endsegment contains elements, it contains them with all preceding endsegments.
> > > > >
> > > > Nope, the intersection of endsegments is empty. only FINITE intersections of endsegments has infintie cardinality.
> > > >
> > > Every finite intersection. All finite intersections amount to the infinite intersection because the infinite intersection is only over all finite sets.
> > nope, infinite intersection yields empty,
> But all endsegments contain numbers? What numbers?
>
> Regards, WM
natural numbers, E_n contains all natural numbers >=n, are you getting dumber?

On Wednesday, 22 September 2021 at 12:36:16 UTC-3, WM wrote:
> zelos...@gmail.com schrieb am Dienstag, 21. September 2021 um 14:02:41 UTC+2:
> > tisdag 21 september 2021 kl. 13:03:30 UTC+2 skrev WM:
> > > zelos...@gmail.com schrieb am Dienstag, 21. September 2021 um 07:11:54 UTC+2:
> > > > fredag 17 september 2021 kl. 22:10:38 UTC+2 skrev WM:
> > >
> > > > > All infinite endsegments have an infinite intersection with each other. This is proved by inclusion monotony! As long as an endsegment contains elements, it contains them with all preceding endsegments.
> > > > >
> > > > Nope, the intersection of endsegments is empty. only FINITE intersections of endsegments has infintie cardinality.
> > > >
> > > Every finite intersection. All finite intersections amount to the infinite intersection because the infinite intersection is only over all finite sets.

Sorry; I have no hyperbole left to describe the insanity of your last comment. You cannot hide your dementia any longer.

> > nope, infinite intersection yields empty,
> But all endsegments contain numbers? What numbers?

Bigly numbers, beautiful numbers, the *BEST* numbers.

Greg Cunt schrieb am Mittwoch, 22. September 2021 um 21:06:25 UTC+2:
> On Wednesday, September 22, 2021 at 8:34:46 PM UTC+2, WM wrote:
> > Greg Cunt schrieb am Mittwoch, 22. September 2021 um 18:45:00 UTC+2:
> > >
> > > The question is whether all natural numbers are used as indices
> The answer is yes, since the set of natural numbers, IN, is adopted as index set for the SEQUENCE of endsegments: (E_n)_(n e IN).

The contents of the endsegments succeeds their indices. What remains if all endsegments are infinite. Why does this contents not appear in the intersection?

> > such that none remains as contents.
> Why do you thing that using the natural numbers as indices PROHIBITS us from using them for other purposes too?

The contents of the endsegments succeeds their indices.

> Look I'll use the natural numbers 1, 2 as indices and also as terms here:
>
> Let k_1 = 1 and k_2 = 2. Then k_1 + k_2 = 3.
>
> A miracle?

No, a diversion.
>
> if E is an endsement, then min(E) is its index and it contains all natural numbers >= min(E).

Here we consider all endsegments. You said all endsegments are infinite.
>
> This holds for each and every ensegments, i.e. for ALL endsegments.

No.

> Let n e IN. Then the endsegment {n, n+1, n+2, ...} contains the (infinitly many) natural numbers n, n+1, n+2, ... (and nothing else).

Every such endsegment E(n) has aleph_0 successors. Let's consider the whole set. It has none.

Regards, WM

zelos...@gmail.com schrieb am Donnerstag, 23. September 2021 um 06:44:42 UTC+2:
> onsdag 22 september 2021 kl. 17:36:16 UTC+2 skrev WM:

> > But all endsegments contain numbers? What numbers?

> natural numbers, E_n contains all natural numbers >=n

Irrelevant sind E(n) has infinitely many successors. Consider the whole set of endsegments. What remains as the contents of endsegments?

Regards, WM

torsdag 23 september 2021 kl. 14:58:52 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 23. September 2021 um 06:44:42 UTC+2:
> > onsdag 22 september 2021 kl. 17:36:16 UTC+2 skrev WM:
>
> > > But all endsegments contain numbers? What numbers?
> > natural numbers, E_n contains all natural numbers >=n
> Irrelevant sind E(n) has infinitely many successors. Consider the whole set of endsegments. What remains as the contents of endsegments?
>
> Regards, WM

What are you talking about?

Greg Cunt schrieb am Mittwoch, 22. September 2021 um 22:08:04 UTC+2:
> On Wednesday, September 22, 2021 at 8:51:26 PM UTC+2, WM wrote:

> > the numbers followed by infinitely many [numbers] can only make up a finite set.
>
> Huh?!

If a set S of natural numbers is followed by infinitely many natural numbers, then the set S is finite.
If the set |N of natural numbers is infinite, then it is not followed by natural numbers.
If the set of endsegments is infinite, then it is followed by infinitely many natural numbers, namely an infinite subset of the contents of all endsegments.

Here we have two contradicting facts.

Regards, 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

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

Greg Cunt schrieb am Mittwoch, 22. September 2021 um 22:24:43 UTC+2:
> On Wednesday, September 22, 2021 at 9:10:39 PM UTC+2, WM wrote:

> Since n is a natural number it has a unique prime decomposition.
> > Specify it. That can be done for every natural number less than 1 billion.

> Can it be done for every natural number less than 10^10^10^10^10^10^10^10^80 too?

In ideal mathematics in principle yes, but for n it can never be done.

Regards, WM

On Thursday, 23 September 2021 at 10:18:46 UTC-3, WM wrote:
> 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?

It would be hard to formulate more concisely your utter incomprehension of all things infinite. "Infinite" means that there is *no end* to the successor operation. Every natural number has an end segment that contains alpha_0 direct and indirect successors. In addition, sets do not have successors, you demented old fool.

On 9/23/2021 8:16 AM, WM wrote:
> Greg Cunt schrieb am Mittwoch, 22. September 2021 um 22:08:04 UTC+2:
>> On Wednesday, September 22, 2021 at 8:51:26 PM UTC+2, WM wrote:
>
>>> the numbers followed by infinitely many [numbers] can only make up a finite set.
>>
>> Huh?!
>
> If a set S of natural numbers is followed by infinitely many natural numbers, then the set S is finite.

how do you write that ? like this ?

set S = {1,2,3,...} , 1,2,3,... ?

> If the set |N of natural numbers is infinite, then it is not followed by natural numbers.

but it could, like above...

set N = {1,2,3,...} , 1,2,3,... ?

> If the set of endsegments is infinite, then it is followed by infinitely many natural numbers, namely an infinite subset of the contents of all endsegments.

>"If the set of endsegments is infinite",

the set of endsegments is infinite.

>"then it is followed by infinitely many natural numbers",

it follows if you write the equations that way.

>"namely an infinite subset of the contents of all endsegments."

which is E(1) and has nothing to do with above

>
> Here we have two contradicting facts.

nope. clean up your statements, and we will review again.

>
> Regards, WM
>

On 9/23/2021 8:23 AM, WM wrote:
> 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.

that is your daffynition. Of course you know you have now eliminated Aleph_0 natural numbers, right ?

What does " known in principle or knowable" mean anyway ?

>
> Regards, WM
>

On 9/23/2021 7:56 AM, WM wrote:
> Greg Cunt schrieb am Mittwoch, 22. September 2021 um 21:06:25 UTC+2:
>> On Wednesday, September 22, 2021 at 8:34:46 PM UTC+2, WM wrote:
>>> Greg Cunt schrieb am Mittwoch, 22. September 2021 um 18:45:00 UTC+2:
>>>>
>>>> The question is whether all natural numbers are used as indices
>> The answer is yes, since the set of natural numbers, IN, is adopted as index set for the SEQUENCE of endsegments: (E_n)_(n e IN).
>
> The contents of the endsegments succeeds their indices.

Wrong.

>What remains if all endsegments are infinite.

no, all endsegments are infinite.

> Why does this contents not appear in the intersection?

there is no element common to all endsegments.

>
>
>>> such that none remains as contents.
>> Why do you thing that using the natural numbers as indices PROHIBITS us from using them for other purposes too?
>
> The contents of the endsegments succeeds their indices.

Wrong.

on 9/23/2021, WM supposed :
> zelos...@gmail.com schrieb am Donnerstag, 23. September 2021 um 06:44:42
> UTC+2:
>> onsdag 22 september 2021 kl. 17:36:16 UTC+2 skrev WM:
>
>>> But all endsegments contain numbers? What numbers?
>
>> natural numbers, E_n contains all natural numbers >=n
>
> Irrelevant sind E(n) has infinitely many successors. Consider the whole set
> of endsegments. What remains as the contents of endsegments?

Use recycled natural numbers, they are not single use items.

Or maybe use natural numbers with zero and say each FISON's cardinality
is the index for the associated endsegment. You shouldn't "run out" of
anything that way. Of course, it doesn't fit your agenda so you won't
do it -- you'll just make up some more lies about set theory and
pretend that you are smarter than the great mathematicians which helped
to develop it.

WM brought next idea :
> Greg Cunt schrieb am Mittwoch, 22. September 2021 um 22:08:04 UTC+2:
>> On Wednesday, September 22, 2021 at 8:51:26 PM UTC+2, WM wrote:
>
>>> the numbers followed by infinitely many [numbers] can only make up a finite
>>> set.
>>
>> Huh?!
>
> If a set S of natural numbers is followed by infinitely many natural numbers,
> then the set S is finite.

"Followed by" seems to ignore order type.

> If the set |N of natural numbers is infinite,

Of course it is.

> then it is not followed by natural numbers.

Of course it isn't.

> If the set of endsegments is infinite,

Of course it is.

> then it is followed by infinitely many natural numbers,

"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.

???

> Here we have two contradicting facts.

No, we only have your muddled thinking on display here.

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?

Is it one of these things? 0,1,2,3,4,5,...
You don't 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.
You avoid needing to support a claim about UPFs
by making it a definition.

>> n can be counted to in principle.
>
> No, not even in principle. You are extremely deranged.

That's my definition, "can be counted to in principle".
It's my description of 0,1,2,3,4,5,...

I look at 0,1,2,3,4,5,... and say
"Can those things be counted to in principle? Hmmm. Yep."

You expect me to believe that _you_ look at 0,1,2,3,4,5,...
and say
"Do each of those things have a knowable UPF? Hmmm. Yep."

Bullshit.

----
When I fill in the details of what "countable-to in principle"
means, we can argue by _truth-preserving inferences_ to the
claim that each one of the countable-to has a unique prime
factorization.

If it is _true_ that n is countable-to in principle,
then _the truth of that_ gets _preserved_ gets handed from claim
to claim to claim to claim until we arrive at the claim
that n has a UPF.

_That_ is how we know that each one of 0,1,2,3,4,5,...
has a UPF as certainly as we know that each one of
0,1,2,3,4,5,... can be counted to in principle.

On Thursday, September 23, 2021 at 3:26:52 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Mittwoch, 22. September 2021 um 22:24:43 UTC+2:
> > On Wednesday, September 22, 2021 at 9:10:39 PM UTC+2, WM wrote:
> >
> > Since n is a natural number > 1 it has a unique prime decomposition.

Period.

> > > Specify it.

There's no need to do so, you silly crank.

Hint: Let n be an element in {3, 25, 99}. Then n is either the natural number 3, or it is the natural number 25, or it is the natural number 99. We just don't know which one. (How could we?) Now n certainly has a unique prime decomposition (since it is a natural number > 1). But we can't "specify" it. (Since we don't know WHICH natural number n is, though we DO KNOW THAT it is a natural number, actually, either 3, 25, or 99.)

Compare it with the following case:

Hint: Let n be an element in {25}. Then n is the natural number 25 (since x e {y} <-> x = y). Now n certainly has a unique prime decomposition (since it is a natural number > 1). And we can "specify" it: n = 5 * 5. (Since we know precisely WHICH natural number n is, actually, it is the natural number 25.)

Hint: In both cases n is a natural number (and not a "placeholder").

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

FromTheRafters schrieb am Donnerstag, 23. September 2021 um 17:48:44 UTC+2:
> You ... pretend that you are smarter than the great mathematicians which helped
> to develop it.

No I am not smarter but I have seen what all of you could have seen or even have seen too, but not understood it in its importance: 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?

Regards, WM

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.
> >
> > The contents of the endsegments succeeds their indices.
> Wrong.

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

Regards, WM

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, ... . 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.

Regards, WM

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.

> 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?

Regards, WM

On Thursday, 23 September 2021 at 16:49:25 UTC-3, WM wrote:
> 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.

Perhaps it escaped your attention that A and B are but *TWO* sets. Two is a *FINITE* number, and the intersection of a finite number of inclusion ordered end segments was never at issue. It is equal to the smallest of these end segments, namely the one with the largest "base". But if the intersection extends over *INFINITELY* many end segments (not even all of them, just infinitely many), then there is *NO* largest "base" and the intersection is empty. That you have not grasped this in over sixteen years of pretending to try speaks volumes about your reasoning capacity, and it's not flattering.

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.

> You avoid needing to support a claim about UPFs
> by making it a definition.
> >> n can be counted to in principle.
> >
> > No, not even in principle. You are extremely deranged.
> That's my definition, "can be counted to in principle".
> It's my description of 0,1,2,3,4,5,...

But try to count to n. Fail.

> _That_ is how we know that each one of 0,1,2,3,4,5,...
> has a UPF

0 has none. 1 had before 1900.

> as certainly as we know that each one of
> 0,1,2,3,4,5,... can be counted to in principle.

0 cannot be counted to. 1,2,3,4,5,... of course, but none is n.

Regards, WM

Greg Cunt schrieb am Donnerstag, 23. September 2021 um 19:45:18 UTC+2:
> On Thursday, September 23, 2021 at 3:26:52 PM UTC+2, WM wrote:
> > Greg Cunt schrieb am Mittwoch, 22. September 2021 um 22:24:43 UTC+2:
> > > On Wednesday, September 22, 2021 at 9:10:39 PM UTC+2, WM wrote:
> > >
> > > Since n is a natural number > 1 it has a unique prime decomposition.

> > > > Specify it.
>
> There's no need to do so,

There is no chance to do so. Fox, grapes.

Regards, WM