{1, 2, 3, ...} = ℕ

is the set of all natural numbers. The order does not matter. It can be chosen arbitrarily but has to remain the same in the following. If every number is prepended by all its predecessors

{1, 1, 2, 1, 2, 3, ...} = ℕ

nothing is changed (since multiple appearance of an element in curly brackets does not matter). But if some more curly brackets are inserted, then ℕ is no longer contained in

{{1}, {1, 2}, {1, 2, 3}, ...}
ℕ is lost. It is no longer present, neither as a subset nor as an element, although no natural number has been removed. . So the mere insertion of braces destroys ℕ – although never two numbers which are neighbours in the original sequence have been separated by braces.

Regards, WM

On Sunday, October 23, 2022 at 1:26:14 PM UTC+2, WM wrote:
> {1, 2, 3, ...} = ℕ
>
> is the set of all natural numbers. The order does not matter. It can be chosen arbitrarily but has to remain the same in the following. If every number is prepended by all its predecessors
>
> {1, 1, 2, 1, 2, 3, ...} = ℕ
>
> nothing is changed (since multiple appearance of an element in curly brackets does not matter). But if some more curly brackets are inserted, then ℕ is no longer contained in
>
> {{1}, {1, 2}, {1, 2, 3}, ...}
>
> ℕ is lost. It is no longer present, neither as a subset nor as an element, although no natural number has been removed. . So the mere insertion of braces destroys ℕ – although never two numbers which are neighbours in the original sequence have been separated by braces.
>
> Regards, WM

This is a good example where a person who can't express himself verbally
produces mathematical nonsense.
These two debilities often run in parallel and, as also demonstrated here, are related
to the inability to distinguish objects and their names.

On Sunday, October 23, 2022 at 7:26:14 AM UTC-4, WM wrote:
> {1, 2, 3, ...} = ℕ
>
> is the set of all natural numbers. The order does not matter. It can be chosen arbitrarily but has to remain the same in the following. If every number is prepended by all its predecessors
>
> {1, 1, 2, 1, 2, 3, ...} = ℕ
>
> nothing is changed (since multiple appearance of an element in curly brackets does not matter). But if some more curly brackets are inserted, then ℕ is no longer contained in
>
> {{1}, {1, 2}, {1, 2, 3}, ...}
>
> ℕ is lost. It is no longer present, neither as a subset nor as an element, although no natural number has been removed. . So the mere insertion of braces destroys ℕ – although never two numbers which are neighbours in the original sequence have been separated by braces.
>
> Regards, WM

This looks pretty good. Ellipses at work and not quite at work.
It is amazing how much mathematics takes place upon the instantiation of three. Seems to be the max element you guys work with.
Getting up to eleven something special happens. Claims of working in large numbers does involve digits. This digital awareness is not a part of standard natural valued analysis. When we regard the large number as a polynomial as in:
123 = 1 x 10^2 + 2 x 10 ^ 1 + 3 x 10 ^ 0
what we see is a stop at 10 ^ 9, for the description of a large number via large numbers would be a logical paradox. This then forms a school of numbers that tops out at 9999999999. It also exposes the early aleph of the first school:
1 2 3 4 5 6 7 8 9 | A
In effect, to resolve the usage of '10' in the above polynomial we require the first school (directly above) carry an aleph or an omega, depending on who you talk to, but this first school is unique in that it is the establishment of the glyphs. As well, formalizing an aleph may help to disambiguate the fact that every number system is modulo 10.

Upon engaging the digits within analysis the slow progression of the ellipsis of the natural numbers is felt. Any who wish to investigate large numbers will be much farther along say, at 33333 versus 5.

On 10/23/2022 6:26 AM, WM wrote:
> {1, 2, 3, ...} = ℕ
>
> is the set of all natural numbers. The order does not matter. It can be chosen arbitrarily but has to remain the same in the following. If every number is prepended by all its predecessors
>
> {1, 1, 2, 1, 2, 3, ...} = ℕ
>
> nothing is changed (since multiple appearance of an element in curly brackets does not matter). But if some more curly brackets are inserted, then ℕ is no longer contained in
>
> {{1}, {1, 2}, {1, 2, 3}, ...}
>
> ℕ is lost. It is no longer present, neither as a subset nor as an element, although no natural number has been removed. . So the mere insertion of braces destroys ℕ – although never two numbers which are neighbours in the original sequence have been separated by braces.
>
> Regards, WM

Ok, so you flunked set theory...

And you are posting here to make up for that ?

or are you trying to spread your malfunction ?

WM wrote :
> {1, 2, 3, ...} = ℕ
>
> is the set of all natural numbers. The order does not matter. It can be
> chosen arbitrarily but has to remain the same in the following. If every
> number is prepended by all its predecessors
>
> {1, 1, 2, 1, 2, 3, ...} = ℕ

This is supposed to be an infinite sequence I assume.

> nothing is changed (since multiple appearance of an element in curly brackets
> does not matter). But if some more curly brackets are inserted, then ℕ is no
> longer contained in

Never was, but all of the elements are there.

> {{1}, {1, 2}, {1, 2, 3}, ...}

|N = {{1}, {1, 2}, {1, 2, 3}, ...} = the set of all FISONs

> ℕ is lost. It is no longer present, neither as a subset nor as an element,
> although no natural number has been removed. . So the mere insertion of
> braces destroys ℕ – although never two numbers which are neighbours in the
> original sequence have been separated by braces.

Sequences and sets are not the same thing.

FromTheRafters schrieb am Sonntag, 23. Oktober 2022 um 19:06:24 UTC+2:
> WM wrote :
> > {1, 2, 3, ...} = ℕ
> >
> > is the set of all natural numbers. The order does not matter. It can be
> > chosen arbitrarily but has to remain the same in the following. If every
> > number is prepended by all its predecessors
> >
> > {1, 1, 2, 1, 2, 3, ...} = ℕ
> This is supposed to be an infinite sequence I assume.

No, it is a set written with superfluous but not forbidden symbols.

> > nothing is changed (since multiple appearance of an element in curly brackets
> > does not matter). But if some more curly brackets are inserted, then ℕ is no
> > longer contained in
> Never was, but all of the elements are there.

Here it was: {1, 2, 3, ...} = ℕ

> > {{1}, {1, 2}, {1, 2, 3}, ...}
> |N = {{1}, {1, 2}, {1, 2, 3}, ...} = the set of all FISONs

No, ℕ = {1, 2, 3, ...}

> > ℕ is lost. It is no longer present, neither as a subset nor as an element,
> > although no natural number has been removed. . So the mere insertion of
> > braces destroys ℕ – although never two numbers which are neighbours in the
> > original sequence have been separated by braces.
> Sequences and sets are not the same thing.

Here no sequences are used. But most sequences without repetitions are well-ordered sets.

Regards, WM

After serious thinking WM wrote :
> FromTheRafters schrieb am Sonntag, 23. Oktober 2022 um 19:06:24 UTC+2:
>> WM wrote :
>>> {1, 2, 3, ...} = ℕ
>>>
>>> is the set of all natural numbers. The order does not matter. It can be
>>> chosen arbitrarily but has to remain the same in the following. If every
>>> number is prepended by all its predecessors
>>>
>>> {1, 1, 2, 1, 2, 3, ...} = ℕ
>> This is supposed to be an infinite sequence I assume.
>
> No, it is a set written with superfluous but not forbidden symbols.

Trying to muddy the waters even more?

>>> nothing is changed (since multiple appearance of an element in curly
>>> brackets does not matter). But if some more curly brackets are inserted,
>>> then ℕ is no longer contained in
>> Never was, but all of the elements are there.
>
> Here it was: {1, 2, 3, ...} = ℕ

|N is a symbol for the set of natural numbers. It is not, itself, a
natural number so cannot be an element of the natural numbers.

On Sunday, October 23, 2022 at 1:26:14 PM UTC+2, WM wrote:

> {1, 2, 3, ...}

=/=

> {{1}, {1, 2}, {1, 2, 3}, ...}

Wow, what a discovery!

A (certain) set containg just natural numbers DIFFERS from a (certain) set containing sets of natural numbers.

An incredible insight!

FromTheRafters schrieb am Sonntag, 23. Oktober 2022 um 19:46:21 UTC+2:
> After serious thinking WM wrote :
> > FromTheRafters schrieb am Sonntag, 23. Oktober 2022 um 19:06:24 UTC+2:
> >> WM wrote :
> >>> {1, 2, 3, ...} = ℕ
> >>>
> >>> is the set of all natural numbers. The order does not matter. It can be
> >>> chosen arbitrarily but has to remain the same in the following. If every
> >>> number is prepended by all its predecessors
> >>>
> >>> {1, 1, 2, 1, 2, 3, ...} = ℕ
> >> This is supposed to be an infinite sequence I assume.
> >
> > No, it is a set written with superfluous but not forbidden symbols.
> Trying to muddy the waters even more?

Doing what is not forbidden.

> >>> nothing is changed (since multiple appearance of an element in curly
> >>> brackets does not matter). But if some more curly brackets are inserted,
> >>> then ℕ is no longer contained in
> >> Never was, but all of the elements are there.
> >
> > Here it was: {1, 2, 3, ...} = ℕ

> |N is a symbol for the set of natural numbers. It is not, itself, a
> natural number so cannot be an element of the natural numbers.

Look above. There you see {1, 2, 3, ...} = ℕ and not {1, 2, 3, ...., ℕ} = ℕ.

Regards, WM

Fritz Feldhase schrieb am Sonntag, 23. Oktober 2022 um 19:55:35 UTC+2:
> On Sunday, October 23, 2022 at 1:26:14 PM UTC+2, WM wrote:
>
> > {1, 2, 3, ...}
> =/> > {{1}, {1, 2}, {1, 2, 3}, ...}
> Wow, what a discovery!
>
> A (certain) set containg just natural numbers DIFFERS from a (certain) set containing sets of natural numbers.

The set ℕ containing all natural numbers has not lost any element but its property to be or contain the set ℕ.

Regards, WM

On Sunday, October 23, 2022 at 8:02:27 PM UTC+2, WM wrote:

> The set ℕ containing all natural numbers has not lost any element but its property to be or contain the set ℕ.

Nope, you psychotic asshole full of shit!

Mathematical sets DON'T CHANGE. IN is IN. It does not lose certain properties or gain some, you silly idiot!

Hint: Appending " ' " to the term "0" does not change 0 to 1, even though 1 = 0'.

WM was thinking very hard :
> FromTheRafters schrieb am Sonntag, 23. Oktober 2022 um 19:46:21 UTC+2:
>> After serious thinking WM wrote :
>>> FromTheRafters schrieb am Sonntag, 23. Oktober 2022 um 19:06:24 UTC+2:
>>>> WM wrote :
>>>>> {1, 2, 3, ...} = ℕ
>>>>>
>>>>> is the set of all natural numbers. The order does not matter. It can be
>>>>> chosen arbitrarily but has to remain the same in the following. If every
>>>>> number is prepended by all its predecessors
>>>>>
>>>>> {1, 1, 2, 1, 2, 3, ...} = ℕ
>>>> This is supposed to be an infinite sequence I assume.
>>>
>>> No, it is a set written with superfluous but not forbidden symbols.
>> Trying to muddy the waters even more?
>
> Doing what is not forbidden.
>
>>>>> nothing is changed (since multiple appearance of an element in curly
>>>>> brackets does not matter). But if some more curly brackets are inserted,
>>>>> then ℕ is no longer contained in
>>>> Never was, but all of the elements are there.
>>>
>>> Here it was: {1, 2, 3, ...} = ℕ
>
>>> N is a symbol for the set of natural numbers. It is not, itself, a
>> natural number so cannot be an element of the natural numbers.
>
> Look above. There you see {1, 2, 3, ...} = ℕ and not {1, 2, 3, ..., ℕ} = ℕ.

A set of elephants might be a circus, but a circus is not an elephant.

söndag 23 oktober 2022 kl. 13:26:14 UTC+2 skrev WM:
> {1, 2, 3, ...} = ℕ
>
> is the set of all natural numbers. The order does not matter. It can be chosen arbitrarily but has to remain the same in the following. If every number is prepended by all its predecessors
>
> {1, 1, 2, 1, 2, 3, ...} = ℕ

Is not a set

>
> nothing is changed (since multiple appearance of an element in curly brackets does not matter). But if some more curly brackets are inserted, then ℕ is no longer contained in
>
> {{1}, {1, 2}, {1, 2, 3}, ...}
>
> ℕ is lost. It is no longer present, neither as a subset nor as an element, although no natural number has been removed. . So the mere insertion of braces destroys ℕ – although never two numbers which are neighbours in the original sequence have been separated by braces.

No, N still exists, you just made a different set.

>
> Regards, WM

söndag 23 oktober 2022 kl. 15:49:52 UTC+2 skrev timba...@gmail.com:
> On Sunday, October 23, 2022 at 7:26:14 AM UTC-4, WM wrote:
> > {1, 2, 3, ...} = ℕ
> >
> > is the set of all natural numbers. The order does not matter. It can be chosen arbitrarily but has to remain the same in the following. If every number is prepended by all its predecessors
> >
> > {1, 1, 2, 1, 2, 3, ...} = ℕ
> >
> > nothing is changed (since multiple appearance of an element in curly brackets does not matter). But if some more curly brackets are inserted, then ℕ is no longer contained in
> >
> > {{1}, {1, 2}, {1, 2, 3}, ...}
> >
> > ℕ is lost. It is no longer present, neither as a subset nor as an element, although no natural number has been removed. . So the mere insertion of braces destroys ℕ – although never two numbers which are neighbours in the original sequence have been separated by braces.
> >
> > Regards, WM
> This looks pretty good. Ellipses at work and not quite at work.
> It is amazing how much mathematics takes place upon the instantiation of three. Seems to be the max element you guys work with.
> Getting up to eleven something special happens. Claims of working in large numbers does involve digits. This digital awareness is not a part of standard natural valued analysis. When we regard the large number as a polynomial as in:
> 123 = 1 x 10^2 + 2 x 10 ^ 1 + 3 x 10 ^ 0
> what we see is a stop at 10 ^ 9, for the description of a large number via large numbers would be a logical paradox. This then forms a school of numbers that tops out at 9999999999. It also exposes the early aleph of the first school:
> 1 2 3 4 5 6 7 8 9 | A
> In effect, to resolve the usage of '10' in the above polynomial we require the first school (directly above) carry an aleph or an omega, depending on who you talk to, but this first school is unique in that it is the establishment of the glyphs. As well, formalizing an aleph may help to disambiguate the fact that every number system is modulo 10.
>
> Upon engaging the digits within analysis the slow progression of the ellipsis of the natural numbers is felt. Any who wish to investigate large numbers will be much farther along say, at 33333 versus 5.
are you stupid? All he said was stupid!

On 10/23/2022 4:26 AM, WM wrote:
> {1, 2, 3, ...} = ℕ
>
> is the set of all natural numbers. The order does not matter. It can be chosen arbitrarily but has to remain the same in the following. If every number is prepended by all its predecessors
>
> {1, 1, 2, 1, 2, 3, ...} = ℕ
>
> nothing is changed (since multiple appearance of an element in curly brackets does not matter). But if some more curly brackets are inserted, then ℕ is no longer contained in
>
> {{1}, {1, 2}, {1, 2, 3}, ...}
>
> ℕ is lost. It is no longer present, neither as a subset nor as an element, although no natural number has been removed. . So the mere insertion of braces destroys ℕ – although never two numbers which are neighbours in the original sequence have been separated by braces.

Imvho, its fun to look at the naturals as a tree with fisons as levels:

{ { 1 }, { 2, 3 }, { 4, 5, 6, 7 }, ... }

1
/ \
/ \
/ \
/ \
2 3
/ \ / \
/ \ / \
4 5 6 7
..............................

On and on forever. There are some nice properties about the this tree.

The children of 3 are: 3*2 and 3*2+1

2 is: 2*2 and 2*2+1

The children of 6 are: 6*2 and 6*2+1

That is just one of its many properties.

On Sunday, 23 October 2022 at 14:06:24 UTC-3, FromTheRafters wrote:
[...]
> |N = {{1}, {1, 2}, {1, 2, 3}, ...} = the set of all FISONs

|N is *NOT* the set of all FISONs. Just a simple hint: the natural number 2 is an element of |N, but it is *NOT* an element of the set of FISONs, because 2 is not a FISON. Why the fuck do you drop your bullshit time and time again? You are a fucking moron, so please piss off.

On Monday, October 24, 2022 at 2:59:41 PM UTC+2, Gus Gassmann wrote:
> On Sunday, 23 October 2022 at 14:06:24 UTC-3, FromTheRafters wrote:
> >
> > |N = {{1}, {1, 2}, {1, 2, 3}, ...} = the set of all FISONs
> >
> |N is *NOT* the set of all FISONs. Just a simple hint: the natural number 2 is an element of |N, but it is *NOT* an element of the set of FISONs, because 2 is not a FISON. Why the fuck do you drop your bullshit time and time again? You are a fucking moron, so please piss off.

Keep cool, man. If we do PROPER set theory ... in the context of ZFC, where the "natural numbers" are defined due to von Neumann... we would indeed have:

IN = {0, 1, 2, 3, ...} = {{}, {0}, {0, 1}, {0, 1, 2}, ...} = Set_of_FISONs (since {} may be considered a FISON too; after all, it's a finite set and there is a natural number, namely 0, such that all numbers in {} are smaller than that natural number). In this case, especially 2 *is* a FISON, namely the FISON {0, 1}. :-P

But WM is not interested in proper set theory, only in bullshit. (Hence his IN is {1, 2, 3, ...}, etc.)

Gus Gassmann pretended :
> On Sunday, 23 October 2022 at 14:06:24 UTC-3, FromTheRafters wrote:
> [...]
>>> N = {{1}, {1, 2}, {1, 2, 3}, ...} = the set of all FISONs
>
>> N is *NOT* the set of all FISONs. Just a simple hint: the natural number 2
>> is an element of |N, but it is *NOT* an element of the set of FISONs,
>> because 2 is not a FISON. Why the fuck do you drop your bullshit time and
>> time again? You are a fucking moron, so please piss off.

Okay.

https://www.bookofproofs.org/graphics/examples/axiom8.jpg

Nesting increase as per Zermelo, cardinality increase as per Frege, and
ordinal numbers by building upon FISONs as per von Neumann.

On Monday, 24 October 2022 at 10:17:00 UTC-3, Fritz Feldhase wrote:
[...]
> But WM is not interested in proper set theory, only in bullshit.

Precisely.