Repeating decimals are irrational (2)

[wij]

If 0.999... is rational, then:
0.999....= p/q (p,q∈ℕ)
<=> 0.999...*q=p
If 0.999...∈ℕ, there exist q∈ℕ such that 0.999...*q∈ℕ
since 0.999... is defined as infinite repeating, and q is finite,
but 0.999...*q is never finite.
Conclusion: Repeating decimals are irrational.

I am kind of reluctant to raise this issue again, but many problems would lead
to a basic issue: what is infinity. It is simple as it merely means "never end".
In this interpretation, 'time'/'steps' is meant existent in algorithms and lies hidden
in math. concept. E.g. Is infinity a natural numbers?
Think about that the Peano model does not reject infinity explicitly. I think
this is the root of many confusions (and errors in math.)

An article is published for this idea.
https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
The article is not aimed to explain the infinity idea (I added some to find out
adding too many such wordings would pollute the original intent)

[Ben Bacarisse]

wij <wyni...@gmail.com> writes:

> If 0.999... is rational, then:
> 0.999....= p/q (p,q∈ℕ)
> <=> 0.999...*q=p

Yes. For example 0.999... = 1/1.

> If 0.999...∈ℕ, there exist q∈ℕ such that 0.999...*q∈ℕ
> since 0.999... is defined as infinite repeating, and q is finite,
> but 0.999...*q is never finite.

0.999...*q is finite for all q in N.

> I am kind of reluctant to raise this issue again,

Really? Where would we be without someone denying (or not knowing) that
... denotes a limit every few weeks?

--
Ben.

[wij]

On Monday, 8 August 2022 at 21:16:39 UTC+8, Ben Bacarisse wrote:
> wij <wyni...@gmail.com> writes:
>
> > If 0.999... is rational, then:
> > 0.999....= p/q (p,q∈ℕ)
> > <=> 0.999...*q=p
> Yes. For example 0.999... = 1/1.

I don't see valid logical derivation.
By definition "repeating decimal" (0.999...) never ends.
"0.999..." and "1/1" is apparently different. So, what is the derivation?
We have gone through this a lot. Nothing from the limit theory is valid (this
is a very fundamental arithmetic problem, any 'higher' level theory will suffer
circular argument problem).

> > If 0.999...∈ℕ, there exist q∈ℕ such that 0.999...*q∈ℕ
> > since 0.999... is defined as infinite repeating, and q is finite,
> > but 0.999...*q is never finite.
> 0.999...*q is finite for all q in N.

ditto. I don't see valid logical derivation.

> > I am kind of reluctant to raise this issue again,
> Really? Where would we be without someone denying (or not knowing) that
> ... denotes a limit every few weeks?
>
> --
> Ben.

I would say it's Emperor's clothes. Innocent child can see it.
You can keep teaching a false thing.

[Ben Bacarisse]

wij <wyni...@gmail.com> writes:

> On Monday, 8 August 2022 at 21:16:39 UTC+8, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>> > If 0.999... is rational, then:
>> > 0.999....= p/q (p,q∈ℕ)
>> > <=> 0.999...*q=p
>> Yes. For example 0.999... = 1/1.
>
> I don't see valid logical derivation.

That's to be expected. I don't think you know what 0.999... means.

> By definition "repeating decimal" (0.999...) never ends.

Yes, it represents a limit. (Technically, it is a never ending sequence
of partial sums, whose limit is the sum.)

> "0.999..." and "1/1" is apparently different.

Yes, and "2/2" and "1+0" and "001" are all apprently different too. But
all denote the same number: 1.

> So, what is the derivation?

0.999... means

Sum n=1 to oo [9/10^n]
= limit n->oo [Sum i=1,n [9/10^i]]
= 1

(Do you know how limits are calculated?)

> We have gone through this a lot.

Yes.

> Nothing from the limit theory is valid (this
> is a very fundamental arithmetic problem, any 'higher' level theory will suffer
> circular argument problem).

Then the notation 0.999... is invalid since it denotes a limit. Don't
write limits if you don't consider them valid.

Please don't play the classic crank game where you use conventional
notation to mean something vague that is not what the notation normally
means. That's so boring:

"The sun in made of cheese!", you say. And after a few rounds it turns
out that dairy products are invalid, so "cheese" must mean something
else.

> I would say it's Emperor's clothes.

Curious that. I am clothed in (a) a definition of what X.YYY... means;
(b) a consistent theory that defines the resulting limits.

You don't even have a definition of what the notation you are misusing
means. One of us certainly naked!

--
Ben.

[Pancho Valvejob]

On 8/8/2022 5:27 AM, wij wrote:
> If 0.999... is rational, then:
> 0.999....= p/q (p,q∈ℕ)
> <=> 0.999...*q=p
> If 0.999...∈ℕ, there exist q∈ℕ such that 0.999...*q∈ℕ
> since 0.999... is defined as infinite repeating, and q is finite,
> but 0.999...*q is never finite.
> Conclusion: Repeating decimals are irrational.

No. Repeating decimals are rational.

[Paul N]

On Monday, August 8, 2022 at 2:37:59 PM UTC+1, wyni...@gmail.com wrote:
> On Monday, 8 August 2022 at 21:16:39 UTC+8, Ben Bacarisse wrote:
> > wij <wyni...@gmail.com> writes:
> >
> > > If 0.999... is rational, then:
> > > 0.999....= p/q (p,q∈ℕ)
> > > <=> 0.999...*q=p
> > Yes. For example 0.999... = 1/1.
> I don't see valid logical derivation.
> By definition "repeating decimal" (0.999...) never ends.
> "0.999..." and "1/1" is apparently different. So, what is the derivation?
> We have gone through this a lot. Nothing from the limit theory is valid (this
> is a very fundamental arithmetic problem, any 'higher' level theory will suffer
> circular argument problem).
> > > If 0.999...∈ℕ, there exist q∈ℕ such that 0.999...*q∈ℕ
> > > since 0.999... is defined as infinite repeating, and q is finite,
> > > but 0.999...*q is never finite.
> > 0.999...*q is finite for all q in N.
> ditto. I don't see valid logical derivation.

I think you may be confusing "finite", meaning the size of the number, with "infinite repeating" where the calculation to determine the number does not stop.

0.999... is clearly more than 0, and clearly less than 2. 0 and 2 are both finite, so 0.999... is also finite.

Likewise, taking q=3 as an example, 3 * 0.999... > 0 and 3 * 0.999... < 6, so 3 * 0.999... is finite.

[Richard Damon]

On 8/8/22 8:27 AM, wij wrote:
> If 0.999... is rational, then:
> 0.999....= p/q (p,q∈ℕ)
> <=> 0.999...*q=p
> If 0.999...∈ℕ, there exist q∈ℕ such that 0.999...*q∈ℕ
> since 0.999... is defined as infinite repeating, and q is finite,
> but 0.999...*q is never finite.
> Conclusion: Repeating decimals are irrational.

Wrong, 9/9 can generate the pattern 0.9999...

This can be proven by 9 = 3+3+3 and the assocative property thus 9/9 =
(3+3+3)/9 = 3/9 + 3/9 + 3/9 = 0.3333... + 0.3333... + 0.3333... =
0.9999....

[wij]

To add more material came up to me (not well ordered):

----------------------------
There are quite a number of proofs of "repeating decimals are irrational".
The basic is the correct equation of 1/3 and its decimal form from long
division (kids understand this 'infinity' with no problem) should be:

1/3= 0.333... + nonzero_remainder.

----------------------------
To translate the 0.999... problem to limit:

Let A= lim(n->∞) 1-1/2^n = 0.999...
B= lim(n->∞) 1-1/10^n = 0.999...

Assume A=B
<=> lim(n->∞) 1-1/2^n = lim(n->∞) 1-1/10^n
<=> lim(n->∞) 1/2^n = lim(n->∞) 1/10^n
<=> lim(n->∞) 1 = lim(n->∞) 1/5^n
<=> 1=0

[Note] I just demonstrate an instance. The limit theory can evolve as it does
(e.g. one-sided limit... There are many slightly different versions of
interpretation of limit as it evolves). Readers might find different
authors use different rules.
Limit is a technic to find its 'limit', it cannot form a logically
consistent theory for real number, e.g. the result of limit in general
must be verified, e.g. numerically, one cannot absolutely trust the
result of limit arithmetic. And at final, lim(x->c) f(c)= L does not
'deduce' f(c)=L (In text book, probably just reads "lim(x->c) f(c)= L, SO
WRITTEN as f(c)=L"). Limit theory only says the limit of 0.999... is 1,
the theory does not say 0.999...=1. There is no equality concept in the
ε-δ theory.
If one resorts to Dedekind-cut-like theories (I did not really read it),
from the knowledge that all the combinations of discrete symbols cannot
represent all the real numbers, I can conclude what those theories
claim are false, let alone I suspect there should be circular arguments
there, because many terms there must be well defined as a fundamental
theory, are undefined (prove me wrong).

The limit example above demonstrated "0.999..." cannot denote a specific number,
which also means "repeating decimal" cannot specify a unique number (A!=B).
Using limit is invalid for me (for this question) but the result is correct,
see the provided reference (I found a typo there).

-----------------------
Simple arithmetic (this should also be a valid way 2.718... is calculated):
(0.999....)^n approaches 1/e
(1.000...1)^n approaches e (or defined as e)
A possible rebuttal might be that the (1-1/n) in lim(n->∞) (1-1/n)^n is an invalid
number (approximated like 0.999...), or it is a 'concept' etc...
But if it is not a number, the whole equation is broken.

-----------------------
A[0]=0
A[n]=(A[n-1]+1)/2

The density property says (implicitly) n can enumerate infinitely (otherwise, it
won't be a rule) and A[∞] never be 1. A[n] infinitely approaches 1 in form
like 0.999.... This is like in the case of the interval [0,1), infinite numbers
of 0.999...s are located near the open end of [0,1).
Can we infinitely refine the scale of a ruler and the last scale never touches
the scale of 1? I think, yes, something like the √2 story, otherwise all numbers
can be 'proved' rational.

[Ben Bacarisse]

wij <wyni...@gmail.com> writes:

> To add more material came up to me (not well ordered):
>
> ----------------------------
> There are quite a number of proofs of "repeating decimals are
> irrational".

I've not seen any. Are any of them published (you know what I mean
here -- properly published) so I can read them clearly presented?

> The basic is the correct equation of 1/3 and its decimal form from long
> division (kids understand this 'infinity' with no problem) should be:
>
> 1/3= 0.333... + nonzero_remainder.

Kids are not very good at mathematics. They tend to generalise without
justification. There is no r =/= 0 such that 1/3 = 0.333... + r.

This is because 0.333... means

lim(n->oo) Sum k=1,n (3/10^k)

which limit is 1/3. Proof of X by "kids think X" is not a sound
argument.

> ----------------------------
> To translate the 0.999... problem to limit:
>
> Let A= lim(n->∞) 1-1/2^n = 0.999...
> B= lim(n->∞) 1-1/10^n = 0.999...

Or, more simply, A=B=1.

> Assume A=B

Or, if you like, one can prove that A=B.

> <=> lim(n->∞) 1-1/2^n = lim(n->∞) 1-1/10^n
> <=> lim(n->∞) 1/2^n = lim(n->∞) 1/10^n
> <=> lim(n->∞) 1 = lim(n->∞) 1/5^n

You can't do basic algebra. For pity's s, just pick up a maths book! I
presume you think you are multiplying both sizes by 2, yes? If so...

2 * lim(n->oo) 1/2^n = lim(n->oo) 2/2^n = lim(n->oo) 2/2^(n-1) = 0

> <=> 1=0

so no.

> Simple arithmetic (this should also be a valid way 2.718... is calculated):
> (0.999....)^n approaches 1/e

No it doesn't. Basic algebra again.

> (1.000...1)^n approaches e (or defined as e

Now this one in unclear. Usually, the ... in the middle like that
denote some constant number of repetitions. If that's what you mean,
then no. For any fixed x=1.000...1, x^n diverges.

> A[0]=0
> A[n]=(A[n-1]+1)/2
>
> The density property says (implicitly) n can enumerate infinitely
> (otherwise, it won't be a rule) and A[∞] never be 1.

This is literal nonsense. Inductive definitions like that for A define
A for all N (and no more). oo is not in N.

> A[n] infinitely approaches 1 in form like 0.999....

Yes, and ∀n A[n] < 1.

Where do you get this stuff? Is there some sort of Info Wars for maths
out there that I've missed?

--
Ben.

[Andy Walker]

On 09/08/2022 13:07, wij wrote:
[Wij still:]
>> I am kind of reluctant to raise this issue again, [...].

Not reluctant enough, apparently!

> There are quite a number of proofs of "repeating decimals are irrational".

There are no /correct/ proofs of that.

> The basic is the correct equation of 1/3 and its decimal form from long
> division (kids understand this 'infinity' with no problem) should be:
> 1/3= 0.333... + nonzero_remainder.

"Kids" perhaps understand "1/3 == 0.333..." where "..." in their
minds means "and so on". They don't, in general, have an understanding
of limits, infinitesimals, non-standard analysis and other relevant
material. If you're going to perpetuate your "correct equation", then
you need at least to understand what you and/or others mean by "...".
It is normally taken, in this context, to mean the limit, in which case
there is no "non-zero remainder". If you want it to mean something else,
then what? You can't legitimately mix "Wij numbers" [ill-defined, thus
far] with the usual [standard] "real" numbers.

> To translate the 0.999... problem to limit:
> Let A= lim(n->∞) 1-1/2^n = 0.999...
> B= lim(n->∞) 1-1/10^n = 0.999...
> Assume A=B
> <=> lim(n->∞) 1-1/2^n = lim(n->∞) 1-1/10^n
> <=> lim(n->∞) 1/2^n = lim(n->∞) 1/10^n

OK so far; this is simply "0 == 0".

> <=> lim(n->∞) 1 = lim(n->∞) 1/5^n

Illegitimate. You seem to have multiplied the equation
on the previous line by "2^n", but that is not a constant, and
indeed is increasing with "n" [which is not "free"]. See

https://en.wikipedia.org/wiki/Free_variables_and_bound_variables

[...]

> If one resorts to Dedekind-cut-like theories (I did not really read it),
> from the knowledge that all the combinations of discrete symbols cannot
> represent all the real numbers, I can conclude what those theories
> claim are false, let alone I suspect there should be circular arguments
> there, because many terms there must be well defined as a fundamental
> theory, are undefined (prove me wrong).

I try very hard to allow for English not being your first
language, but I cannot make sense of the above [or what you wrote
later]. But it does suggest that before making claims about numbers,
limits and so on, you ought at least to try to understand the basics
of analysis. It would help you a /lot/ if you looked up things like
"accumulation point":

https://en.wikipedia.org/wiki/Accumulation_point

and followed some of the material there, about limit points of sets,
boundary points, closed/open sets, neighbourhoods and similar.
Nothing on that page about Dedekind!

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Ravel

[Mr Flibble]

Repeating decimals in base 10 will terminate in some other base thus
are rational numbers and not irrational numbers.

/Flibble

[wij]

I purposely omitted comment to this SO SIMPLE derivation.
https://tutorial.math.lamar.edu/classes/calci/limitsproperties.aspx
You and Ben really showed you don't understand what you think understand.
When things come to 'research level', you 'teachers' are all failed.
Ben is worse, like to play clever talks as if it seems a clever proof.

> [...]
> > If one resorts to Dedekind-cut-like theories (I did not really read it),
> > from the knowledge that all the combinations of discrete symbols cannot
> > represent all the real numbers, I can conclude what those theories
> > claim are false, let alone I suspect there should be circular arguments
> > there, because many terms there must be well defined as a fundamental
> > theory, are undefined (prove me wrong).
> I try very hard to allow for English not being your first
> language, but I cannot make sense of the above [or what you wrote
> later]. But it does suggest that before making claims about numbers,
> limits and so on, you ought at least to try to understand the basics
> of analysis. It would help you a /lot/ if you looked up things like
> "accumulation point":
>
> https://en.wikipedia.org/wiki/Accumulation_point
>
> and followed some of the material there, about limit points of sets,
> boundary points, closed/open sets, neighbourhoods and similar.
> Nothing on that page about Dedekind!
>
> --
> Andy Walker, Nottingham.
> Andy's music pages: www.cuboid.me.uk/andy/Music
> Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Ravel

Keep reciting your 'trained' dogma to your students.

[wij]

Rational number means the ratio of two integer numbers p/q.

[Mr Flibble]

On Tue, 9 Aug 2022 10:42:48 -0700 (PDT)

I know what rational numbers are: not sure why you felt the need to
state the bleeding obvious. Numbers are either rational or irrational.

/Flibble

[wij]

Save craps, just prove what you say.
I can input 0.999... to my computer to any desired length and expect (0.999...)^n
can approach 1/e to any precision I desire.
I can input 1.000..1 to my computer to any desired length and expect (1.000..1)^n
can approach e to any precision I desire.

Note that lim(n->∞) (1+k/n)^n is a strictly increasing or decreasing function.
This is my early "exponential amplifier" proof.
Try your computer to see if it agree with your "bleeding obvious".
(I assume you cannot see there are infinite number of 0.999... are at the open
end of interval [0,1), and 1.00..1 also equals to 1)

[Skep Dick]

On Tuesday, 9 August 2022 at 17:15:14 UTC+2, Ben Bacarisse wrote:
> This is because 0.333... means
>
> lim(n->oo) Sum k=1,n (3/10^k)

I notice that you think yourself very skilled in telling other people what stuff means.
So I am curious whether you know what a limit means.

Are you using the epsilon-delta definition, or the infinitesimal definition?

[Skep Dick]

You are raising all the usual objections to these issues and you are going to be brushed away by all the dogmatists who know the “correct way” to think about the problem.

Asking what infinity is doesn’t give you any useful answers either.
Go the other way - start with infinitesimals and the hyperreal number line…

https://www.khanacademy.org/college-careers-more/bjc/2015-challenge/2015-math/v/infinitesimals-non-standards-analysis

[Skep Dick]

On Monday, 8 August 2022 at 15:16:39 UTC+2, Ben Bacarisse wrote:
> Yes. For example 0.999... = 1/1.

Ok. So it needs saying so I will say it. You are an ass.

Few posts ago (on another thread) you were giving me a lecture about functions needing domains/context.

Whether 0.999... = 1/1 is a theorem or not absolutely depends on the number system. The domain of your symbols.

It is a theorem in R.
It isn’t a theorem in *R

[Skep Dick]

On Tuesday, 9 August 2022 at 02:30:19 UTC+2, richar...@gmail.com wrote:
> On 8/8/22 8:27 AM, wij wrote:
> > If 0.999... is rational, then:
> > 0.999....= p/q (p,q∈ℕ)
> > <=> 0.999...*q=p
> > If 0.999...∈ℕ, there exist q∈ℕ such that 0.999...*q∈ℕ
> > since 0.999... is defined as infinite repeating, and q is finite,
> > but 0.999...*q is never finite.
> > Conclusion: Repeating decimals are irrational.
> Wrong, 9/9 can generate the pattern 0.9999...
>
> This can be proven by 9 = 3+3+3 and the assocative property thus 9/9 =
> (3+3+3)/9 = 3/9 + 3/9 + 3/9 = 0.3333... + 0.3333... + 0.3333... =
> 0.9999....

Ooooh! The associative property! I know how to abuse that too.

S = 1 + 2 + 3 + 4 + 5 ...
S = 1 + (2+3+4) + (5+6+7) + (8+9+10)...
S-1 = 9 + 18 + 28 ...
S-1 = 9(1 + 2 + 3 ...)
S-1 = 9S
-8S = 1
S = -1/8
1+2+3+4+5... = -1/8

[Richard Damon]

Which is why you need to be clear what domain you are talking about,
especially if you are not talking about the domain that people will assume.

That just shows that you believe it is ok to be deceptive by not being
clear.


The number, 0.9999... if not otherwise indicated by context, is a Real
Number in the Real Number system. To later say you are talking about
some other system *R, is just showing you are being deceptive.

Virtually ANY statement can be proven or disproven if you allow it to be
moved into an arbitrary logic system crafted for that purpose, so
context is vital for communication.

[Richard Damon]

Except that you did it with non-convergent sums, I didn't it with a
finite value.

The value of the sum 1 + 2 + 3 + 4 + 5 + ... doesn't HAVE a value, so
you can't apply associativity.

The value 1 / 3 as 0.3333... IS a value (since you propose that
0.9999... is something that can be evaluated)


The key here is the assocative property applies to the domain of the REALS.

1/3 is a real

1 + 2 + 3 + 4 + 5 .... is not, so you can't apply it there.

It is a well established property of infinite sums that many of the
conventional operations only "work" if the sum is in fact, convergent.

The thing you run into in your proof is that the your first line we get
the value of "infinity" for S.

The problem is infinity-1 is the same as infinity (at least for some
versions of infinity), so everything breaks down. The number systems
that handle the new values, like infinity, have this annoying problem of
losing characteristics that we like to assume we have.

[Skep Dick]

On Thursday, 11 August 2022 at 13:55:09 UTC+2, richar...@gmail.com wrote:
> Except that you did it with non-convergent sums, I didn't it with a
> finite value.

So what? You did it with an infinite representation.

If you are going to substitute a finite representation with an infinite one it's on you to prove that they are equivalent representations of the object.

> 1/3 is a real
Why do you say that?

1/3 ∈ ℚ
1/3 ∈ *R
1/3 ∈ R

It's whatever you want it to be.

> 1 + 2 + 3 + 4 + 5 .... is not, so you can't apply it there.

That's a lot of talk what I can and can't do. Where's this rulebook of yours?

You sure as hell haven't produces anything formal yet!

> It is a well established property of infinite sums that many of the
> conventional operations only "work" if the sum is in fact, convergent.
> The thing you run into in your proof is that the your first line we get
> the value of "infinity" for S.

Which is a perfectly valid thing to do in some number systems.
Why are you discriminating against those number systems?

> The problem is infinity-1 is the same as infinity (at least for some
> versions of infinity), so everything breaks down.

Not in my model. Maybe your model is incomplete?

> that handle the new values, like infinity, have this annoying problem of
> losing characteristics that we like to assume we have.

That's a really easy problem to solve! Stop making stupid assumptions.

[wij]

The video seems blocked to where I live. I had skimmed a little what so called
"non-standard analysis", I did not like the impression read to me (too artificial).

[wij]

I think you should provide a solution to really solve the issue.

My idea of infinity is simple. '∞' denotes a unique number like the i (unit of
imaginary number).

Definition of ∞:
1. ∀n∈ℕ, n<∞
2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞

The meaning of ∞ in 'thinking' is merely (a process/procedure) 'never end'.

I think I solved the basic 'paradoxes' of infinite series. The basics is that
the addend of an infinite series cannot be rearranged.
Everybody seems to agree this point, but really as did?, or I formalized the idea.
Snippet from https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download

+-----------------+
| Infinite Series |
+-----------------+
Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
a(n) is called the general term, addend, summand. n is referred as the
index. Series S is the sum from the first term a(0) to the last term a(k).
The sum of those first terms (n<k) is called the partial sum.
"a(0)+...+a(k)" is called expanded form.

Infinite Series::= If the series S refers to infinite terms (n=∞), S is called
an infinite series. Note that there are infinite addend. The sum cannot be
completed by enumeration (∞ means unreachable, by definition).

In the concept that number-is-an-expression-of-computation, infinite series is
a number, no such concern of converge/diverge (statement when number converges
is a number, diverges is not, is self-controdictory). The computaion rule of
infinite series is based on the expanded form and concepts mentioned above.
Noteworthy difference is that the interpretation of "..." in the expanded form
is a "fixed/unique" number of terms, i.e. "∞+1≠∞" (not the notion of Cantor's
infinite correspondence).

Arithmetic of expanded form:
Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
<=> S= 1+a*S-a^(∞+1)
<=> S(1-a)=1-a^(∞+1)
<=> S= (1-a^(∞+1))/(1-a)

Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
S= 1+2+3+...+n // (1)
S= n+...+3+2+1 // (2)
2S= n*(n+1) // (1)+(2)
<=> S= n*(n+1)/2

∴ Basically, formula for 'finite' series is applicable to infinite series.
(note that mathematical inducion cannot prove such formulas because by
definition, ∞ is unreachable by counting.)

Rule: Handling of the expanded form of infinite series must list the last
addend. Otherwise, the expanded form is ill-formed (obscure semantics and
information being lost cannot conduct valid deduction).

Ex.1 (the last addend is omitted):
A=1+2+3+4+5+...
=(1+2)+(3+4)+5+...
=3+7+5+... // ill-formed, obscure semantics.

Last addend listed:
A=1+2+3+4+5+...+∞ // well-formed, the exanded form of Σ(n=1,∞) {n}

Ex.2:
S=1+2+4+8+... // ill-formed
<=> S=1+2(1+2+4+8+...)
<=> S=1+2S
<=> S=-1

Last addend listed:
S=1+2+4+8+...+2^∞
<=> S=1+2(1+2+4+...+2^(∞-1))
<=> S=1+2S-2^(∞+1)
<=> S=2^(∞+1)-1 // Lots of similar "magic calculation" deriving the result
// S=-1 can be found in youtube. (the term containing the
// last addend ∞ is ignored)

Ex.3:
"f(n)= Σ(k=0,n) 1/k! => f(∞)=e(The base of natural logarithm)"?
We know for sure ∀n∈ℕ, f(n)∈ℚ. To get the result f(n)=e (f(n)∉ℚ), the only
current option is n=∞. But the issue whether or not f(∞)=e (exact equal by
definition) can only be decided via definition, e.g. e≡f(∞). Otherwise, we
can only say f(∞)≈e. (In considering the definition of the equal sign '=',
other forms of e are likely not mutually replaceable with f(∞))

Ex.4: x= Σ(n=1,∞) 1/n²
A common expression is x= Σ(n=1,∞) 1/n²= π²/6, therefore, π=√(6*x)
The issue here is: Lots of π can be derived from various infinite serieses.
But, according to the definition of '=', the result of mutual substitution
may become inconsistent.
For now, the uncontroversial definition of π is the ratio of the
circumference of a circle to its diameter (no computable definition), it is
more correct to use '≈'.
Therefore, Σ(n=1,∞) 1/n² ≈ π²/6 is what it is.

[Note1] "..." in expression is normally indeterminant, of vague semantic.
"0.999..." is also indeterminant before the "..." is eliminated, the
number "0.999..." represents is uncertain, must be removed to ensure
what the number is.
Ex1: Let x=0.999...
10*x= 9+x // This is the result of x after interpreted, not necessarily
// the result followed from "x=0.999..."
// This equation must be given to define x (eliminate the
// ambiguous "...")
Ex2: Let x=√(2+√(2+√(2+...))). Then, possible interpretation of x are:
x=√(2+x)
x=√(2+√(2+x))
x=√(2+√(2+√(2+x)))
...

Ex3: "0.999..." usual 'repeating decimal' cannot denote a unique number.
Let A= Σ(n=1,∞) 1/2^n = 0.999...
B= Σ(n=1,∞) 9/10^n = 0.999...

Let A=B
<=> 1-1/2^∞= 1-1/10^∞ // converted from the formula of geometric series
<=> 1/2^∞= 1/10^∞
<=> 10^∞= 2^∞
<=> 5^∞=1
<=> false

[Note2] Expanded form is prone to magic tricks, perhaps owing to conceptional
generalization of visual illusion too easy to form. It is an error
because the regrouping of the expanded form hides the fact that the
original way of computation is reformulated.
Ex: S can be the sum of any sequence of natural numbers.
S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
= Σ(n=1,∞) n+1 // S is modified

Axiom: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
= a(∞)+ Σ(n=0,∞-1) a(n)
Theorem1: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
Theorem2: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
Proof: Omitted (Can be derived from the expanded form)

Ex1: Σ 2*n =Σ (n+n) =Σ n + Σ n
If Σ 2*n is said the sum of all even numbers, Σ n the sum of all natural
numbers, the notion that the whole is greater than the part is conflicted
by this rule (many paradoxical and current text book arithmetic have the
same issue using Theorem2 like in Ex3).
But, how do we express "the sum of even numbers"? Or Σ(n=0,∞/2) 2*n ?
An idea that using C-language's for loop expression might solve this
problem (or, at least, better than the traditional Σ notation):
for(n=0;;++n) n; or f(n=0;;n+=2) n;
Benefit of such a notation is 1.the symbol '∞' can be omitted 2. the
meaning is more concrete, reducing mathematical imagination of 'Σ'.

Temporary Conclusion: The essence of an infinite series may be a number whose
computation never terminates because of infinite number of non-zero
addends), or could be imagined as a 'running' number (density property
requires the existence of such an 'irrational' number).
------------------------- End of Quote

[wij]

Definition man, allow me to call you such (another meaning of definition is
that your understanding stops there, you can't go anything beyond).
Your R is only valid in 'standard + academic + examination' reinforced environments.
'Real number' is actually quite physical and global, in ancient time or present.
If a theory of 'Real number' does not fit 'what is observed', it gets modified.
There may be a standard R, like in physics there is a standard model, it is just
a standardized resolution (AND, still in revision process) among many.

The real number 0.999... is a number roughly

A[0]=0
A[n]=(A[n-1]+1)/2

When n goes infinitely large. And it could be 'visualized' in thought logically.
If possible, it could also be the experimental reality, like the scale marks.

[Skep Dick]

On Thursday, 11 August 2022 at 13:43:10 UTC+2, richar...@gmail.com wrote:
> Which is why you need to be clear what domain you are talking about,
> especially if you are not talking about the domain that people will assume.

I have no way of knowing what "people" will assume. I am not a mind reader.

> That just shows that you believe it is ok to be deceptive by not being
> clear.

If your first conclusion is that I am being 'deceptive'; instead of you being presumptious then I have some really bad news for you.

There's a principle you should study; and learn to apply: https://en.wikipedia.org/wiki/Principle_of_charity

> The number, 0.9999... if not otherwise indicated by context, is a Real
> Number in the Real Number system.

Well, how about you just use a clearer notation!

>To later say you are talking about some other system *R, is just showing you are being deceptive.

See Principle of charity. You really need it.

> Virtually ANY statement can be proven or disproven if you allow it to be
> moved into an arbitrary logic system crafted for that purpose, so
> context is vital for communication.

I know that. It seems like you know that too. Why then are you so outraged by my theorems?

[Skep Dick]

On Thursday, 11 August 2022 at 16:48:28 UTC+2, wyni...@gmail.com wrote:
> The video seems blocked to where I live.

OK... same video on YouTube: https://youtu.be/SoU2ePlqG5M

> I had skimmed a little what so called. "non-standard analysis", I did not like the impression read to me (too artificial).
It's Mathematics - everything is artificial!

There's no beauty here - only utility.

[Ben Bacarisse]

wij <wyni...@gmail.com> writes:

> The real number 0.999... is a number roughly
> A[0]=0
> A[n]=(A[n-1]+1)/2

Yes, roughly. Fortunately there is no need for rough analogies as
0.999... has a precise meaning already.

I'm curious... You posted some "proofs" with simple school-level
algebra errors. You ignored the replies. Why? Do you accept that your
algebra was wrong? You can accept that these arguments were wrong
without having to concede the main point.

--
Ben.

[wij]

Just point what is wrong. I am glad to know.
No need to show your idiotic 'smart' again and again.

[wij]

So, the next question is simplicity. I think my solution is the possibly simple
ones, easy to explain to computers.

[Ben Bacarisse]

Two people pointed out the basic mistakes in the algebra. You ignored
both. I am curious as to why given you now claim that you'd be glad to
know?

I'm not going to go find the post, copy out what you write and then
repeat what I wrote just so you can ignore it again. I just want to
know why you ignored it originally.

--
Ben.

[Keith Thompson]

Skep Dick <skepd...@gmail.com> writes:
> On Thursday, 11 August 2022 at 13:43:10 UTC+2, richar...@gmail.com wrote:

[...]

>> The number, 0.9999... if not otherwise indicated by context, is a Real
>> Number in the Real Number system.
> Well, how about you just use a clearer notation!

What clearer notation would you suggest?

My experience matches Richard's: the notation "0.9999..." refers to a
real number in ℝ unless otherwise specified. I, and most people, find
the existing notation perfectly clear. Anyone who means something else
by "0.9999..." needs to specify what they mean if they want to be
understood.

And since it's a real number in ℝ, the proof that it's equal to 1.0
applies.

Do you *not* assume that "0.9999..." refers to a real number in ℝ when
you see it without some other explanation?

I think most of us here understand and agree that 0.9999... is equal to
1.0 when interpreted in the real numbers, and may or may not be equal to
1.0 in some other system.

--
Keith Thompson (The_Other_Keith) Keith.S.T...@gmail.com
Working, but not speaking, for Philips
void Void(void) { Void(); } /* The recursive call of the void */

[Ben Bacarisse]

Skep Dick <skepd...@gmail.com> writes:

> On Monday, 8 August 2022 at 15:16:39 UTC+2, Ben Bacarisse wrote:
>> Yes. For example 0.999... = 1/1.
>
> Ok. So it needs saying so I will say it. You are an ass.
>
> Few posts ago (on another thread) you were giving me a lecture about
> functions needing domains/context.

A few words words is not a lecture! In your case, there was no way to
determine the domain, but here there is. Wij is not making claims about
*R. He or she is claiming that "ordinary arithmetic" proves him or her
right.

> Whether 0.999... = 1/1 is a theorem or not absolutely depends on the
> number system. The domain of your symbols.

Yes (though that's a different use of the word). This is, of course,
the point I'm making. If Wij wants to dispute that 0.999... = 1 he or
she must explain what new meaning is being referred to. But cranks
never do that because learning about infinitesimal numbers is too much
work.

> It is a theorem in R.

Indeed.

> It isn’t a theorem in *R

Actually, I think it /is/ a theorem in *R isn't it? It's been a while,
but I remember series like this converge in *R just as the do in R. Is
there anyone here who knows about limits like this in *R (the
hyperreals) and can give the governing theorem? There will be some
convergence criterion theorem. I remember something to do with infinite
hyperinteger sums being approximately zero. Euler's criterion, I
think...

--
Ben.

[Ben Bacarisse]

Skep Dick <skepd...@gmail.com> writes:

> On Tuesday, 9 August 2022 at 17:15:14 UTC+2, Ben Bacarisse wrote:
>> This is because 0.333... means
>>
>> lim(n->oo) Sum k=1,n (3/10^k)
>
> I notice that you think yourself very skilled in telling other people
> what stuff means.

Thank you (I suppose). It's better when people use agreed meanings.

--
Ben.

[Richard Damon]

Definitions are Definitions.

The "Real Number System" is a well defined entity, calling anything else
by that name is just a LIE.

If you want to talk about some other number system, go ahead, just use a
proper name for it, and make it clear what you are doing, or you are
being decietful.

The essence of communication is agreed definitions. To use words in a
manner that they are not agreed on, is to not be actually trying to
communicate.

[Richard Damon]

On 8/11/22 11:29 AM, Skep Dick wrote:
> On Thursday, 11 August 2022 at 13:43:10 UTC+2, richar...@gmail.com wrote:
>> Which is why you need to be clear what domain you are talking about,
>> especially if you are not talking about the domain that people will assume.
> I have no way of knowing what "people" will assume. I am not a mind reader.

Then you fail the first step of communication, and thus fail to communicate.

>
>> That just shows that you believe it is ok to be deceptive by not being
>> clear.
> If your first conclusion is that I am being 'deceptive'; instead of you being presumptious then I have some really bad news for you.
>
> There's a principle you should study; and learn to apply: https://en.wikipedia.org/wiki/Principle_of_charity

I gave you charity at first. You then showed that you didn't care, and
just turned abusive.

Since YOU refuse to try to meet others interpretations, you lose the
right to claim yours are special.

You have proved what sort of person you are, and thus have lost the
right to charity.

>
>> The number, 0.9999... if not otherwise indicated by context, is a Real
>> Number in the Real Number system.
> Well, how about you just use a clearer notation!

What clear notation needs to be used? It is presumed that a number that
is written is part of the simplest number system that its representation
needs.

>
>> To later say you are talking about some other system *R, is just showing you are being deceptive.
> See Principle of charity. You really need it.

Maybe YOU need to use it.

>
>> Virtually ANY statement can be proven or disproven if you allow it to be
>> moved into an arbitrary logic system crafted for that purpose, so
>> context is vital for communication.
> I know that. It seems like you know that too. Why then are you so outraged by my theorems?
>
>

Because you refuse to put them into the context of the system they
belong in but claim they are part of a domain that they aren't in.

THAT makes them a LIE.

The fact that you ADMIT that you don't intend of following conventions,
says that nothing you say is apt to actually be meaningful, because you
won't label the system that it actually needs to be understood in.

[Richard Damon]

On 8/11/22 8:41 AM, Skep Dick wrote:
> On Thursday, 11 August 2022 at 13:55:09 UTC+2, richar...@gmail.com wrote:
>> Except that you did it with non-convergent sums, I didn't it with a
>> finite value.
> So what? You did it with an infinite representation.
>
> If you are going to substitute a finite representation with an infinite one it's on you to prove that they are equivalent representations of the object.
>
>> 1/3 is a real
> Why do you say that?
>
> 1/3 ∈ ℚ
> 1/3 ∈ *R
> 1/3 ∈ R
>
> It's whatever you want it to be.
>
>> 1 + 2 + 3 + 4 + 5 .... is not, so you can't apply it there.
> That's a lot of talk what I can and can't do. Where's this rulebook of yours?

The value of 1 + 2 + 3 + 4 + 5 + ... does not result in a value that is
a Real Number.

So, when you are talking about things you SAY are Real Numbers, there is
no such value.

>
> You sure as hell haven't produces anything formal yet!
>
>> It is a well established property of infinite sums that many of the
>> conventional operations only "work" if the sum is in fact, convergent.
>> The thing you run into in your proof is that the your first line we get
>> the value of "infinity" for S.
>
> Which is a perfectly valid thing to do in some number systems.
> Why are you discriminating against those number systems?

Because, until specifically declaired, there are rules to determine the
assumed number system.

>
>
>> The problem is infinity-1 is the same as infinity (at least for some
>> versions of infinity), so everything breaks down.
> Not in my model. Maybe your model is incomplete?

Then your model isn't the Real Number System. If you have redefined
that, then your system just becomes a deception.

>
>> that handle the new values, like infinity, have this annoying problem of
>> losing characteristics that we like to assume we have.
> That's a really easy problem to solve! Stop making stupid assumptions.
>
>

Stop making statement that imply them then. I guess your GOAL is to be
ambiguous, and thus you are speaking deceptions.

You are just full of EGO and lack of concern about others, and thus
putting yourself outside of the community you seem to want to be part of.

[Richard Damon]

Yes, you can start to build a number system like this, and in fact there
are a number of them based on slightly different details.

The key is that none of them are "The Real Number System", but an
expansion of it, and when you do this to express the concept of
"infinity" you tend to also lose some of the properties of the Real
Number system. (Which properties you lose depends on details of how you
flesh out the details of this new number).

Note, it takes some work to fully define how your "infinity" works, and
one of the problems is that some (many) definitions turn out to actually
end up with inconsistent systems (which is one of the reasons you lose
some of the normal basic properties, to eliminate those inconsistencies).

[Skep Dick]

On Friday, 12 August 2022 at 04:54:15 UTC+2, richar...@gmail.com wrote:
> On 8/11/22 11:29 AM, Skep Dick wrote:
> > On Thursday, 11 August 2022 at 13:43:10 UTC+2, richar...@gmail.com wrote:
> >> Which is why you need to be clear what domain you are talking about,
> >> especially if you are not talking about the domain that people will assume.
> > I have no way of knowing what "people" will assume. I am not a mind reader.
> Then you fail the first step of communication, and thus fail to communicate.

Is that like you failed to communicate 0.999...'s domain?
Or how you failed to communicate 3 and 9's domains.

Or are we only counting my communication failures but not yours?

> I gave you charity at first. You then showed that you didn't care, and
> just turned abusive.

I turned "abusive" and "stopped caring" the moment I realised you are uncharitable.

Which was roughly the moment you began insiting that you are "right" and other people are "wrong".

It was roughly the moment I realised that you don't know Mathematics is absolutely relative.

> Since YOU refuse to try to meet others interpretations, you lose the
> right to claim yours are special.

Wait. What?!? I never claimed that I am special. In fact! It is my primary axiom that there are NO privileged models/interpretations of computation!

You are the one gatekeeping YOUR interpretation.

> You have proved what sort of person you are, and thus have lost the
> right to charity.

You are equivocating "prove" you uncharitable twat!

The "sort of person I am" is beyond the techniques of Mathematics.

> What clear notation needs to be used?

So clarity in communication is no longer your goal? I thought so!

> is written is part of the simplest number system that its representation
> needs

Representations don't have needs. Humans have needs.

You are projecting needs onto your representations.

> >> To later say you are talking about some other system *R, is just showing you are being deceptive.
> > See Principle of charity. You really need it.
> Maybe YOU need to use it.

I am using it. That's why I am questioning your Mathematical dogma!

> Because you refuse to put them into the context of the system they
> belong in but claim they are part of a domain that they aren't in.

You fucking hypocrite!

WHere in this sentence did YOU put them in the context of the system they belong?!?

> This can be proven by 9 = 3+3+3 and the assocative property thus 9/9 =
> (3+3+3)/9 = 3/9 + 3/9 + 3/9 = 0.3333... + 0.3333... + 0.3333... =
> 0.9999....

> THAT makes them a LIE.

OK. Then I shall call you a Liar from now on.

> The fact that you ADMIT that you don't intend of following conventions.
There you go, being uncharitable again.

I have absolutely no idea whether "I am following conventions" or not!
I have no idea what YOUR cultural biases are - I am not a mind reader!

I am an autodidact - I rely on my intuition. That is to say - I am an intuitionist in the footsteps of Brouwer.
My intuition doesn't tell me which system it's operating in - if any!

In fact, the only time the question of systems/context even matters is when I encounter Mathematical statements which assault my intuitions.
Statements like "0.999... = 1". Or calculus done the epsilon-delta way.

> says that nothing you say is apt to actually be meaningful, because you
> won't label the system that it actually needs to be understood in.

That's just gaslighting! I can play the exact same game, you uncharitable ass.

You've said nothng meaningful every time you failed to label your system.

[Skep Dick]

Oh, I disagree wholeheartedly. Negotiating disagreement in meanings is part of being a skilled communicator - we know a bunch of consensus protocols and they converge rapidly.

That's why you have all of those error-detection and error-correction facilities wired in your brain.
I would highly recommend you using them, but you seem far more interested in strongly-normalizing reductions than effective communication.

[Skep Dick]

On Friday, 12 August 2022 at 05:00:23 UTC+2, richar...@gmail.com wrote:
> The value of 1 + 2 + 3 + 4 + 5 + ... does not result in a value that is
> a Real Number.

So what? If you want 1+2+3+4+5... to result in a value then DON'T use the Real Number System.

Use another system.

> So, when you are talking about things you SAY are Real Numbers, there is
> no such value.

I am not saying what they are or aren't. I am just manipulating symbols - I am computing!

Depending on the semantic properties I am interested in I may switch systems. So what?

> Because, until specifically declaired, there are rules to determine the
> assumed number system.

There are rules to mind-reading!?! Please could you explain those rules to a Turing Machine!

I would love to program via telepathy!

> Then your model isn't the Real Number System. If you have redefined
> that, then your system just becomes a deception.

I haven't redefined anything. I am just using symbols and constructions in a way that makes sense to me.

> Stop making statement that imply them then. I guess your GOAL is to be
> ambiguous, and thus you are speaking deceptions.

That is a wrong guess. But it's funny that this idea/goal should pop in in YOUR mind.

Maybe that's what YOU are doing? Are you trying to deceive other people?

After all - there's no empirical way to verify that ANYTHING you say is correct.

> You are just full of EGO and lack of concern about others, and thus
> putting yourself outside of the community you seem to want to be part of.

Are you psychoanalyzing yourself or...?

I have no ego. Which is why I am pointing out THAT Mathematics is relative.

There's no right and wrong. There's only right and wrong within the rules of a system.

But who created the system and why? Does their creation suit my purpose?

[Richard Damon]

On 8/12/22 1:07 AM, Skep Dick wrote:
> On Friday, 12 August 2022 at 04:54:15 UTC+2, richar...@gmail.com wrote:
>> On 8/11/22 11:29 AM, Skep Dick wrote:
>>> On Thursday, 11 August 2022 at 13:43:10 UTC+2, richar...@gmail.com wrote:
>>>> Which is why you need to be clear what domain you are talking about,
>>>> especially if you are not talking about the domain that people will assume.
>>> I have no way of knowing what "people" will assume. I am not a mind reader.
>> Then you fail the first step of communication, and thus fail to communicate.
>
> Is that like you failed to communicate 0.999...'s domain?
> Or how you failed to communicate 3 and 9's domains.
>
> Or are we only counting my communication failures but not yours?

But I DID, by following the conventions.

IT isn't MY fault that you are ignorant of the standard
social-mathematical conventions.

Note, earlier in the conversation, and in the topic of the message, that
domain was actually specifically mentioned.

By making a rational/irrational comment, the domain being the Real is
made clear, as that is the smplest system that has both of them.

>
>> I gave you charity at first. You then showed that you didn't care, and
>> just turned abusive.
> I turned "abusive" and "stopped caring" the moment I realised you are uncharitable.

Nope, I pointed out that your statements were incorret in the impllied
system. I didn't say they couldn't be right if you actually specified a
domain, giving you the option to correct you statement and include the
domain.

YOUR reply was that you

>
> Which was roughly the moment you began insiting that you are "right" and other people are "wrong".

IF you ARE Right about something, you are right.

>
> It was roughly the moment I realised that you don't know Mathematics is absolutely relative.

So, you don't know that Truth is ABSOLUTE.

>
>> Since YOU refuse to try to meet others interpretations, you lose the
>> right to claim yours are special.
> Wait. What?!? I never claimed that I am special. In fact! It is my primary axiom that there are NO privileged models/interpretations of computation!
>
> You are the one gatekeeping YOUR interpretation.

WRONG.

>
>> You have proved what sort of person you are, and thus have lost the
>> right to charity.
> You are equivocating "prove" you uncharitable twat!
>
> The "sort of person I am" is beyond the techniques of Mathematics.

You are just proving the sort of person you are,

[Richard Damon]

On 8/12/22 1:42 AM, Skep Dick wrote:
> On Friday, 12 August 2022 at 05:00:23 UTC+2, richar...@gmail.com wrote:
>> The value of 1 + 2 + 3 + 4 + 5 + ... does not result in a value that is
>> a Real Number.
> So what? If you want 1+2+3+4+5... to result in a value then DON'T use the Real Number System.
>
> Use another system.

The actually USE the other system, but you need to specify it.

>
>> So, when you are talking about things you SAY are Real Numbers, there is
>> no such value.
> I am not saying what they are or aren't. I am just manipulating symbols - I am computing!
>
> Depending on the semantic properties I am interested in I may switch systems. So what?

If you don't define the system you are in, your symbols are meaningless.

That seems to be where you are stuck,

>
>> Because, until specifically declaired, there are rules to determine the
>> assumed number system.
> There are rules to mind-reading!?! Please could you explain those rules to a Turing Machine!
>
> I would love to program via telepathy!

Who needs mind-reading. That is YORU false assumption.

>
>> Then your model isn't the Real Number System. If you have redefined
>> that, then your system just becomes a deception.
> I haven't redefined anything. I am just using symbols and constructions in a way that makes sense to me.

And that is your problem, you should be using them in the way that makes
sense to the people you are talking about by being clear about the
system you are working in.

You are just proving your EGO problem.

>
>> Stop making statement that imply them then. I guess your GOAL is to be
>> ambiguous, and thus you are speaking deceptions.
> That is a wrong guess. But it's funny that this idea/goal should pop in in YOUR mind.
>
> Maybe that's what YOU are doing? Are you trying to deceive other people?
>
> After all - there's no empirical way to verify that ANYTHING you say is correct.
>
>> You are just full of EGO and lack of concern about others, and thus
>> putting yourself outside of the community you seem to want to be part of.
> Are you psychoanalyzing yourself or...?
>
> I have no ego. Which is why I am pointing out THAT Mathematics is relative.

So you are just ignorant.

>
> There's no right and wrong. There's only right and wrong within the rules of a system.

Incorrect. Yes, a system can define what it will treat as correct, but
there are fundamental rules that must be followed or the system is just
broken (and thus "incorrect")

>
> But who created the system and why? Does their creation suit my purpose?
>

That is one of the big questions.

[Skep Dick]

On Friday, 12 August 2022 at 13:36:36 UTC+2, richar...@gmail.com wrote:
> > Or are we only counting my communication failures but not yours?
> But I DID, by following the conventions.

Precisely! You failed to communicate THAT you are following the convention.

> IT isn't MY fault that you are ignorant of the standard
> social-mathematical conventions.

It is your fault. You failed to communicate which convention you are using. Not me.

> Note, earlier in the conversation, and in the topic of the message, that
> domain was actually specifically mentioned.
> By making a rational/irrational comment, the domain being the Real is
> made clear, as that is the smplest system that has both of them.

What does that have to do with the point that the truth of 0.999... = 1 depends on an arbitrary choice?

It's true if you want it to be true.
It's false if you want it to be false.

What do you want it to be?

> >> I gave you charity at first. You then showed that you didn't care, and
> >> just turned abusive.
> > I turned "abusive" and "stopped caring" the moment I realised you are uncharitable.
> Nope, I pointed out that your statements were incorret in the impllied system.

See! You are uncharitable.

The statement is whatever we want it to be! You want it to be true. I want it to be false.

That doesn't make me incorrect. It makes you an ass.

Mathematics is relative!

>I didn't say they couldn't be right if you actually specified a
> domain, giving you the option to correct you statement and include the
> domain.

Idiot. Which part of choice-dependent truth is lost upon you?

> IF you ARE Right about something, you are right.

One more time - for the idiot in the classroom. There is no such thing as "right" or "wrong" in Mathematics!

0.999... = 1 is neither right nor wrong!

If you want to MAKE it wrong you use *R.
If you want to MAKE it right you use R.

> So, you don't know that Truth is ABSOLUTE.

Ahahahahahahahahahahahahahahahahaha!
Ahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahaha!
Ahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahaha!
Ahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahaha!

Shame. Somebody doesn't understand Tarski's theorems.

https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem

> > You are the one gatekeeping YOUR interpretation.
> WRONG.

There is no right and wrong in Mathematics.

> >> You have proved what sort of person you are, and thus have lost the
> >> right to charity.
> > You are equivocating "prove" you uncharitable twat!
> >
> > The "sort of person I am" is beyond the techniques of Mathematics.
> You are just proving the sort of person you are,

Q.E.D using the concept of "proof" outside of its applicable domain.

Does that make you an idiot? I wonder.

> >> What clear notation needs to be used?
> > So clarity in communication is no longer your goal? I thought so!

Q.E.D Ucharitable twat.

[Mr Flibble]

0.999... = 1.0, proof:

S = 0.999...
S * 10 = 9.999...
S * 10 - S = 9.0
S * 9 = 9.0
S = 1.0
ergo 0.999... = 1.0 QED

/Flibble

[Skep Dick]

On Friday, 12 August 2022 at 13:41:32 UTC+2, richar...@gmail.com wrote:
> > Use another system.
> The actually USE the other system, but you need to specify it.

I am USING the other system. And no - I don't have to fucking specify it.
I'll USE whichever damn system I want to use.

As long as it works.

> If you don't define the system you are in, your symbols are meaningless.
> That seems to be where you are stuck,

Ooooh! Are you sure you want to play that game?

If you don't define "Truth" then "Truth" is meaningless.
If you don't define "define" then "define" is meaningless.
If you don't define "meaningless" then "meaningless" is meaningless.

> Who needs mind-reading. That is YORU false assumption.

My assumption is neither true, nor false - it's just a fact.

It's a fact THAT I can't read your mind.
It's a fact THAT you failed to communicate which system you are using.

> And that is your problem, you should be using them in the way that makes
> sense to the people you are talking about by being clear about the
> system you are working in.

So when are you going to make clear which system of Truth you are working in?

> You are just proving your EGO problem.

In so far as one of us has an EGO problem. It's the guy who always has to be right...

> > I have no ego. Which is why I am pointing out THAT Mathematics is relative.
> So you are just ignorant.

Ahhh. The irony!

IF Mathematics is relative and you don't know that. Which one of us is ignorant?

Remind me again whether the truth of 0.999...= 1 is absolute; or relative.


> Incorrect.
You are incorrect about me being incorrect 😂😂😂😂

>Yes, a system can define what it will treat as correct, but
> there are fundamental rules that must be followed or the system is just
> broken (and thus "incorrect")

Shame. We have a fundamentalist on our hands.

> > But who created the system and why? Does their creation suit my purpose?
> >
> That is one of the big questions.

So you don't actually know the answer, but you blindly follow the rules?

Good computer!

[Skep Dick]

On Friday, 12 August 2022 at 15:47:51 UTC+2, Mr Flibble wrote:
> 0.999... = 1.0, proof:
>
> S = 0.999...
> S * 10 = 9.999...
> S * 10 - S = 9.0
> S * 9 = 9.0
> S = 1.0
> ergo 0.999... = 1.0 QED
>
> /Flibble

if 0.999.. is convergent then the proof is valid.
if 0.999... is divergent then the proof is invalid.

Is 0.999... convergent or divergent?

That's undecidable. Subject to choice.

[Mr Flibble]

Bullshit. 0.999... = 1.0 as I have just shown.

/Flibble

[Skep Dick]

OK, genius.

S = 1 + 2 + 3 + 4...
S = 1 + (2 + 3 + 4) + (5 + 6 + 7)...
S = 1 + 9 + 18 + 27...
S = 1 + 9(1 + 2 + 3 + 4...)
S = 1 + 9S
S = -1/8

ergo 1 + 2 + 3 + 4... < 0

[Mr Flibble]

On Fri, 12 Aug 2022 07:11:37 -0700 (PDT)

false analogy strawman logical fallacy QED

/Flibble

[wij]

An assumption is hidden here (deliberate?)
10*S= 9+S
This line specifies what the property of THE "0.999..." you mean (remove "...").
This property is 'unknown', not a 'given'.

[wij]

See my solution in previous post about infinite series. All (hope) are solved.
(It seems people didn't read my post? just talking about their beliefs and pretend
they are talking about fact and truth.)

[Malcolm McLean]

On Friday, 12 August 2022 at 15:11:39 UTC+1, Skep Dick wrote:
> On Friday, 12 August 2022 at 16:04:33 UTC+2, Mr Flibble wrote:
> > On Fri, 12 Aug 2022 06:58:27 -0700 (PDT)
> > Skep Dick <skepd...@gmail.com> wrote:
> >
> > > On Friday, 12 August 2022 at 15:47:51 UTC+2, Mr Flibble wrote:
> > > > 0.999... = 1.0, proof:
> > > >
> > > > S = 0.999...
> > > > S * 10 = 9.999...
> > > > S * 10 - S = 9.0
> > > > S * 9 = 9.0
> > > > S = 1.0
> > > > ergo 0.999... = 1.0 QED
> > > >
> > > > /Flibble
> > > if 0.999.. is convergent then the proof is valid.
> > > if 0.999... is divergent then the proof is invalid.
> > >
> > > Is 0.999... convergent or divergent?
> > >
> > > That's undecidable. Subject to choice.
> > Bullshit. 0.999... = 1.0 as I have just shown.
> OK, genius.
>
> S = 1 + 2 + 3 + 4...

S = infinity

> S = 1 + (2 + 3 + 4) + (5 + 6 + 7)...

S = infinity

> S = 1 + 9 + 18 + 27...

S = infinity

> S = 1 + 9(1 + 2 + 3 + 4...)

S = 1 + 9 * infinity

> S = 1 + 9S

S (infinity) =1 + 9 * infinity
> S = -1/8
infinity = -1/8. That's the point at which you go wrong. infinity = 1 + infinity,
but we can't subtract infinity from both sides of the equation to obtain 1 = 0
then use that result to prove theorems about the natural numbers.

>
> ergo 1 + 2 + 3 + 4... < 0
>

As you are obviously well aware. The algebra is good as a high school sort
of joke demonstration, but it's just based on a sleight of hand.

[Skep Dick]

On Friday, 12 August 2022 at 17:00:52 UTC+2, malcolm.ar...@gmail.com wrote:
> That's the point at which you go wrong. infinity = 1 + infinity,

That's not true in *R ! Infinitesimals (ε) and infinities (ω) exist.

1/ε = ω/1

> but we can't subtract infinity from both sides of the equation to obtain 1 = 0

In your number system! Not in mine.

1/ε + 1 < 1/ε
ω < ω + 1

This is obvious to children but not to over-educated idiot Mathematicians.

X + 1 is ALWAYS more than X !

> As you are obviously well aware. The algebra is good as a high school sort
> of joke demonstration, but it's just based on a sleight of hand.

You have been lied to all your life.

The sleight of hand is by the Establishment of Mathematics.

0.999... = 1 is false, and infinity + 1 > infinity in any system where x+1 > x

[Mr Flibble]

0.999... = 1.0 as I have shown and demonstrated, QED.

/Flibble

[Mr Flibble]

No.

infinity + 1 = infinity.

/Flibble

[Skep Dick]

On Friday, 12 August 2022 at 17:25:16 UTC+2, Mr Flibble wrote:
> No.
>
> infinity + 1 = infinity.
>
> /Flibble

Fucking idiot.

∀x : x + 1 > x

[Skep Dick]

You are wrong. As I have shown.

There goes your God.

[Python]

What non-standard system of numbers are you using?

[Skep Dick]

On Friday, 12 August 2022 at 17:29:50 UTC+2, Python wrote:
> > ∀x : x + 1 > x
> What non-standard system of numbers are you using?

I am using the standard system!

Adding 1 to ANYTHING makes it more.

Any system that says otherwise is the bullshit system.

[Python]

Skep Dick wrote:
> On Friday, 12 August 2022 at 17:29:50 UTC+2, Python wrote:
>>> ∀x : x + 1 > x
>> What non-standard system of numbers are you using?
>
> I am using the standard system!
>
> Adding 1 to ANYTHING makes it more.

There is no infinity in the standard system for real numbers,
so you cannot say you're using the standard system.

> Any system that says otherwise is the bullshit system.

Have you heard about transifinite cardinals (or ordinals)? In
such systems ∀x : x + 1 > x is wrong. And they are not
"bullshit systems".

[Skep Dick]

On Friday, 12 August 2022 at 17:45:44 UTC+2, Python wrote:
> There is no infinity in the standard system for real numbers,
> so you cannot say you're using the standard system.

You confused about WHICH system we are talking about.

I am not talking about the standard system for real numbers.
I am talking about the standard system of Mathematics.

In the standard system of Mathematics : ∞ + 1 = ∞

Which is exactly the same thing as saying.
x + 1 = x when x is bound to ∞

But if infinity is not a number then how the hell are you binding it to a variable?!?!?

> Have you heard about transifinite cardinals (or ordinals)? In
> such systems ∀x : x + 1 > x is wrong. And they are not "bullshit systems".

They are bullshit systems.

The ω in "1/ε = ω/1" is Omega. Cantor's Ω - the Absolute Infinite!
https://en.wikipedia.org/wiki/Absolute_Infinite

∀x : x + 1 > x !!!!!!!!!!!!
let x = Ω

THEN Ω + 1 > Ω

[Python]

Skep Dick wrote:
> On Friday, 12 August 2022 at 17:45:44 UTC+2, Python wrote:
>> There is no infinity in the standard system for real numbers,
>> so you cannot say you're using the standard system.
> You confused about WHICH system we are talking about.
>
> I am not talking about the standard system for real numbers.
> I am talking about the standard system of Mathematics.

"standard system of mathematics" means what? The usual number
system? There is no ∞ there. Something else, then WHAT?

> In the standard system of Mathematics : ∞ + 1 = ∞
>
> Which is exactly the same thing as saying.
> x + 1 = x when x is bound to ∞
>
> But if infinity is not a number then how the hell are you binding it to a variable?!?!?

So again, when dealing with ∞, what system do you conside ? What is
the definition of ∞ there?

>> Have you heard about transifinite cardinals (or ordinals)? In
>> such systems ∀x : x + 1 > x is wrong. And they are not "bullshit systems".
> They are bullshit systems.

No they are not. You just don't know them.

And now I know that you are a crank (not that I doubted much...)

> The ω in "1/ε = ω/1" is Omega. Cantor's Ω - the Absolute Infinite!
> https://en.wikipedia.org/wiki/Absolute_Infinite
>
> ∀x : x + 1 > x !!!!!!!!!!!!

Not necessarily. You say so, that's not a proof.

[Mr Flibble]

On Fri, 12 Aug 2022 09:03:08 -0700 (PDT)
Skep Dick <skepd...@gmail.com> wrote:

> On Friday, 12 August 2022 at 17:45:44 UTC+2, Python wrote:
> > There is no infinity in the standard system for real numbers,
> > so you cannot say you're using the standard system.
> You confused about WHICH system we are talking about.
>
> I am not talking about the standard system for real numbers.
> I am talking about the standard system of Mathematics.
>
> In the standard system of Mathematics : ∞ + 1 = ∞
>
> Which is exactly the same thing as saying.
> x + 1 = x when x is bound to ∞
>
> But if infinity is not a number then how the hell are you binding it
> to a variable?!?!?

Well obviously that was your mistake, you cannot assign a value of
infinity to any variable in the standard system of Mathematics, so:

infinity + 1 = infinity

*and*

x + 1 > x

/Flibble

[Skep Dick]

On Friday, 12 August 2022 at 18:20:12 UTC+2, Mr Flibble wrote:
> infinity + 1 = infinity
>
> *and*
>
> x + 1 > x
>
> /Flibble

Translation: x + 1 > x

EXCEPT when x = infinity

[Mr Flibble]

Nope, x can never be infinity, only infinity can be infinity.

/Flibble

[Python]

So you have infinity as an "object" in the system, but you cannot
bound it to a name. Weird.

> /Flibble

>>> x = float("inf")
>>> x + 1 > x
False

(nothing specific to Python here, this is IEE754)

also:

>>> x = float("nan")
>>> x is x
True
>>> x == x
False

[Mr Flibble]

On Fri, 12 Aug 2022 18:27:41 +0200
Python <pyt...@example.invalid> wrote:

> Mr Flibble wrote:
> > On Fri, 12 Aug 2022 09:22:27 -0700 (PDT)
> > Skep Dick <skepd...@gmail.com> wrote:
> >
> >> On Friday, 12 August 2022 at 18:20:12 UTC+2, Mr Flibble wrote:
> >>> infinity + 1 = infinity
> >>>
> >>> *and*
> >>>
> >>> x + 1 > x
> >>>
> >>> /Flibble
> >> Translation: x + 1 > x
> >>
> >> EXCEPT when x = infinity
> >
> > Nope, x can never be infinity, only infinity can be infinity.
>
> So you have infinity as an "object" in the system, but you cannot
> bound it to a name. Weird.
>
> > /Flibble
>
> >>> x = float("inf")
> >>> x + 1 > x
> False

This because for any sane definition of infinity adding 1 to it doesn't
change it from being infinity, i.e.

infinity = infinity + 1

*and*

infinity + 1 = infinity

Similarly,

infinity - 1 = infinity
infinity * 2 = infinity
infinity / 2 = infinity

It is a shame the amateurish troll "Skep Dick" doesn't get this as it
seems quite simple to me.

/Flibble

[Skep Dick]

On Friday, 12 August 2022 at 18:11:16 UTC+2, Python wrote:
> So again, when dealing with ∞

We are not dealing with ∞. That symbol is colloquial. It's what lazy Mathematicians and the general public use to represent a bunch of different infinities.

I could ask the exact same question when you write stuff like: x = float("inf")
What's the definition of "inf"? It doesn't have one! It's just the identity functon

f(x) = x
float("inf") = inf

https://en.wikipedia.org/wiki/Identity_function

Further more... you said that infinity is not a number. OK!
(NaN == NaN) ⇔ False

In [1]: float("nan") == float("nan")
Out[1]: False

So then... if infinity is not a number then... float("inf") == float("inf") must be False!

Uh! Oh! Contradiction!

In [1]: float("inf") == float("inf")
Out[1]: True

>what syste do you conside ?
https://en.wikipedia.org/wiki/Hyperreal_number

>What is the definition of ∞ there?

It doesn't have one!

What is defined (axiomatically) is the identity 1/ε = ω/1
ε is infinitesimal
ω is infinity

In English: dividing 1 by really small quantity produces an really big quantity.

> No they are not. You just don't know them.

Whatever floats your boat.

> And now I know that you are a crank (not that I doubted much...)

And now I know that you are an indoctrinated conformist who has never had an original thought in their life (not that I doubted much...)

> Not necessarily. You say so, that's not a proof.

Who needs to say it in order for it become a proof?

[Skep Dick]

On Friday, 12 August 2022 at 18:32:55 UTC+2, Mr Flibble wrote:
> This because for any sane definition of infinity adding 1 to it doesn't
> change it from being infinity, i.e.

Au contraire! I think it's totally insane. Batshit crazy. Intellectually tonedeaf and outright crazy to pretend that adding something (however small) to something else (howvery big) changes nothing whatsoever.

It's even crazier and infinitely stupid to pretend that doubling or halving something leaves it unchanged.

> infinity = infinity + 1

Translation: x = x + 1

> infinity - 1 = infinity

Translation: x - 1 = x

> infinity * 2 = infinity

Translation: x * 2 = x

> infinity / 2 = infinity

Translation: x / 2 = x

> It is a shame the amateurish troll "Skep Dick" doesn't get this as it
> seems quite simple to me.

Everything is simple to a simple mind.

[Skep Dick]

On Monday, 8 August 2022 at 15:16:39 UTC+2, Ben Bacarisse wrote:
> wij <wyni...@gmail.com> writes:
>
> > If 0.999... is rational, then:
> > 0.999....= p/q (p,q∈ℕ)
> > <=> 0.999...*q=p
> Yes. For example 0.999... = 1/1.
> > If 0.999...∈ℕ, there exist q∈ℕ such that 0.999...*q∈ℕ
> > since 0.999... is defined as infinite repeating, and q is finite,
> > but 0.999...*q is never finite.
> 0.999...*q is finite for all q in N.
> > I am kind of reluctant to raise this issue again,
> Really? Where would we be without someone denying (or not knowing) that
> ... denotes a limit every few weeks?
>
> --
> Ben.

Yeah so it's worth repeating. Ben doesn't actually know what a limit is either.

Ask him to work out the derrivative of f(x) = x^2 and watch him make a fool out of himself by regurgitating all the nonsense he's been taught about dy and dx.

But he will feel vindicated that he has "knowledge" because the entire status quo is behind him.

[Richard Damon]

On 8/12/22 9:38 AM, Skep Dick wrote:
> On Friday, 12 August 2022 at 13:36:36 UTC+2, richar...@gmail.com wrote:
>>> Or are we only counting my communication failures but not yours?
>> But I DID, by following the conventions.
> Precisely! You failed to communicate THAT you are following the convention.
>

But the conventions says it doesn't need to be stated.

>> IT isn't MY fault that you are ignorant of the standard
>> social-mathematical conventions.
> It is your fault. You failed to communicate which convention you are using. Not me.

Nope. Just shows that you failed to learn the established conventions.

Ignorance of the rules is not an excuse.

>
>> Note, earlier in the conversation, and in the topic of the message, that
>> domain was actually specifically mentioned.
>> By making a rational/irrational comment, the domain being the Real is
>> made clear, as that is the smplest system that has both of them.
>
> What does that have to do with the point that the truth of 0.999... = 1 depends on an arbitrary choice?

But an choice that HAS been made by the convention, that applies unless
explicitly stated to not be in force.

Are you going to say that EVERYTHING needs to be explicit?

Why didn't you say you were going to talk in English?
And before you say that, you need to say you are going to anounce you
definition of the lanuage you are going to use in English.

....

Conventions establish the base rules we follow to allow us to work with
each other, and to establish how we indicate we are going off in
"non-standard" directions, but stating that.

>
> It's true if you want it to be true.
> It's false if you want it to be false.

Not if you want to work with other.

>
> What do you want it to be?
>
>>>> I gave you charity at first. You then showed that you didn't care, and
>>>> just turned abusive.
>>> I turned "abusive" and "stopped caring" the moment I realised you are uncharitable.
>> Nope, I pointed out that your statements were incorret in the impllied system.
> See! You are uncharitable.
>
> The statement is whatever we want it to be! You want it to be true. I want it to be false.

And you are thus wrong.

>
> That doesn't make me incorrect. It makes you an ass.

Nope.

By YOUR logic, I get to define what I mean, and you need to accept it.
Right? Isn't that what you are doing?

That just shows your logic fails the EGO check.

>
> Mathematics is relative!

Nope. It CAN be relative once you establish a reference point.

>
>> I didn't say they couldn't be right if you actually specified a
>> domain, giving you the option to correct you statement and include the
>> domain.
> Idiot. Which part of choice-dependent truth is lost upon you?

Truth is NOT choice-dependent.

>
>> IF you ARE Right about something, you are right.
> One more time - for the idiot in the classroom. There is no such thing as "right" or "wrong" in Mathematics!
>
> 0.999... = 1 is neither right nor wrong!
>
> If you want to MAKE it wrong you use *R.
> If you want to MAKE it right you use R.
>
>> So, you don't know that Truth is ABSOLUTE.
> Ahahahahahahahahahahahahahahahahaha!
> Ahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahaha!
> Ahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahaha!
> Ahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahaha!
>
> Shame. Somebody doesn't understand Tarski's theorems.


>
> https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem
>>> You are the one gatekeeping YOUR interpretation.
>> WRONG.
> There is no right and wrong in Mathematics.
>
>

Maybe YOU don't understand the meaning of that.

You can't define Truth within the system, because the actual meaning of
Truth is established before the system existed.

>>>> You have proved what sort of person you are, and thus have lost the
>>>> right to charity.
>>> You are equivocating "prove" you uncharitable twat!
>>>
>>> The "sort of person I am" is beyond the techniques of Mathematics.
>> You are just proving the sort of person you are,
> Q.E.D using the concept of "proof" outside of its applicable domain.
>
> Does that make you an idiot? I wonder.

No, it makes me smarter that you as I understand stuff you do not.

>
>>>> What clear notation needs to be used?
>>> So clarity in communication is no longer your goal? I thought so!
> Q.E.D Ucharitable twat.

Nope, charity is to be given to those that honestly need it, not those
who get themselves into trouble by their own devices. They should suffer
the consequences of their actions, until they see their need for actual
charity.

[Malcolm McLean]

y = x^2

now when we add a tiny amount to x, which we call dx, y will change, so call that dy.

y + dy = (x + dx)^2

expand

y + dy = x^2 + 2x*dx + (dx)^2

by y = x^2, so subtract from both sides

dy = 2x*dx + (dx)^2.

Now all we've said is that dx is a tiny amount we add to x. Let's say that it's so small that, when
squared, we cannot measure it at all.

(dx^2) = 0 (or ignore)
dy = 2x*dx

dy/dx = 2x

Now you can ask, "what does it mean to have a slope at a point?" Surely a slope is a characteristic
of two points. So we don't say that dx goes to zero. We say it goes arbitrarily small, so small that we
cannot measure (dx)^2. You can say that dx is an infinitesimal, but that opens up whole questions
about what the infinitesimal means. We don't need to go there. It's just arbitrarily small.

[Mr Flibble]

On Fri, 12 Aug 2022 13:17:26 -0700 (PDT)
Skep Dick <skepd...@gmail.com> wrote:

> On Friday, 12 August 2022 at 18:32:55 UTC+2, Mr Flibble wrote:
> > This because for any sane definition of infinity adding 1 to it
> > doesn't change it from being infinity, i.e.
> Au contraire! I think it's totally insane. Batshit crazy.
> Intellectually tonedeaf and outright crazy to pretend that adding
> something (however small) to something else (howvery big) changes
> nothing whatsoever.

You don't seem to understand what infinity actually is: infinity is a
concept not a number, it isn't "very big" as "very big" implies some
finite size and of course infinity isn't finite, by definition.

>
> It's even crazier and infinitely stupid to pretend that doubling or
> halving something leaves it unchanged.

Not when dealing with infinity it isn't.

>
>
> > infinity = infinity + 1
> Translation: x = x + 1
>
> > infinity - 1 = infinity
> Translation: x - 1 = x
>
> > infinity * 2 = infinity
> Translation: x * 2 = x
>
> > infinity / 2 = infinity
> Translation: x / 2 = x

No, no, no and no. Infinity cannot be assigned to a variable in a
formula or equation as infinity is not a number.

>
> > It is a shame the amateurish troll "Skep Dick" doesn't get this as
> > it seems quite simple to me.
> Everything is simple to a simple mind.

You are the one over simplifying things, so that would be a projection,
dear.

/Flibble

[Richard Damon]

On 8/12/22 11:09 AM, Skep Dick wrote:
> On Friday, 12 August 2022 at 17:00:52 UTC+2, malcolm.ar...@gmail.com wrote:
>> That's the point at which you go wrong. infinity = 1 + infinity,
> That's not true in *R ! Infinitesimals (ε) and infinities (ω) exist.
>
> 1/ε = ω/1
>
>> but we can't subtract infinity from both sides of the equation to obtain 1 = 0
> In your number system! Not in mine.
>
> 1/ε + 1 < 1/ε
> ω < ω + 1
>
> This is obvious to children but not to over-educated idiot Mathematicians.
>
> X + 1 is ALWAYS more than X !

Nope. Trans-finite math DOES seem strange to some, as some common
properties get lost, like x + 1 > x.

After all:
2 + 3 + 4 + 5 + 6 + ... = Infinity
and
1 + 2 + 3 + 4 + 5 + 6 + ... = Infinity.

but the first + 1 equals the second, so Infinity + 1 = Infinity.

>
>> As you are obviously well aware. The algebra is good as a high school sort
>> of joke demonstration, but it's just based on a sleight of hand.
> You have been lied to all your life.
>
> The sleight of hand is by the Establishment of Mathematics.
>
> 0.999... = 1 is false, and infinity + 1 > infinity in any system where x+1 > x
>
>

And "x + 1 > x" as an always true statement doesn't hold for many
systems that have the value of infinity in them.

[Skep Dick]

On Friday, 12 August 2022 at 23:30:22 UTC+2, Mr Flibble wrote:
> You don't seem to understand what infinity actually is: infinity is a
> concept not a number, it isn't "very big" as "very big" implies some
> finite size and of course infinity isn't finite, by definition.

Regurgitating definitions doesn't seem to amount to any understanding on your behalf.

> Not when dealing with infinity it isn't.

Infinity is not exempt from change - that's just a sacred cow.

> >
> >
> > > infinity = infinity + 1
> > Translation: x = x + 1
> >
> > > infinity - 1 = infinity
> > Translation: x - 1 = x
> >
> > > infinity * 2 = infinity
> > Translation: x * 2 = x
> >
> > > infinity / 2 = infinity
> > Translation: x / 2 = x
> No, no, no and no. Infinity cannot be assigned to a variable in a
> formula or equation as infinity is not a number.

I know, you idiot.

Do you actually know what a performative contradiction is?

You SAY that infinity cannot be assigned to a variable in a formula or an equation.... and THEN you assign it to a variable in a formula/equation.

The expression "infinity = infinity + 1" means the exact same thing as the expression "x = x + 1" (with infinity being assigned to the variable x)
The expression "infinity = infinity - 1" means the exact same thing as the expression "x = x - 1" (with infinity being assigned to the variable x)
The expression "infinity = infinity * 2" means the exact same thing as the expression "x = x * 1" (with infinity being assigned to the variable x)
The expression "infinity = infinity /2" means the exact same thing as the expression "x = x / 1" (with infinity being assigned to the variable x)

> You are the one over simplifying things, so that would be a projection,
> dear.

Yes, well. This is very much like explaining colors to the colourblind.

[Richard Damon]

On 8/12/22 9:50 AM, Skep Dick wrote:
> On Friday, 12 August 2022 at 13:41:32 UTC+2, richar...@gmail.com wrote:
>>> Use another system.
>> The actually USE the other system, but you need to specify it.
>
> I am USING the other system. And no - I don't have to fucking specify it.
> I'll USE whichever damn system I want to use.
>
> As long as it works.

But be WORNG because without specifying it, your statement is put into
another system that it doesn't work in.

Failure to specify the domain of a statement, unless the domain is the
domain that is assumed by the conventions, is an error.

You are just proving that you aren't going to work with others.

>
>> If you don't define the system you are in, your symbols are meaningless.
>> That seems to be where you are stuck,
> Ooooh! Are you sure you want to play that game?
>
> If you don't define "Truth" then "Truth" is meaningless.
> If you don't define "define" then "define" is meaningless.
> If you don't define "meaningless" then "meaningless" is meaningless.

Nope, since there IS a convention of definitions that apply unless
stated otherwise, you can build on the conventions without needed to
explicitly respecify those definitions.

That is why you need to specifiy if you are using something else,
otherwise you DO need to fully specify EVERYTHING from the beginning
every time.

>
>> Who needs mind-reading. That is YORU false assumption.
> My assumption is neither true, nor false - it's just a fact.
>
> It's a fact THAT I can't read your mind.
> It's a fact THAT you failed to communicate which system you are using.

No, as I said, you don't need to be a mind-reader, just know the
socially established conventions and follow them, so be clear that you
are deviating from them, and how.

>
>> And that is your problem, you should be using them in the way that makes
>> sense to the people you are talking about by being clear about the
>> system you are working in.
> So when are you going to make clear which system of Truth you are working in?
>
>> You are just proving your EGO problem.
> In so far as one of us has an EGO problem. It's the guy who always has to be right...

It is the guy who says *I* have to be right, instead of *WE* are right.

>
>>> I have no ego. Which is why I am pointing out THAT Mathematics is relative.
>> So you are just ignorant.
> Ahhh. The irony!
>
> IF Mathematics is relative and you don't know that. Which one of us is ignorant?
>
> Remind me again whether the truth of 0.999...= 1 is absolute; or relative.

0.999... = 1 IN THE DEFUALT NUMBER SYSTEM OF THE REAL NUMBER SYSTEM, is
a ABSOLUTE truth.


>
>
>> Incorrect.
> You are incorrect about me being incorrect 😂😂😂😂
>
>> Yes, a system can define what it will treat as correct, but
>> there are fundamental rules that must be followed or the system is just
>> broken (and thus "incorrect")
> Shame. We have a fundamentalist on our hands.
>
>>> But who created the system and why? Does their creation suit my purpose?
>>>
>> That is one of the big questions.
> So you don't actually know the answer, but you blindly follow the rules?
>
> Good computer!

I am not a Computer, I am a person. I am not bound by the limitation of
a computational device, as I have Free-Will.

Maybe you should learn about that.

[Skep Dick]

On Friday, 12 August 2022 at 23:34:26 UTC+2, richar...@gmail.com wrote:
> Nope. Trans-finite math DOES seem strange to some, as some common
> properties get lost, like x + 1 > x

I am doing Transfinite math!

Without losing the most important property of all - common sense.

Seems you sold yours.

> 2 + 3 + 4 + 5 + 6 + ... = Infinity
> and
> 1 + 2 + 3 + 4 + 5 + 6 + ... = Infinity.

Which infinity? There are so many of them!

> but the first + 1 equals the second, so Infinity + 1 = Infinity.

Only in systems lacking common sense.

> And "x + 1 > x" as an always true statement doesn't hold for many
> systems that have the value of infinity in them.

And yet it holds in mine.

Almost as if I don't have to treat infinities as if they are special.

[Mr Flibble]

No, no, no and no. "infinity" is a symbol representing a concept it is
NOT equivalent to a variable, x.

>
> > You are the one over simplifying things, so that would be a
> > projection, dear.
> Yes, well. This is very much like explaining colors to the
> colourblind.

/Flibble

[Skep Dick]

On Friday, 12 August 2022 at 23:44:17 UTC+2, richar...@gmail.com wrote:
> But be WORNG because without specifying it, your statement is put into
> another system that it doesn't work in.

What are you 5 year old? Is everything about right and wrong about you?

Wait... Never mind. That's a rhetorical question.

> Failure to specify the domain of a statement, unless the domain is the
> domain that is assumed by the conventions, is an error.

Oh, ok. Then I am wrong.

But you are still much wronger than me.

> You are just proving that you aren't going to work with others.

I don't like working with people who are much wronger than me.

> Nope, since there IS a convention of definitions that apply unless
> stated otherwise, you can build on the conventions without needed to
> explicitly respecify those definitions.

That's called an bandwagon fallacy. I thought you said you care about logic?

https://en.wikipedia.org/wiki/Argumentum_ad_populum

> That is why you need to specifiy if you are using something else,
> otherwise you DO need to fully specify EVERYTHING from the beginning
> every time.

Or I can just ignore idiots who can't keep up with the times?

Planck was right you know... A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it ...

> No, as I said, you don't need to be a mind-reader, just know the
> socially established conventions and follow them

At which point did I give you the impression that I care about convention?

>so be clear that you are deviating from them, and how.

I wouldn't know that I am deviating from "social convention" now, would I?

Since I don't follow conventions.

> It is the guy who says *I* have to be right, instead of *WE* are right.

In both cases - you are the only one in this conversation who is talking about being right...

I just want better tools. I have better tools.

> 0.999... = 1 IN THE DEFUALT NUMBER SYSTEM OF THE REAL NUMBER SYSTEM, is
> a ABSOLUTE truth.

If your truth is ABSOLUTE truth, then my truth is more ABSOLUTE than yours.

infinity + 1 > infinity

> I am not a Computer, I am a person. I am not bound by the limitation of
> a computational device, as I have Free-Will.

I haven't seen any free will. You just follow rules. Exactly like a computer.

> Maybe you should learn about that.

I have learned about that. I have free will.

That's why I can tell you don't.

I have freely chosen to give up my belief in R and accept R* as a better theory.

You haven't - the Real Numbers are "absolute truth" to you - they are your God. Your sacred cow.

[Skep Dick]

On Friday, 12 August 2022 at 23:48:07 UTC+2, Mr Flibble wrote:
> No, no, no and no. "infinity" is a symbol representing a concept it is
> NOT equivalent to a variable, x.

Golly gosh. You don't even understand how bound and unbound variable work.

Start there. Infinity is a bit above your praygrade.

[Mr Flibble]

Infinity is *not* a variable.

/Flibble

[Richard Damon]

And you get that:

> 1+2+3+4+5... = -1/8

So, it shows how useful your system is.

[Richard Damon]

On 8/12/22 5:58 PM, Skep Dick wrote:
> On Friday, 12 August 2022 at 23:44:17 UTC+2, richar...@gmail.com wrote:
>> But be WORNG because without specifying it, your statement is put into
>> another system that it doesn't work in.
> What are you 5 year old? Is everything about right and wrong about you?
>
> Wait... Never mind. That's a rhetorical question.
>
>> Failure to specify the domain of a statement, unless the domain is the
>> domain that is assumed by the conventions, is an error.
> Oh, ok. Then I am wrong.
>
> But you are still much wronger than me.
>
>> You are just proving that you aren't going to work with others.
> I don't like working with people who are much wronger than me.
>
>> Nope, since there IS a convention of definitions that apply unless
>> stated otherwise, you can build on the conventions without needed to
>> explicitly respecify those definitions.
> That's called an bandwagon fallacy. I thought you said you care about logic?
>
> https://en.wikipedia.org/wiki/Argumentum_ad_populum

That isn't what I said, so you are wrong.

>
>
>> That is why you need to specifiy if you are using something else,
>> otherwise you DO need to fully specify EVERYTHING from the beginning
>> every time.
> Or I can just ignore idiots who can't keep up with the times?
>
> Planck was right you know... A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it ...

Yes, a new system CAN become the accepted default,

>
>> No, as I said, you don't need to be a mind-reader, just know the
>> socially established conventions and follow them
> At which point did I give you the impression that I care about convention?
>
>> so be clear that you are deviating from them, and how.
> I wouldn't know that I am deviating from "social convention" now, would I?
>
> Since I don't follow conventions.

And thus are anti-social (by definition)?

When entering a group, your first responsibility is to learn there

conventions.

>
>
>> It is the guy who says *I* have to be right, instead of *WE* are right.
> In both cases - you are the only one in this conversation who is talking about being right...

No, you keep on saying that you are right because you have chosen to be.

>
> I just want better tools. I have better tools.
>
>> 0.999... = 1 IN THE DEFUALT NUMBER SYSTEM OF THE REAL NUMBER SYSTEM, is
>> a ABSOLUTE truth.
> If your truth is ABSOLUTE truth, then my truth is more ABSOLUTE than yours.
>
> infinity + 1 > infinity

Which gets you that 1 + 2 + 3 + 4 + 5 = infinity = -1/8

>
>
>> I am not a Computer, I am a person. I am not bound by the limitation of
>> a computational device, as I have Free-Will.
> I haven't seen any free will. You just follow rules. Exactly like a computer.
>
>> Maybe you should learn about that.
> I have learned about that. I have free will.
>
> That's why I can tell you don't.
>
> I have freely chosen to give up my belief in R and accept R* as a better theory.
>
> You haven't - the Real Numbers are "absolute truth" to you - they are your God. Your sacred cow.

I never said you can't beleive in *R as better, just that unless you are
talking in a group that HAS also accepted that as the default, it is an
error to assume it is the default.

You are just proving you are anti-social.

You confuse free-will with license.

WHen we have free-will, we CHOOSE to limit ourselves to that which
actually is benificial, and works with the group.

Otherwise, we are shown to be controlled by our EGO.

[Skep Dick]

You don’t even understand sarcasm, do you?

Speaking of utility… how useful is a system in which x + 1 = x -1; and x / 2 = x * 2 ?

[Skep Dick]

On Saturday, 13 August 2022 at 00:17:19 UTC+2, richar...@gmail.com wrote:
> >> That is why you need to specifiy if you are using something else,
> >> otherwise you DO need to fully specify EVERYTHING from the beginning
> >> every time.
> > Or I can just ignore idiots who can't keep up with the times?
> >
> > Planck was right you know... A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it ...
> Yes, a new system CAN become the accepted default

I know! It has become my default. When will the rest of society catch up? I don’t care…

> >> No, as I said, you don't need to be a mind-reader, just know the
> >> socially established conventions and follow them
> > At which point did I give you the impression that I care about convention?
> >
> >> so be clear that you are deviating from them, and how.
> > I wouldn't know that I am deviating from "social convention" now, would I?
> >
> > Since I don't follow conventions.
> And thus are anti-social (by definition)?

I will be social just as soon as the convention catches up to me.

> When entering a group, your first responsibility is to learn there
> conventions.

Who is entering what group? I don’t give a shit about your conventions.

> No, you keep on saying that you are right because you have chosen to be.

Is that you you are understanding my words?

That sure sounds like your vocabulary.

I am anti-social, remember? Being right doesn’t get me any social status.

> Which gets you that 1 + 2 + 3 + 4 + 5 = infinity = -1/8

Can’t take a joke, eh?

But it is funny how your don’t think the proof for 0.999… = 0 isn’t as stupid as the proof for 1 + 2 + 3 + 4 + 5 … = -1/8

> WHen we have free-will, we CHOOSE to limit ourselves to that which
> actually is benificial, and works with the group.

And what is it that you think I am doing when I choose to use a system in which x +1 > x ?

I am working with the group of humans who find them to be their default intuition about how stuff works.
Mind you, I am absolutely working AGAINST the group of ivory tower Mathematicians who have mistaken themselves as being representative of the group’s intellectuals.

I am just kicking then high Priests of Mathematics out of the temple.

> Otherwise, we are shown to be controlled by our EGO.

An egoist is not mentally equipped to tell that others aren’t like them.

[Richard Damon]

Depends what you are using it for.

Note, that many of these system you can assert that x+1 > x for a
significant subset of the system, you just need to keep track of which
types of numbers things are.

You get the ability to talk about infinities, at the cost of losing some
common properties unless you know that in THIS instance you aren't
talking about infinities, because they DO behave differently.

If you know the rules, and are clear about them, you can do a lot.

Sort of like division has this problem with 0, but as long as we watch
out for it, it is important.

So, if we have that x*y == x*z, IF we know that x isn't 0, and we are
working in the Reals, then we can say that y == z,

We can then progress to the notion that if we have x*y == x*z that
either x == 0 or y == z (or possibly both, that or is inclusive).

[Richard Damon]

maybe because the proof that 0.9999... == 1 is corret (if we are in a
system like the Reals), and gives us a nice consistant system.

The system that shows that 1 + 2 + 3 + 4 + 5 + ... == -1/8 doesn't

>
>> WHen we have free-will, we CHOOSE to limit ourselves to that which
>> actually is benificial, and works with the group.
> And what is it that you think I am doing when I choose to use a system in which x +1 > x ?
>
> I am working with the group of humans who find them to be their default intuition about how stuff works.
> Mind you, I am absolutely working AGAINST the group of ivory tower Mathematicians who have mistaken themselves as being representative of the group’s intellectuals.
>
> I am just kicking then high Priests of Mathematics out of the temple.

Then talk with them in the places where they are,

All you are showing is you don't know where you are.

>
>> Otherwise, we are shown to be controlled by our EGO.
> An egoist is not mentally equipped to tell that others aren’t like them.
>
>

As you are proving.

[dklei...@gmail.com]

On Tuesday, August 9, 2022 at 10:42:50 AM UTC-7, wyni...@gmail.com wrote:
>
> Rational number means the ratio of two integer numbers p/q.
>
Not quite. Rational numbers are the set of all ordered pairs of two
signed integers modulo the equivalence {two pairs [a,b] and [c,d]
are equivalent if and only if a*d = b*c}.

[Skep Dick]

Are you SURE you are you are using ℝ? Are you sure sure?

Because if dx^2 = 0 then dx = 0. There is no quantity in ℝ such that x^2 = 0

if dx = 0 => dy = 2x*dx => dy = 2x * 0 => dy = 0

[Skep Dick]

On Friday, 12 August 2022 at 23:30:15 UTC+2, malcolm.ar...@gmail.com wrote:

> Now all we've said is that dx is a tiny amount we add to x. Let's say that it's so small that, when
> squared, we cannot measure it at all.

That right there tells me everything I need to know. You are actually using *ℝ, but you think you are using ℝ.

The quantity you are speaking about is the infinitesimal ε such that 0 <= ε, ε^2 = 0

[Skep Dick]

Imbecille. Nobody is saying that x is a variable.

Lets start with simple things, because infinity is definitely swimming in the deep end for you.

5 is not a variable either. 5 is ASSINGED to the variable.
Once you ASSIGN 5 to x you can use x and 5 interchangeably.

>>> x
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
NameError: name 'x' is not defined
>>> x = 5
>>> x + 1
6
>>>5+1
6
>>>5+1 == x + 1
True

[Mr Flibble]

On Sat, 13 Aug 2022 01:16:00 -0700 (PDT)

Skep Dick <skepd...@gmail.com> wrote:

> On Saturday, 13 August 2022 at 00:01:38 UTC+2, Mr Flibble wrote:
> > On Fri, 12 Aug 2022 14:59:07 -0700 (PDT)
> > Skep Dick <skepd...@gmail.com> wrote:
> >
> > > On Friday, 12 August 2022 at 23:48:07 UTC+2, Mr Flibble wrote:
> > > > No, no, no and no. "infinity" is a symbol representing a
> > > > concept it is NOT equivalent to a variable, x.
> > > Golly gosh. You don't even understand how bound and unbound
> > > variable work.
> > >
> > > Start there. Infinity is a bit above your praygrade.
> > Infinity is *not* a variable.
>
> Imbecille. Nobody is saying that x is a variable.

Imbecille? Of course x is a variable, stupid.

>
> Lets start with simple things, because infinity is definitely
> swimming in the deep end for you.
>
> 5 is not a variable either. 5 is ASSINGED to the variable.
> Once you ASSIGN 5 to x you can use x and 5 interchangeably.

Ergo x is a variable, stupid.

Infinity cannot be assigned to a variable in the same way as 5 can
because unlike 5 infinity is NOT a number.

/Flibble

[Skep Dick]

On Saturday, 13 August 2022 at 13:38:12 UTC+2, Mr Flibble wrote:
> > Imbecille. Nobody is saying that x is a variable.
> Imbecille? Of course x is a variable, stupid.

**sigh** You uncharitable imbecille. Let me fix it for you (since it seems your brains's error correction algorithm is faulty)

Nobody is saying that ̶x̶ infinity is a variable.

> Infinity cannot be assigned to a variable in the same way as 5 can because unlike 5 infinity is NOT a number.

I know, you imbcecille. Which is preciesly why I am pointing out that YOU are violating your own rule. I am charging you with hyporcisy.

When you bind ∞ to x in ANY equation YOU are treating ∞ as a number! This contradicts your own rule.

IF ∞ cannot be assigned to x (a variable)
THEN you cannot replace x for ∞ in the expression "x = x"
THEREFORE the expression "∞ = ∞" is meaningless!

IF ∞ cannot be assigned to x (a variable)
THEN you cannot replace x for ∞ in the expression "x + 1 = x"
THEREFORE the expression "∞ + 1 = ∞" is meaningless!

IF ∞ cannot be assigned to x (a variable)
THEN you cannot replace x for ∞ in the expression "x - 1 = x"
THEREFORE the expression "∞ - 1 = ∞" is meaningless!

IF ∞ cannot be assigned to x (a variable)
THEN you cannot replace x for ∞ in the expression "x / 2 = x"
THEREFORE the expression "∞ /2 = ∞" is meaningless!

IF ∞ cannot be assigned to x (a variable)
THEN you cannot replace x for ∞ in the expression "x * 2 = x"
THEREFORE the expression "∞ * 2 = ∞" is meaningless!

[Mr Flibble]

No, no, no, no and no. Infinity can be used in any formula but it is
NOT acting as a variable in the formula just as a constant is not
acting as a variable in the formula.

/Flibble

[Skep Dick]

On Saturday, 13 August 2022 at 14:09:50 UTC+2, Mr Flibble wrote:
> No, no, no, no and no. Infinity can be used in any formula but it is
> NOT acting as a variable in the formula just as a constant is not
> acting as a variable in the formula.

Imbecile. When 5 is assigned to x then x also acts as a constant in the equation "1 + x = 6"

[Mr Flibble]

You must have skipped math lessons at school. x is NOT constant in "1 +
x = 6", it is a VARIABLE.

/Flibble

[Skep Dick]

On Saturday, 13 August 2022 at 14:37:21 UTC+2, Mr Flibble wrote:o x then x also acts as a constant in

> > the equation "1 + x = 6"
> You must have skipped math lessons at school. x is NOT constant in "1 +
> x = 6", it is a VARIABLE.

I told you that you don't undrstand the difference between bound and unbound variables, but you didn't even understand what that means.

The x in "1 + x = 6" is a variable when x is unbound.
The x in "1 + x = 6" is NOT a variable when x is bound to 5

Similarly:

If you give the name x to the symbol ∞;
and you give the name y to the symbol 1

Then you can write the following expression:

(x + y = x) ≡ (∞ + 1 = ∞)

Name binding. You don't understand it.

https://en.wikipedia.org/wiki/Name_binding

[Mr Flibble]

Now you are confusing/conflating mathematics with programming
languages. You really are hopeless.

/Flibble

[Skep Dick]

If I am hopeless then you are way worse. I understand exactly what Programming and Mathematics is all about.

In [1]: expression="1 + x == 6" # 1 + x = 6 is a Mathematical expression.

In [2]: eval(expression) # The expression cannot be evaluated if x is undefined. Obviously!
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
Input In [2], in <cell line: 1>()
----> 1 eval(expression)
File <string>:1, in <module>

NameError: name 'x' is not defined

In [3]: x = 5 # As soon as we define x...

In [4]: eval(expression) # the expression evaluates to True (as expected)
Out[4]: True

In [5]: x = 9 # ... and with x=9

In [6]: eval(expression) # it evaluates to False (as expected)
Out[6]: False