Re: Counterexample

[Serg io]

On 9/13/2021 7:32 AM, WM wrote:
> Gus Gassmann schrieb am Sonntag, 12. September 2021 um 21:36:24 UTC+2:
>> On Sunday, 12 September 2021 at 13:42:34 UTC-3, Transfinity wrote:
>>> Gus Gassmann schrieb am Sonntag, 12. September 2021 um 02:37:38 UTC+2:
>>>> On Saturday, 11 September 2021 at 17:22:50 UTC-3, WM wrote:
>>>>> Gus Gassmann schrieb am Samstag, 11. September 2021 um 17:16:18 UTC+2:
>>>>>> On Saturday, 11 September 2021 at 10:20:38 UTC-3, WM wrote:
>>>>>>> William schrieb am Samstag, 11. September 2021 um 02:18:54 UTC+2:
>>>
>>>>>> And why should it? For any n in N, the *FINITE* (got that?) intersection of end segments {k, k+1, k+2, ...} over k <= n is just E(n).
>>>>> And there is no finite n leading to an infinite intersection. But there are only finite n. Hence there is no empty intersection.
>>>> For the record, there are infinitely many natural numbers. each of them finite.
>>> But infinitely many must be spared for the contents of the endsegments. Therefore only finitely many n can be used to index the endsegments E(n). Note that in order omega two consecutive infinite sets of card aleph_0 are not possible.
> Every natural number n can *OF COURSE* serve two purposes: To be an element of an end segment and to be the starting element (what you call the "index") of an end segment.

what else can a natural number n serve ?

>
> But if there are infinitely many starting elements, then there are no successors. You cannot count beyond all infinitely many starting elements.

Wrong, you state that infinity is used up.

>
> How the *FUCK* do you think that E(1) and E(2) can exist simultaneously? Note: 2 is both an element of E(1) and the "index" of E(2).

All endsegments exist simultaneously. Those with small brains cannot think of such things, and must stick to counting rocks.

>
> No problem. Only if there were infinitely many endsegments, no consecutive infinite set would be possible, not even a single finite natural number.


obviously thousands of brain cells were liquidated during the generation of that sentence

>
> Regards, WM
>



Starting Ants
Simultaneously Ants

[Greg Cunt]

On Monday, September 13, 2021 at 4:04:40 PM UTC+2, Serg io wrote:

> what else can a natural number n serve ?

See? You are calling it (n) a natural number, here. :-)

[WM]

If we call n a natural number, then we mean the natural number inserted at n.
"Let n be a natural number" does not put n in the sequence of ordinals but means that any natural number can be chosen to fill the place occupied by n.

Regards, WM

[Serg io]

what "place occupied" by n ??

Leave n alone, and let n grow up the way he wants too!

[Greg Cunt]

On Tuesday, September 14, 2021 at 10:00:23 PM UTC+2, WM wrote:

> If we call /n/ a natural number, then we mean

that n is a natural number, naturally.

> "Let n be a natural number" does not put n in the sequence of ordinals

*lol* That's EXACTLY what it does, Mucke.

Hint: After stating this definition we are able to prove

There is an ordinal x such that x = n ,

or simpler

n is an ordinal.


Hope this helps.

[WM]

Greg Cunt schrieb am Mittwoch, 15. September 2021 um 07:46:55 UTC+2:

> Hint: After stating this definition we are able to prove
>
> There is an ordinal x such that x = n ,
>
> or simpler
>
> n is an ordinal.
>

What is its place in the sequence of ordinals?

Regards, WM

[Gus Gassmann]

It follows its immediate predecessor and is followed by its successor S(n).

[WM]

Here is the sequence: 1, 2, 3, 4, 5, 6, ...

Please point to the position of n.

Regards, WM

[Greg Cunt]

On Wednesday, September 15, 2021 at 3:08:54 PM UTC+2, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 15. September 2021 um 14:03:00 UTC+2:
> > On Wednesday, 15 September 2021 at 08:18:20 UTC-3, WM wrote:
> > > Greg Cunt schrieb am Mittwoch, 15. September 2021 um 07:46:55 UTC+2:
> > > >
> > > > Hint: After stating this definition we are able to prove
> > > >
> > > > | There is an ordinal x such that x = n ,
> > > >
> > > > or simpler
> > > >
> > > > | n is an ordinal.
> > > >
> > > What is its place in the sequence of ordinals?
> > >

> > It follows its immediate predecessor [if it has one --GC] and is followed by its successor S(n).

Right.

> Here is the sequence: (1, 2, 3, 4, 5, 6, ...)

>
> Please point to the position of n.

No problem: The position of n in the sequence (1, 2, 3, 4, 5, 6, ...) is n.

Now THAT was easy!

[Jim Burns]

We don't know. At least, we apparently haven't been told yet.

However, we already know other things about n.

There is an order-relation '<' between n and other individuals
such that, for any collection B of these (ordinal) individuals,
B contains a first element, unless B is empty.
('<' is a well-order.)

We know facts that can be derived from this fact about n
It's interesting that there are a lot of these derived facts,
but, interesting or not, we know them.

Not-knowing which individual n is does not remove from our
brains anything we know about n.
"You don't know X" is not in itself an argument for
"You don't know Y".

[Greg Cunt]

On Wednesday, September 15, 2021 at 4:32:37 PM UTC+2, Jim Burns wrote:
> On 9/15/2021 7:18 AM, WM wrote:

> > Greg Cunt wrote:
> > >
> > > Hint: After stating this definition we are able to prove
> > >
> > > | There is an ordinal x such that x = n ,
> > >
> > > or simpler
> > >
> >> | n is an ordinal.
> > >
> > What is its place in the sequence of ordinals?
> >
> We don't know.

Oh, but we *do* know! It's n (!).

Really!

I mean, it's not n-k for all k e {m e {1, 2, 3, ...} : m <= n} and it is not n+k for all k e IN = {1, 2, 3, ...}, it's just n.

[WM]

There is no n visible yet. Try again.

Regards, WM

[WM]

Jim Burns schrieb am Mittwoch, 15. September 2021 um 16:32:37 UTC+2:
> On 9/15/2021 7:18 AM, WM wrote:
> > Greg Cunt schrieb
> > am Mittwoch, 15. September 2021 um 07:46:55 UTC+2:
>
> >> Hint: After stating this definition we are able to prove
> >> There is an ordinal x such that x = n ,
> >> or simpler
> >> n is an ordinal.
> >
> > What is its place in the sequence of ordinals?
> We don't know. At least, we apparently haven't been told yet.

But you have been told, long ago, the position of 3 and 17 and even 10^10.

>
> However, we already know other things about n.

Yes, it should be the placeholder for a natural number which satisfies the following conditions:

>
> There is an order-relation '<' between n and other individuals
> such that, for any collection B of these (ordinal) individuals,
> B contains a first element, unless B is empty.
> ('<' is a well-order.)
>
> We know facts that can be derived from this fact about n
> It's interesting that there are a lot of these derived facts,
> but, interesting or not, we know them.

Of course.

>
> Not-knowing which individual n is does not remove from our
> brains anything we know about n.

But it should remove the silly idea that it is a natural number. On the other hand this simple example explains how fanatical matheologians can defend the obviously wrong results of set theory. To mention only the most striking ones which every sober mind can identify as nonsense:
- All fractions can be enumerated.
- There are uncountably many real numbers although only countably many can be distinguished.
- There are more paths in the Binary Tree than nodes.
- The bankruptcy of McDuck.
- aleph_0 rational points separate uncountably many irrational points such that never two are existing without rational between them.

It is painful to see so much fool's crap blocking your minds. I will really enjoy my lectures next semester. It will be relieving to address an audience of unbiased thinkers.

Regards, WM

[Greg Cunt]

On Wednesday, September 15, 2021 at 5:52:04 PM UTC+2, WM wrote:

> There is no n visible yet.

Look, you psychotic asshole, NO ONE has ever s e e n a number. (Well except you, it seems.)

[Greg Cunt]

On Wednesday, September 15, 2021 at 6:08:36 PM UTC+2, WM wrote:

> But you have been told, long ago, the position of 3 and 17 and even 10^10.

Yeah, the position of 3 is 3, the position of 17 is 17, the position of 10^10 is 10^10, and the position of n is n.

> > it should be <bla>

Shut up, you psychotic asshole full of shit.

Since we "introduced" the term (actually a constant) "n" the following way:

| Let n e IN.

n is a natural number (i.e. an element in IN).

Hence we know quite a lot about the natural number n (since we know quite a lot about "the natural numbers").

> But it should remove the silly idea that it is a natural number.

Only if suffering from a psychosis, just like you.

[Jim Burns]

On 9/15/2021 12:08 PM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 15. September 2021 um 16:32:37 UTC+2:
>> On 9/15/2021 7:18 AM, WM wrote:
>>> Greg Cunt schrieb
>>> am Mittwoch, 15. September 2021 um 07:46:55 UTC+2:

>>>> Hint: After stating this definition we are able to prove
>>>> There is an ordinal x such that x = n ,
>>>> or simpler
>>>> n is an ordinal.
>>>
>>> What is its place in the sequence of ordinals?
>>
>> We don't know. At least, we apparently haven't been told yet.
>
> But you have been told, long ago,
> the position of 3 and 17 and even 10^10.

The difference between
"n is 3", "n is 17", "n is 10^10"
and
"n is one of these well-ordered things"
is that we know
"n is one of these well-ordered things".

From things we know, we can derive more things we know,
by truth-preservation.

From things we don't know, not so much.

( An exception pops into my mind.
( https://en.wikipedia.org/wiki/Sum_and_Product_Puzzle
( Two perfect logicians S and P reason to the value of
( a pair of numbers starting from what the other
( _doesn't_know.
(
( Still, the opposite is pretty rare.

>> However, we already know other things about n.
>
> Yes, it should be the placeholder for a natural number which
> satisfies the following conditions:

We know that that number satisfies the following conditions.
And, from that, we can derive a lot more.

>> There is an order-relation '<' between n and other individuals
>> such that, for any collection B of these (ordinal) individuals,
>> B contains a first element, unless B is empty.
>> ('<' is a well-order.)
>>
>> We know facts that can be derived from this fact about n
>> It's interesting that there are a lot of these derived facts,
>> but, interesting or not, we know them.
>
> Of course.
>
>> Not-knowing which individual n is does not remove from our
>> brains anything we know about n.
>
> But it should remove the silly idea that it is a natural number.

| Something which cannot be counted to, even in principle,
| is not a natural number.
| If it is a natural number, it can be counted to, in principle.

Is that silly, in your opinion?

[Jim Burns]

[ Sorry, I unintentionally sent this to FF's email.
[ I did not intend to make this a private conversation.

[ This is my third try to post this to sci.math.
[ (@FF sorry^2)
[ Some days I need a keeper.

Natural language is not the best tool for this discussion.

We have some facts about n.

If all we know about n is that n is a natural number,
we _don't_ know anything about n that we don't also know
about any other natural number.
Then, there are a lot of things we don't know about n.
However, what's left, things we know about n anyway,
is not nothing. What's left are some facts about n.

And, starting from some facts, we derive further facts with
truth-preserving inferences.

This is all a bland, nodding-off description of mathematics.
What puts WM's guts in an uproar is the claim that we can
still do this even if we don't have as one of our facts
"n is one of some individuals for which there is a last".

[Greg Cunt]

On Thursday, September 16, 2021 at 1:11:33 AM UTC+2, Jim Burns wrote:

> Then, there are a lot of things we don't know about n.

Indeed!

> [...] What's left are some facts about n.

Right. For example, we don't even know if n is even or if it is odd.

On the other hand, we know for certain that it is EITHER even OR odd. :-P

> What puts WM's guts in an uproar is the claim that we can
> still do this even if we don't have as one of our facts
> "n is one of some individuals for which there is a last".

:-)

Right.

I still like Russell's claim:

"Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

[... "true" in an "absolute" sense.]

[zelos...@gmail.com]

before n+1 but after n-1

[WM]

Greg Cunt schrieb am Donnerstag, 16. September 2021 um 01:42:41 UTC+2:

> I still like Russell's claim:
>
> "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."
>

I prefer what he says about such fools: "This is an instance of the amazing power of desire in blinding even very able men to fallacies which would otherwise be obvious at once."

Regards, WM

[WM]

Jim Burns schrieb am Mittwoch, 15. September 2021 um 21:28:32 UTC+2:
> On 9/15/2021 12:08 PM, WM wrote:

> >>> What is its place in the sequence of ordinals?
> >>
> >> We don't know. At least, we apparently haven't been told yet.
> >
> > But you have been told, long ago,
> > the position of 3 and 17 and even 10^10.
> The difference between
> "n is 3", "n is 17", "n is 10^10"
> and
> "n is one of these well-ordered things"
> is that we know
> "n is one of these well-ordered things".

No is a place holder. In place of n one of these things can be inserted.

Regards, WM

[Greg Cunt]

On Thursday, September 16, 2021 at 3:02:56 PM UTC+2, WM wrote:
> Jim Burns schrieb am Mittwoch, 15. September 2021 um 21:28:32 UTC+2:

Let n e IN.

> > The difference between "n is 3", "n is 17", "n is 10^10"
> > and "n is one of these well-ordered things" is that we know
> > "n is one of these well-ordered things".
> >

> No, [n] is a place holder. [bla bla]

No, n is a natural number, not a "place holder".

[Serg io]

and your class is not required for any real Math Degree, just a fillin elective for those majoring in non science.

[Serg io]

no wonder you never progressed in Math.

You are missing the foundations in Math, and you resent Math.

[Jim Burns]

It is not the case that _anything_ can be inserted.
No flying rainbow sparkle ponies allowed, for example.

Suppose that we want to discuss individuals that can be
counted to, in principle.

We could say
"n can be counted to, in principle."

I assume that we have, at that point, excluded flying
rainbow sparkle ponies. Because flying rainbow sparkle
ponies cannot be counted to, not even in principle.

If it were flying rainbow sparkle ponies we intend to
discuss, the description of n would be different.

The description of what we are discussing is the essence,
the keystone of this whole style of argumentation.

----
( If you (WM) want to add another layer of verbiage,
( that's irritating, but nothing worse than that.
( What do you want to say instead?
( "One of those individuals we are discussing can be
( inserted for n."
(
( I don't see any advantage to the extra verbiage.
( But, once we've formalized everything, all we see
( is "n". I don't see any serious disadvantage, either.

----
Note that that being one of those individuals is not
the sort of thing that can be "checked".
Essentially, it's true because _we say it's true_
(This is a rare and valuable situation.)

Assuming we _know_ that we say it's true, we _know_ that
n is (n can be replaced by)(n refers to)(...)
a natural number. A point in the line. A flying rainbow
sparkle pony. Whatever it is we are discussing.

[WM]

Jim Burns schrieb am Freitag, 17. September 2021 um 02:00:49 UTC+2:

> We could say
> "n can be counted to, in principle."

Yes n is or is representig the typical natural number. That means n is not a concrete natural number, but every natural number can be put at its place. Let n ∈ ℕ, then n has a uniqe prime decomposition. You cannot know what this prime decomposition is. But you know it in principle from every natural number.

Regards, WM

[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?

Regards, WM

[Serg io]

Please refer to "Highly Composite Numbers" by S. Ramanujan, Proc London Mathematical Society 2, XIV, 1915, 347-409

[Greg Cunt]

On Friday, September 17, 2021 at 9:43:23 PM UTC+2, WM wrote:

Let n e IN.

> n is not a concrete natural number

It's certainly a "concrete" natural number, though we don't KNOW which one.

> but every natural number can be put at its place.

Nope it can't. Hint n+1 =/= n.

> Let n ∈ ℕ, then n has a unique prime decomposition.

Exactly (if n > 1).

> You cannot know what this prime decomposition is.

Right.

> But you know [there is one for each and] every natural number [> 1].

Right.

[Greg Cunt]

I/we don't *know*.

Though we DO KNOW that n has a unique prime factorization (if n > 1), SINCE n is a natural number. See?!

(Hint: Does a "place holder"have a unique prime factorization ...and can it even be > 1?)

[WM]

Greg Cunt schrieb am Samstag, 18. September 2021 um 01:55:24 UTC+2:
> On Friday, September 17, 2021 at 9:43:23 PM UTC+2, WM wrote:
>
> Let n e IN.
> > n is not a concrete natural number
> It's certainly a "concrete" natural number, though we don't KNOW which one.

For every natural number we know it. Concrete means known.
Matheologians are simplistic minds who believe even the most stupid ideas like
- There are unknown known numbers

- All fractions can be enumerated.
- There are uncountably many real numbers although only countably many can be distinguished.
- There are more paths in the Binary Tree than nodes.
- The bankruptcy of McDuck.
- aleph_0 rational points separate uncountably many irrational points such that never two are existing without rational between them.

But the matter with "number" n is even clearer than all the other nonsense.

> > but every natural number can be put at its place.
> Nope it can't. Hint n+1 =/= n.

n+1 is not a natural number too.

>
> > Let n ∈ ℕ, then n has a unique prime decomposition.
>
> Exactly (if n > 1).
> > You cannot know what this prime decomposition is.
> Right.
>

But for every natural number we know it.
>
> Right.

Regards, WM

[Greg Cunt]

On Saturday, September 18, 2021 at 5:56:43 PM UTC+2, WM wrote:

Let n e IN.

> > > n is not a concrete natural number [WM]

> > >
> > It's certainly a "concrete" natural number, though we don't KNOW which one.
> >
> For every natural number we know it.

Huh?! So you personally know ALL natural numbers? Wow - that's rather cool!

> Concrete means known.

Nope.

You may feel a coin in your pocket without knowing which one it is. This coin isn't "concret"?

> > Hint: n+1 =/= n.
> >
> n+1 is not a natural number too.

Oh, I stand corrected!

So n is in IN and n+1 is in IN, but neither n nor n+1 are natural numbers? Is that your claim? So what are they? Pink elephants?

[WM]

Greg Cunt schrieb am Samstag, 18. September 2021 um 20:20:00 UTC+2:
> On Saturday, September 18, 2021 at 5:56:43 PM UTC+2, WM wrote:
>
> Let n e IN.
> > > > n is not a concrete natural number [WM]
> > > >
> > > It's certainly a "concrete" natural number, though we don't KNOW which one.
> > >
> > For every natural number we know it.
> Huh?! So you personally know ALL natural numbers? Wow - that's rather cool!

I know, in principle, every number that you can mention. n does not belong to that set.

> > > Hint: n+1 =/= n.
> > >
> > n+1 is not a natural number too.
> Oh, I stand corrected!
>
> So n is in IN and n+1 is in IN, but neither n nor n+1 are natural numbers? Is that your claim? So what are they? Pink elephants?

They are placeholders. "Let n in |N, then P(n)" is an abbreviation for: "Any natural number can be put in place of n and will satisfy P."

Regards, WM

[Greg Cunt]

On Sunday, September 19, 2021 at 4:40:47 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Samstag, 18. September 2021 um 20:20:00 UTC+2:

> > Let n e IN.
> >

> n does not belong to that set.

To the set of all natural numbers, IN?

So we have n e IN, but in your psychotic kingdom, n !e IN at the same time?

Cool!

I mean, THAT would explain A LOT!

> > > > Hint: n+1 =/= n. [GC]
> > > >
> > > n+1 is not a natural number too. [WM]

See?!

> > So n is in IN and n+1 is in IN, but neither n nor n+1 are natural numbers? Is that your claim?

No answer?

> > So what are they? Pink elephants?
> >
> They are placeholders.

So placeholders are elements in IN?

Cool!

So if these placeholders aren't natural numbers, there are objects in IN which aren't natural numbers? THAT'S FASCINATING, MÜCKENHEIM!

"Let n in IN, then ...n..." is an abbreviation for <bla bla>

It's not a abbreviation of anything, dumbo. But It might be the beginning of a proof.

[Python]

Which in turns (following your "logic") is an abréviation for "Any
natural number can be put in place of n and will satisfy 'Any natural
number can be put in place of n and will satisfy'", ... Crank Wolfgang
Mueckenheim from Hochschule Augsburg.

[WM]

Greg Cunt schrieb am Sonntag, 19. September 2021 um 19:18:09 UTC+2:
> On Sunday, September 19, 2021 at 4:40:47 PM UTC+2, WM wrote:
> > Greg Cunt schrieb am Samstag, 18. September 2021 um 20:20:00 UTC+2:
>
> > > Let n e IN.
> > >
> > n does not belong to that set.
> To the set of all natural numbers, IN?

No. Every natural number has a unique prime decomposition. n has none.

>
> So we have n e IN, but

that is only an abbreviation for: A natural number is element of |N.

> > > So n is in IN and n+1 is in IN, but neither n nor n+1 are natural numbers? Is that your claim?
> No answer?

Of course, since it is fact.

> > They are placeholders.
> So placeholders are elements in IN?

No. They are place holders for elements of |N.

>
>
> It's not a abbreviation of anything

It is well-known that matheologians claim counterfactual nonsense like:

- All fractions can be enumerated.

- There are uncountably many real numbers which can be well-ordered although only countably many can be distinguished. (This amounts to: there are many even prime numbers, but they cannot be constructed.)
- There are more paths in the Binary Tree than nodes. (This amounts to: there are more houses than bricks.)
- The bankruptcy of McDuck. (He earns 1000 $ every day and spends only one, but in the limit he will be bancrupt.)

- aleph_0 rational points separate uncountably many irrational points such that never two are existing without rational between them.

- All endsegments are infinite but their intersection is empty.

They are easily recognizable as fools. But your claim schlägt dem Fass die Krone ins Gesicht.

> But It might be the beginning of a proof.

Yes, but the proof shall be valid for all natural numbers like 1, 2, 3, ..., not only for n.

Regards, WM

[zelos...@gmail.com]

It has as many as it does.

[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

[Serg io]

Extra Credit Problems!!: Prove or show these statements are false.


1 - All fractions can be enumerated.

2 - There are uncountably many real numbers which can be well-ordered although only countably many can be distinguished. (This amounts to: there are

many even prime numbers, but they cannot be constructed.)

3 - There are more paths in the Binary Tree than nodes. (This amounts to: there are more houses than bricks.)

4 - The bankruptcy of McDuck. (He earns 1000 $ every day and spends only one, but in the limit he will be bancrupt.)

5 - aleph_0 rational points separate uncountably many irrational points such that never two are existing without rational between them.

6 - All endsegments are infinite but their intersection is empty.

[Greg Cunt]

On Tuesday, September 21, 2021 at 12:57:12 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Sonntag, 19. September 2021 um 19:18:09 UTC+2:
> > On Sunday, September 19, 2021 at 4:40:47 PM UTC+2, WM wrote:
> > > Greg Cunt schrieb am Samstag, 18. September 2021 um 20:20:00 UTC+2:

Let n e IN.

> Every natural number has a unique prime decomposition. n has none.

Huh?! I mean... errr?

Since every natural number has a unique prime decomposition and n e IN (i.e. n is a natural number), n has a unique prime decomposition.

What's wrong with you, Mückenheim?

> > So we have
> >
> > n e IN
> >

> that is only an abbreviation for: A natural number is element of IN.

No, it says/expresses: "n is a natural number" (well, actually it says/expresses "n is an element in IN").

> > > > So n is in IN and n+1 is in IN, but neither n nor n+1 are natural numbers? Is that your claim?
> > > >
> > No answer?
> >
> Of course, since it is fact.

What is a fact?

That n isn't a natural number even though n e IN?

Are you SERIOUS?

> > > They are placeholders.
> > >
> > So placeholders are elements in IN?
> >

> No. They are place holders for elements of IN.

No, n and n+1 aren't "place holders for elements of IN", BUT _elements in IN_.

That's what the expressions

n e IN

and

n+1 e IN

"say" or "express", namely:

n is an element in IN

and

n+1 is an element in IN .

> > It is well-known that matheologians claim counterfactual nonsense

In this case you clearly qualify for a matheologian.

> They are easily recognizable as fools.

Indeed! :-)

Again: 'Let n e IN.'

> > might be the beginning of a proof.
> >
> Yes, but the proof shall be valid for all natural numbers like 1, 2, 3, ..., not only for n.

Holy shit! :-(

[Serg io]

whoa, that last one is a keeper!

Let n e IN
n = 1 + 2
therefore n = 3

however, "proof shall be valid for all natural numbers like 1, 2, 3, ..., not only for n"

[Jim Burns]

On 9/21/2021 6:57 AM, WM wrote:
> Greg Cunt schb

> am Sonntag, 19. September 2021 um 19:18:09 UTC+2:
>> On Sunday, September 19, 2021 at 4:40:47 PM UTC+2,
>> WM wrote:
>>> Greg Cunt schrieb
>>> am Samstag, 18. September 2021 um 20:20:00 UTC+2:

>>>> Let n e IN.
>>>
>>> n does not belong to that set.

n is in N if and only if
n can be counted to in principle..

>> To the set of all natural numbers, IN?
>
> No.
> Every natural number has a unique prime decomposition.
> n has none.

We know some things about n. We don't know other things.
We know n has a unique prime factorization.
We don't know what its UPF is.

We know that,
if P(n),
then a _first_ n1 exists,
P(n1) and, for each k < n1, ~P(k)

We know that,
if non-trivial n doesn't have a UPF,
then a _first_ non-trivial n1 exists such that
n1 doesn't have a UPF, and
for each non-trivial k < n1, k has a UPF.

We know that non-trivial n1 has a non-trivial factor, n1.
So, we know n1 has a _first_ non-trivial factor, j.

We know that j does not have any non-trivial factor i < j
If j did, i would also be a non-trivial factor of n1,
and j would not be first.
So, we know j is a prime factor of n1.

For some k, j*k = n1.
k < n1, so k has a UPF
j has a UPF, which is j.
n1 = j*k has a UPF.
But n1 is the first non-trivial without a UPF.
Contradiction.

Therefore,
we know that non-trivial n has a UPF.

We don't know what the UPF is that n has,
but we don't need to know what the UPF is
in order to know n has it. See above.

In some ways, knowing n has a UPF _without_ knowing
what n is is better.

What is the UPF of floor(pi*(10^(10^(10^(10^(10))))))?
I don't know, and I'm not sure that it's physically
possible to calculate.

Does floor(pi*(10^(10^(10^(10^(10)))))) have a UPF?
Yes. That's trivial to show. I just did, in a paragraph.

----
Also, having a UPF is not part of the _definition_ of an
element of N.

The _definition_ is that n can be counted to in principle.
More facts follow from the definition.
N is well-ordered.
(i*j)*k = i*(j*k)
for non-trivial i,j,k, i*j = k -> i < k and j < k
And from those facts, more facts,
like that n has a UPF.

[Greg Cunt]

Right. And since n was an arbitrary natural number, we get that for all x e IN: x = 3.

Well done, Serg io! (Only slightly worse than Mückenheim.)

[Serg io]

Thank you! Quite an improvement! and now only slightly (within ϵ ?) within range of the Professor !

[WM]

Serg io schrieb am Dienstag, 21. September 2021 um 15:34:58 UTC+2:
> On 9/21/2021 5:57 AM, WM wrote:

> 6 - All endsegments are infinite but their intersection is empty.

All endsegments which have an infinite intersection with all endsegments have an empty intersection with each other.

Regards, WM

[WM]

Greg Cunt schrieb am Dienstag, 21. September 2021 um 17:13:38 UTC+2:
> On Tuesday, September 21, 2021 at 12:57:12 PM UTC+2, WM wrote:

> Let n e IN.
> > Every natural number has a unique prime decomposition. n has none.

> Since every natural number has a unique prime decomposition and n e IN (i.e. n is a natural number), n has a unique prime decomposition.

Specify it. That can be done for every natural number less than 1 billion. Or is n larger?

Regards, WM

[WM]

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?

> 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. You don't. Therefore n is not a natural number.

Regards, WM

[Jim Burns]

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.

>> 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.

> You don't.
> Therefore n is not a natural number.

n can be counted to in principle.

Therefore, n is a natural number.

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.

No doubt, for your next trick, you will "disprove" the
Pythagorean theorem by changing "triangle" to include
four-cornered figures.

[Greg Cunt]

Look, you silly asshole:

There are some coins in your pocket. You put your hand in your pocket and just grab one (not pulling it out).

| For a coin you would know what it is. You don't. Therefore the thing in your hand is not a coin.

Holy shit!

[Greg Cunt]

On Wednesday, September 22, 2021 at 9:10:39 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Dienstag, 21. September 2021 um 17:13:38 UTC+2:
> > On Tuesday, September 21, 2021 at 12:57:12 PM UTC+2, WM wrote:
> >
> > Let n e IN.
> >
> > > Every natural number has a unique prime decomposition. n has none.
> >
> > Since every natural number has a unique prime decomposition and n e IN (i.e. n is a natural number), n has a unique prime decomposition.

DID YOU GET THAT, ASSHOLE?!

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.

So what?

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

So this is certainly not a sensible criterium.

Hint: There are many "precisely specified" natural numbers, the prime decomposition of which we do not know (yet). :-)

Please specify the unique prime decomposition for:

2519590847565789349402718324004839857142928212620403202777713783604366202070
7595556264018525880784406918290641249515082189298559149176184502808489120072
8449926873928072877767359714183472702618963750149718246911650776133798590957
0009733045974880842840179742910064245869181719511874612151517265463228221686
9987549182422433637259085141865462043576798423387184774447920739934236584823
8242811981638150106748104516603773060562016196762561338441436038339044149526
3443219011465754445417842402092461651572335077870774981712577246796292638635
6373289912154831438167899885040445364023527381951378636564391212010397122822
120720357

See: https://en.wikipedia.org/wiki/RSA_numbers#RSA-2048

[Greg Cunt]

On Wednesday, September 22, 2021 at 9:14:35 PM UTC+2, WM wrote:

Oh, really?

Please specify the unique prime decomposition for RSA-2048:

2519590847565789349402718324004839857142928212620403202777713783604366202070
7595556264018525880784406918290641249515082189298559149176184502808489120072
8449926873928072877767359714183472702618963750149718246911650776133798590957
0009733045974880842840179742910064245869181719511874612151517265463228221686
9987549182422433637259085141865462043576798423387184774447920739934236584823
8242811981638150106748104516603773060562016196762561338441436038339044149526
3443219011465754445417842402092461651572335077870774981712577246796292638635
6373289912154831438167899885040445364023527381951378636564391212010397122822
120720357

For a natural number you would know what it is. You don't. Therefore RSA-2048 (the decimal expansion of which is given above) is not a natural number.

See: https://en.wikipedia.org/wiki/RSA_numbers#RSA-2048

[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

[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

[Serg io]

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
>

[Jim Burns]

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.

[Greg Cunt]

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").

[WM]

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

[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

[Jim Burns]

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.

[FromTheRafters]

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?

[Greg Cunt]

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.

[zelos...@gmail.com]

It has as many as it does. That is how many it has.

Stop your idiocy for fuck sake.

[zelos...@gmail.com]

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

[WM]

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

[FromTheRafters]

After serious thinking WM wrote :

n.000...

[Greg Cunt]

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

[Serg io]

(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. )

[WM]

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

[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

[WM]

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

Regards, WM

[WM]

FromTheRafters schrieb am Freitag, 24. September 2021 um 16:01:20 UTC+2:
> After serious thinking WM wrote :

> > 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...

Decimal means digits 0 to 9.

Regards, WM

[WM]

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:

> if I say n is a natural number, it is a natural number.

If you say you have proved the Riemann hypothesis, then you have proved it? If you say that set theory is mathematics, then it is mathematics? You need not say that you are a fool.

Regards, WM

[Greg Cunt]

On Friday, September 24, 2021 at 6:39:51 PM UTC+2, WM wrote:
> 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.

n does not need to have a KNOWN prime factorization.

Hint: The Schnirelmann constant, which certainly is a natural number, doesn't have a KNOWN prime factorization either, you silly idiot!

[Greg Cunt]

On Friday, September 24, 2021 at 6:43:02 PM UTC+2, 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:
> >

> > If

Zelos defines /n/ to be

> > a[n arbitrary] natural number, it is a natural number

in that [i.e. his] context.

Don't you get that you silly crank?

[FromTheRafters]

WM formulated the question :

Yes, and the fractional part of the expansion uses zeros.

How about using nines:

(n-1).999...

for *another* CDE representation of the same exact number for n > zero.

[FromTheRafters]

WM was thinking very hard :

You are unaware of what a statement is?

You are the fool.

[Transfinity]

Contrary to n which is no number with certainty.

Regards, WM

[FromTheRafters]

WM used his keyboard to write :

Non sequitur. I'm just saying that natural numbers have a unique prime
factorization, but only "composite numbers" can have "decompositions".

[Transfinity]

Greg Cunt schrieb am Freitag, 24. September 2021 um 18:55:47 UTC+2:
> On Friday, September 24, 2021 at 6:43:02 PM UTC+2, 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:
> > >
> > > If
>
> Zelos defines /n/ to be
>
> > > a[n arbitrary] natural number, it is a natural number
>
> in that [i.e. his] context.

Nonsense. n is neither a number nor a circle or a cube. It can denote a number or something else if this is uniquely defined.

Regards, WM

[Jim Burns]

On 9/24/2021 3:17 PM, Transfinity wrote:
> Greg Cunt schrieb
> am Freitag, 24. September 2021 um 18:55:47 UTC+2:
>> On Friday, September 24, 2021 at 6:43:02 PM UTC+2,
>> WM wrote:
>>> zelos...@gmail.com schrieb
>>> am Freitag, 24. September 2021 um 07:08:11 UTC+2:

>>>> If
>>
>> Zelos defines /n/ to be
>>
>>>> a[n arbitrary] natural number, it is a natural number
>>
>> in that [i.e. his] context.
>
> Nonsense.
> n is neither a number nor a circle or a cube.

It's possible that n is either, neither, or both.

Which case holds might be discoverable later.
Or it might not be.

> It can denote a number or something else
> if this is uniquely defined.

Your requirement that n transform from placeholder to number
when certain information about n is revealed
is incoherent.

Consider three mathematicians, Ingrid, Janice, and Kathleen.
Each one has different information about i,j,k.
Ingrid knows that i + 2j + 3k = 14.
Janice knows that 2i + 3j + k = 11.
Kathleen knows that 3i + j + 2k = 11.

To Ingrid, i,j,k are not numbers?
To Janice, i,j,k are not numbers?
To Kathleen, i,j,k are not numbers?

*UNLESS* the mathematicians compare notes.
Then, i,j,k are uniquely identified,
and, *POOF*, they turn into numbers.

This raises a host of questions.
What are the implications of special relativity to
the POOF? Suppose Ingrid, Janice and Kathleen are
in motion relative to one another and the placeholders
being POOFed into numbers are in the Andromeda Galaxy?

Whose reference frame here determines, way out there,
at what local time the placeholders are POOFed?
Don't you (WM) claim placeholders and numbers must
have a physical reality? Then there must be an answer.
What could it be?

----
Your placeholders are a pointless complication.
Unless the point is to complicate, in order to hide
your ignorance. I think that is the point.

[Serg io]

no, in math we say "let n be a natural number" that means that n is a natural number.

then in math we say "n = 1 + 1", therefore n is 2

n is called a "variable",

and since we said "let n be a natural number", we know throughout the problem that n is a natural number.


factorization is diversion.

[Serg io]

you POOFed WM's SPOOF

[Serg io]

wrong, you have it backwards again.

The definition of a natural number is not in any way dependent upon prime factorization.

your factorization is your intentional diversion, like Framed Achilles and that poor turtle, you keep tossing out red herrings to obscure your very poor
math, but that does not work at all. ZOD says that JG has better math skills than WM.

>
> Regards, WM
>

[Jim Burns]

> you POOFed WM's SPOOF

relativistically poofed placeholder ants

[Chris M. Thomasson]

Good one! :^D

[WM]

Jim Burns schrieb am Freitag, 24. September 2021 um 22:36:54 UTC+2:
> On 9/24/2021 3:17 PM, Transfinity wrote:

> > n is neither a number nor a circle or a cube.
> It's possible that n is either, neither, or both.

Of course. If n is defined to denote 3, then n is a number.

>
> > It can denote a number or something else
> > if this is uniquely defined.
> Your requirement that n transform from placeholder to number
> when certain information about n is revealed
> is incoherent.

It is fact.

>
> Consider three mathematicians, Ingrid, Janice, and Kathleen.
> Each one has different information about i,j,k.
> Ingrid knows that i + 2j + 3k = 14.
> Janice knows that 2i + 3j + k = 11.
> Kathleen knows that 3i + j + 2k = 11.
>
> To Ingrid, i,j,k are not numbers?
> To Janice, i,j,k are not numbers?
> To Kathleen, i,j,k are not numbers?

Yes.

>
> *UNLESS* the mathematicians compare notes.
> Then, i,j,k are uniquely identified,
> and, *POOF*, they turn into numbers.

Yes.

>
> This raises a host of questions.
> What are the implications of special relativity to
> the POOF? Suppose Ingrid, Janice and Kathleen are
> in motion relative to one another and the placeholders
> being POOFed into numbers are in the Andromeda Galaxy?
>
> Whose reference frame here determines, way out there,
> at what local time the placeholders are POOFed?

Interesting questions. They do not change the fact that n with no further specifications than n ∈ ℕ is not a number.

> Don't you (WM) claim placeholders and numbers must
> have a physical reality? Then there must be an answer.
> What could it be?

The possible numbers represented by n can be all or some or only one. In this case n represents a number.

Regards, WM

[Serg io]

wrong-O,

did you know that "n ∈ ℕ" says "n is a element of the set of natural numbers" ?

[FromTheRafters]

WM formulated on Saturday :

Elements of the set of natural numbers are not numbers? No wonder your
matheology fails.

[Jim Burns]

On 9/25/2021 4:39 PM, WM wrote:
> Jim Burns schrieb
> am Freitag, 24. September 2021 um 22:36:54 UTC+2:
>> On 9/24/2021 3:17 PM, Transfinity wrote:

>>> n is neither a number nor a circle or a cube.
>> It's possible that n is either, neither, or both.
>
> Of course.
> If n is defined to denote 3, then n is a number.

If n can be counted to in principle,
then n is a natural number.

If n cannot be counted to in principle,
then n is not a natural number.

n can be counted to in principle iff
the FISON {0,...,n} from 0 to n exists.

{0,...,n} is the FISON from 0 to n iff
{0,...,n} is a collection
with a transitive and connected order '<' such that
{0,...,n} begins at 0, ends at n, and
for each _split_ BEFORE,AFTER of {0,...,k},
a _crossing-pair_ j,j+1 exists.

j+1 is the successor of j
Each successor has a unique successor
0 has a unique successor.
0 is not a successor.
(j+1 = k+1) iff (j = k)

>>> It can denote a number or something else
>>> if this is uniquely defined.

In the example below, i,j,k did not change.
To suggest that they changed is bizarre and incoherent.
They are not physical.

Ingrid, Janice, and Kathleen changed.
Their knowledge of i,j,k changed from
(individually) not enough to determine i,j,k to
(jointly) enough to determine i,j,k.

>> Consider three mathematicians, Ingrid, Janice, and Kathleen.
>> Each one has different information about i,j,k.
>> Ingrid knows that i + 2j + 3k = 14.
>> Janice knows that 2i + 3j + k = 11.
>> Kathleen knows that 3i + j + 2k = 11.
>>
>> To Ingrid, i,j,k are not numbers?
>> To Janice, i,j,k are not numbers?
>> To Kathleen, i,j,k are not numbers?
>
> Yes.
>
>> *UNLESS* the mathematicians compare notes.
>> Then, i,j,k are uniquely identified,
>> and, *POOF*, they turn into numbers.
>
> Yes.
>
>> This raises a host of questions.
>> What are the implications of special relativity to
>> the POOF? Suppose Ingrid, Janice and Kathleen are
>> in motion relative to one another and the placeholders
>> being POOFed into numbers are in the Andromeda Galaxy?
>>
>> Whose reference frame here determines, way out there,
>> at what local time the placeholders are POOFed?
>
> Interesting questions.
> They do not change the fact that
> n with no further specifications than n ∈ ℕ is not a number.

We reason about n starting from the fact that
n can be counted to in principle.
Imagine, for the sake of argument that, instead of
a natural number, n was a hyper-intelligent shade of blue
or n was a flying rainbow sparkle pony *AND*
n could be counted to in principle.

Then nothing of significance has changed.
No argument from n being able to be counted to in principle
needs to change one iota.

[Greg Cunt]

On Friday, September 24, 2021 at 10:36:54 PM UTC+2, Jim Burns wrote:
> On 9/24/2021 3:17 PM, Transfinity wrote:

> > n is neither a number nor a circle or a cube.

Depends on the definition of /n/.

> It's possible that n is either, neither, or both.

I beg to differ: n can't be "both" (in one and the same context). I mean, if n is a (natural) number (due to von Neumann), it certainly won't be a circle or a cube, and if it is a circle or a cube, it certainly won't be a number (due to von Neumann).

> Your placeholders are a pointless complication.
> Unless the point is to complicate, in order to hide
> your ignorance. I think that is the point.

He's mixing up (1) n with "n" and (2) "n" used/adopted as a /constant/ with "n" used/adopted as a variable.

Lit. concerning (1): https://en.wikipedia.org/wiki/Use%E2%80%93mention_distinction

(2):
(a) Let n be a natural number. << Here "n" is a /constant/ (used in a certain context ("namespace"), usually a proof).
(b) For all n e IN: n >= 0. << Here "n" is a (bound) variable (WM: "placeholder")

[Greg Cunt]

On Saturday, September 25, 2021 at 11:39:13 PM UTC+2, Serg io wrote:

> Did you know that "n ∈ ℕ" says "n is a element of the set of natural numbers" ?

He's just too dumb to get that.

[Greg Cunt]

On Saturday, September 25, 2021 at 10:40:03 PM UTC+2, WM wrote:

> [It's a] fact that n with no further specifications than ["Let] n ∈ ℕ["] is not a number.

Oh really?! On the other hand "n ∈ ℕ" is just a formal expression for "n is an element in IN". And since IN is the set of all natural numbers, that just means "n is a natural number". (You may look that up in Halmos' "Naive Set Theory".)

You see, if I state (in a mathematical text): "Let n be a natural number." THEN n IS a natural number (in that context). In other words, then "n" refers to a natural number.

Hint: You are mixing up (1) n with "n" and (2) "n" used/adopted as a /constant/ with "n" used/adopted as a variable.

Lit. concerning (1): https://en.wikipedia.org/wiki/Use%E2%80%93mention_distinction

(2):
(a) Let n be a natural number. << Here "n" is a /constant/ (used in a certain context ("namespace"), usually a proof).

(b) For all n e IN: n >= 0. << Here "n" is a (bound) variable (your term: "placeholder")

[Greg Cunt]

On Saturday, September 25, 2021 at 4:58:58 AM UTC+2, Serg io wrote:

> in math we say "let n be a natural number" that means that n is a natural number.

Right.

> n is called a "variable",

Nope. At least not IN THIS context.

Actually, "n" in math is used for both purposes, i.e. as a constant or a variable. Not really "clean" (from a logcal point of view), but that's math (in practice).

Again,

(a) "Let n be a natural number." << Here "n" is a /constant/ (used in a certain context ("namespace"), usually a proof).

(b) "For all n e IN: n >= 0." << Here "n" is a (bound) variable (WM: "placeholder")

> and since we said "let n be a natural number", we know throughout the problem that n is a natural number.

Exactly. From a technical point of view "Let n be a natural number" is a "definition" which introduces the constant "n". (In a formal ND proof "n" might be called an "arbitrary name", due to Lemmon.)

[Greg Cunt]

On Sunday, September 26, 2021 at 12:12:40 AM UTC+2, FromTheRafters wrote:
> WM formulated on Saturday :
> >
> > [It's a] fact that n with no further specifications than n ∈ ℕ is not a number.
> >
> Elements of the set of natural numbers are not numbers? [...]

Note that "n+1 is not a natural number too" [WM].

[Jim Burns]

On 9/25/2021 8:51 PM, Greg Cunt wrote:
> On Friday, September 24, 2021 at 10:36:54 PM UTC+2,
> Jim Burns wrote:
>> On 9/24/2021 3:17 PM, Transfinity wrote:

>>> n is neither a number nor a circle or a cube.
>
> Depends on the definition of /n/.
>
>> It's possible that n is either, neither, or both.
>
> I beg to differ: n can't be "both" (in one and the same context).
> I mean, if n is a (natural) number (due to von Neumann),
> it certainly won't be a circle or a cube, and if it is a circle
> or a cube, it certainly won't be a number (due to von Neumann).

Hooo-boy. My bad.
I wasn't paying enough attention. I thought it said
"n is either a number or a square or a cube."

( I thought that I was actually being a bit _clever_ by
( mentioning numbers that are both squares and cubes.
( Such as sixth powers.
( Oops.

One point, though.

In the context of geometric figures, yes,
if it is circle or a cube,then it's not a number.
However, in the context of arithmetic operations,
it could be both a cube and a number.

A very small point, almost entirely useless except to
underline the importance of context.

[Jim Burns]

On 9/24/2021 12:42 PM, 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, it is a natural number.
>
> If you say you have proved the Riemann hypothesis,
> then you have proved it?
> If you say that set theory is mathematics,
> then it is mathematics?
> You need not say that you are a fool.

| n is a natural number

is merely a better way of saying
| That is a natural number.

Suppose I say
| That is a natural number.
Are you going to object and say
| No! "That" is a pronoun!

Well, maybe _you_ will.

The sort of work "n" does for us can be done without variables.
Before they were developed , it _was_ done without variables.
However, it's done better (more clearly, more easily) _with_
variables.

For example.

When Pierre de Fermat wrote his Last Theorem in the margin of
Diophantus's _Arithmetica_ variables were in development
in Europe at that time. What he wrote was

| Cubum autem in duos cubos, aut quadratoquadratum in duos
| quadratoquadratos & generaliter nullam in infinitum ultra
| quadratum potestatem in duos eiusdem nominis fas est dividere

In English,

| It is impossible to separate a cube into two cubes, or a
| fourth power into two fourth powers, or in general, any power
| higher than the second, into two like powers.

This is a much better way to express that:

| An > 2, ~Ex,y,z: x^n + y^n = z^n

"Cubum autem in duos cubos" refers to natural numbers.
n,x,y,z are used to say the same as the Latin.
They also refer to natural numbers.

"That" is a pronoun.
That is a natural number, when I refer to a natural number.

"n" is variable name, or a placeholder, or call it what you like.
n is a natural number, if I say it is.

[Greg Cunt]

On Sunday, September 26, 2021 at 7:14:45 PM UTC+2, Jim Burns wrote:

> The sort of work "n" does for us can be done without variables.

Indeed, espercially, when "n" is used as a constant.

Example: "Let n be a natural number."

[This amounts to a introduction of a constant, namely, "n".]

Hint: In Hilbert's epsilon calculus there's actually an operator which could be used in a FORMAL definition introducing "n":

n = epsilon x(x e IN) .

See: https://en.wikipedia.org/wiki/Epsilon_calculus

On the other hand, HERE

An e IN: n + n = 2*n

"n" is indeed a variable.

> "n" is variable [...], or a placeholder, or call it what you like.

> n is a natural number, if I say it is.

Indeed.

[Jim Burns]

On 9/24/2021 12:37 PM, WM wrote:
> 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.

<WM>
> My definition:
> Any natnumbers must have its UPF known in principle
> or knowable.

Bullshit.
A natural number has a unique prime factorization,
whether it's known, whether it's not known,
whether it can be known, whether it cannot be known.

>> 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.

Try to keep up.

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.

( 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.

[Greg Cunt]

On Sunday, September 26, 2021 at 7:44:50 PM UTC+2, Jim Burns wrote:
> On 9/24/2021 12:37 PM, WM wrote:
> >
> > some fools claim that the expression "natural number" is a natural number.

Really? Who are these fools?

Clearly, WM is a fool himself.

Though HE claims that the natural number n is a "placeholder". :-)

Assuming that IN only contains natural numbers as elements, the question arises how an n with n e IN can be a placeholder. Are some natural numbers placeholders?