The Emperor's Clothes

[wij]

1/3= 0.333... + nonzero_remainder ... The HONEST IDENTITY
Hey kids, number too small is zero!

0.999...∉ [0,1)
You need to learn ADVANCED MATH to understand.

Infinitesimal does not exist?
e=lim(n->∞) (1+1/n)^n ... Is not e defined by infinitesimal and infinity?
(0.999....)^n passes 1/e, toward 0.
(1.000...1)^n passes e, toward infinity.

0.999...= lim(n->∞) (10^n-1)/10^n = lim(n->∞) 1 - 1/10^n = 1
numerator and denominator will eventually equal (10^n-1=10^n <=> -1=0).

PS. I believe TM (program,algorithm) should/would be the foundation of math.
These are for programmers.

[Ben Bacarisse]

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

> 1/3= 0.333... + nonzero_remainder ... The HONEST IDENTITY
> Hey kids, number too small is zero!

No.

> 0.999...∉ [0,1)

Yes.

> You need to learn ADVANCED MATH to understand.

I don't think it's really very advanced. You may not have come across
it yet, but I am sure you could lean it.

> Infinitesimal does not exist?

Mathematical objects exist if we define them and the results are
self-consistent. There are several number systems with things you'd
call infinitesimals. Have you considered learning about any on them?
It seems odd to post so frequently about something you don't want to
learn about.

> e=lim(n->∞) (1+1/n)^n ... Is not e defined by infinitesimal and
> infinity?

There are lots of definitions of e.

> (0.999....)^n passes 1/e, toward 0.

(0.999....)^n does not "pass" anything. It's 1. You might want to use
a different notation if you want to talk about a number infinitely close
to 1 but not equal to 1. Until you do, the people who read you will
take the usual meaning and will assert that 0.999... = 1.

> (1.000...1)^n passes e, toward infinity.

Yes, (1.000...1)^n grows without bound for increasing n > 0.

> 0.999...= lim(n->∞) (10^n-1)/10^n = lim(n->∞) 1 - 1/10^n = 1

Yes.

> numerator and denominator will eventually equal (10^n-1=10^n <=>
> -1=0).

No. 10^n-1 =/= 10^n for any n.

--
Ben.

[wij]

I just bought [Analysis 3rd Edition, Terence Tao] to see the smart guy's idea
of real number, a bit tragic to read. First chapters are high school stuff.
He starts the teaching of his students by emphasize what proof is and how to
prove, like proving 4!=0, proving a+b=b+a, a+(b+c)=(a+b)+c,... (4 chapters !!!)
Because he found many enough college students don't know how to prove.

Indeed, many 'educated' people do not really know what a proof is.
Why such a question "0.999...∉ [0,1)" needs the math. taught in college to prove?
If these questions (along with "0.999...∉ [0,1)") cannot be proved by elementary math.,
the math. is really in trouble, a big hole exists.

Correction: Title of the post should be "The Emperor's New Clothes"

[Skep Dick]

Proofs/programs (themselves) are Mathematical objects. They can be manipulated using Mathematical means.

It is the essence of reflection and “metamathematics”

[wij]

"1/3= 0.333... + nonzero_remainder" can be proved by contradiction.
If nonzero_remainder is zero, the dividing (algorithm) procedure must complete,
0.333... cannot not be infinitely long. (the proof-by-contradiction should
also apply to the proof that 4!=0 as in Tao's exercise)

I saw (skim) many of your replies about limit. Just thought that
lim(x->c) <expr> is really a formalized statement of what people actually do.
Nothing special except those ADDED by 'designer/user'.

lim(x->c) x = c is a dangerous expression, it DEFINES EQUALITY. (e.g. every
natural number is redefined in ℝ). Circular argument issue exists in many
theory/proof of real numbers.

I had a book about type theory. In recent cleaning up of my house, I threw it
away (approx. 1 ton, with several early C++ 'classic' books I thought I could
buy back anytime). I cannot talk 'type' things and probably “metamathematics”.

[Ben Bacarisse]

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

> I just bought [Analysis 3rd Edition, Terence Tao]

I thought the 3rd edition was in two volumes. Are you references to
chapters and so on about volume 1, or is there a singe volume 3rd
edition?

> to see the smart guy's idea
> of real number, a bit tragic to read. First chapters are high school
> stuff.

You can skip stuff you know.

> He starts the teaching of his students by emphasize what proof is and how to
> prove, like proving 4!=0, proving a+b=b+a, a+(b+c)=(a+b)+c,... (4 chapters !!!)
> Because he found many enough college students don't know how to prove.
> Indeed, many 'educated' people do not really know what a proof is.

Sadly true. You should pay particular attention to what he says about
proof because your arguments of often quite weak.

> Why such a question "0.999...∉ [0,1)" needs the math. taught in
> college to prove?

More precisely, 0.999... = 1.

> If these questions (along with "0.999...∉ [0,1)") cannot be proved by
> elementary math., the math. is really in trouble, a big hole exists.

How do you prove 0.999... = 1 using elementary maths? You would have to
start with what 0.999... means, and that's not usually covered (in
enough detail make a proof at least) in elementary courses.

--
Ben.

[Ben Bacarisse]

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

> "1/3= 0.333... + nonzero_remainder" can be proved by contradiction.

No it can't. With the usual meanings 1/3 = 0.333...

--
Ben.

[Richard Damon]

No, because the ... represent and unbounded number of digits, so the
division goes on forever, so the dividing algorithm never completes at a
final answer.

[wij]

Your only source of knowledge is the limit theory in the book (if dig deeper,
like most book-worm, lack concrete meaning, just words).
You keep evading the foundamental question and talk irrelevant superstition:
A logical deduction that lim(x->c) f(c)=L yields the conclusion f(c)=L (EQUAL).
Answer this question, everybody will be convinced.
Make sure you really solve it, because all such problems since at least
calculus was invented, will be solved by you !!!

[Ben Bacarisse]

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

> On Tuesday, 30 August 2022 at 23:30:28 UTC+8, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>> > "1/3= 0.333... + nonzero_remainder" can be proved by contradiction.
>> No it can't. With the usual meanings 1/3 = 0.333...
>>
>

> Your only source of knowledge is the limit theory in the book (if dig deeper,
> like most book-worm, lack concrete meaning, just words).

Yes. My source of knowledge is what the symbols are taken to mean.
Communication would be impossible if we did not have some agreed
meanings.

My dispute with the dot-deniers if that they (i.e. you) won't do the
work to define a consistent meaning. That any your refusal to use a new
notation because you don't want to be right a new meaning of an infinite
sum, you want everyone else to be wrong about the usual meaning.

> You keep evading the foundamental question and talk irrelevant superstition:
> A logical deduction that lim(x->c) f(c)=L yields the conclusion f(c)=L
> (EQUAL).

No. As I said before, if you bother to learn something you would see
that that is not the case. It is not needed for the definition of the
limit, nor can it be concluded from definition of the limit.

> Answer this question, everybody will be convinced.

There was no question that I could see. I saw in incorrect assertion
that "lim(x->c) f(c)=L yields the conclusion f(c)=L (EQUAL)". That
conclusion is invalid (though I admit to not knowing what you mean by
adding (EQUAL) to an equation).

Note also, that "lim(x->c) is a different operation to the limit
operation needed to define an infinite sum like 0.333... = 1/3.

> Make sure you really solve it, because all such problems since at
> least calculus was invented, will be solved by you !!!

Solve what? The modern theory of the reals works very well for limits
and calculus.

--
Ben.

[Keith Thompson]

wij <wyni...@gmail.com> writes:
> On Tuesday, 30 August 2022 at 23:30:28 UTC+8, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>> > "1/3= 0.333... + nonzero_remainder" can be proved by contradiction.
>> No it can't. With the usual meanings 1/3 = 0.333...
>

> Your only source of knowledge is the limit theory in the book (if dig deeper,
> like most book-worm, lack concrete meaning, just words).
> You keep evading the foundamental question and talk irrelevant superstition:
> A logical deduction that lim(x->c) f(c)=L yields the conclusion f(c)=L (EQUAL).
> Answer this question, everybody will be convinced.
> Make sure you really solve it, because all such problems since at least
> calculus was invented, will be solved by you !!!

Answer what question?

If that were a question, the answer would be no.

You wrote:
lim(x->c) f(c)=L yields the conclusion f(c)=L (EQUAL).

I think you meant:
lim(x->c) f(x)=L yields the conclusion f(c)=L (EQUAL).
And I assume the added "(EQUAL)" was just for emphasis.

No, lim(x->c) f(x)=L does not imply f(c)=L.

Let f(x) be (x^2-x)/(x-1).

lim(x->1) f(x) = 1, but f(1) is undefined.

If you want an example that doesn't involve division by 0,
let f(x) be:
42 if x = 1
x if x ≠ 1
limit(x->1) f(x) = 1, but f(1) = 42.

--
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 */

[wij]

On Thursday, 1 September 2022 at 03:05:16 UTC+8, Keith Thompson wrote:
> wij <wyni...@gmail.com> writes:
> > On Tuesday, 30 August 2022 at 23:30:28 UTC+8, Ben Bacarisse wrote:
> >> wij <wyni...@gmail.com> writes:
> >>
> >> > "1/3= 0.333... + nonzero_remainder" can be proved by contradiction.
> >> No it can't. With the usual meanings 1/3 = 0.333...
> >
> > Your only source of knowledge is the limit theory in the book (if dig deeper,
> > like most book-worm, lack concrete meaning, just words).
> > You keep evading the foundamental question and talk irrelevant superstition:
> > A logical deduction that lim(x->c) f(c)=L yields the conclusion f(c)=L (EQUAL).
> > Answer this question, everybody will be convinced.
> > Make sure you really solve it, because all such problems since at least
> > calculus was invented, will be solved by you !!!
> Answer what question?
>
> If that were a question, the answer would be no.
> You wrote:
> lim(x->c) f(c)=L yields the conclusion f(c)=L (EQUAL).
> I think you meant:
> lim(x->c) f(x)=L yields the conclusion f(c)=L (EQUAL).
> And I assume the added "(EQUAL)" was just for emphasis.
>
> No, lim(x->c) f(x)=L does not imply f(c)=L.

Exactly what I mean.
This is my point: Limit cannot yield an equal conclusion .'lim' should
always stick to its expression, 'lim' cannot be removed. Otherwise, I ask for a
valid logic removing it from the limit expression to form an identity expression.
Or, I request a CONCLUSION suitable for all kind of further application,
e.g. lim(x->0) 1-x cannot yield 1-0=1 conclusion (by limit's δ-ϵ definition).
Same as lim 0.999...=1 cannot (logically) yield the conclusion 0.999...=1.

> Let f(x) be (x^2-x)/(x-1).
>
> lim(x->1) f(x) = 1, but f(1) is undefined.
>
> If you want an example that doesn't involve division by 0,
> let f(x) be:
> 42 if x = 1
> x if x ≠ 1
> limit(x->1) f(x) = 1, but f(1) = 42.
>
> --
> 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 */

A little confusion to read. So, I just say: The first kind of "lim(x-c)" I read
is "x approaches c, but can never be c". Recent replies from internet people says
this is not required (they have their 'own revised' reasons like always).
All I know is that the normal rule of arithmetic/math in limit no more apply.
All rules are implicitly changed. There is no arithmetic rule in limit (some
special,sub-optimal rules, yes, like L’Hôpital’s Rule). The limit theory cannot
even say which of the normal arithmetic rule apply, which doesn't apply in
'limit's own arithmetic'. Along with Infinity, all are mess inside.
I don't know which of the possible variations I should respond to.
(If you, or others, believe 0.999...∈ [1,2] or 0.999...∉ [0,1), I don't deny your vision,
don't reply this post, we 'too obviously' cannot have consensus)

[Ben Bacarisse]

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

> On Thursday, 1 September 2022 at 03:05:16 UTC+8, Keith Thompson wrote:

<cut>

>> No, lim(x->c) f(x)=L does not imply f(c)=L.
>
> Exactly what I mean.
> This is my point: Limit cannot yield an equal conclusion.

It can if someone wants it to. The reals are closed under taking the
least upper bounds of sets of rationals. That's how they are defined.
It's the whole point of the set. There are other ways to express the
same definition -- Cauchy sequences converge in the reals, every Dedkind
cut is a real, and so on -- but it's the basic definition of the set we
use to do calculus.

(There's a mass of very interesting history here with the motivation
that numbers like pi and e should be actual numbers. They are not
algebraic, so the set we want needs to be closed under some operation
other than taking roots of polynomials.)

> 'lim' should always stick to its expression, 'lim' cannot be
> removed.

In the reals, that just gives us lots of ways to write numbers. The
rules of real arithmetic can not distinguish between

lim_{n->oo} 1/n!

and

lim_{n->oo) (1+1/n)^n

and so we say they are equal. Do you say they are equal? (Please
answer this question. You are not very diligent when it comes to
answering clarifying questions.)

Similarly, the rules of real arithmetic don't let us distinguish between
lim_{n->oo} Sum_{k=1,n} 9/10^k and 1. That's why we write 0.999... = 1.

No one objects to "leaving the lim there" except on grounds of
convenience. After all, people have told you repeatedly that
0.999... is just another way to write a limit. We could ditch the
... and always write the limit, but there would still be no way to tell
the difference between that long form number "lim_{n->oo} Sum_{k=1,n}
9/10^k" and "1". People will just write "1" because it's shorter.

You need to say, explicitly, that you are not talking about the real
numbers. And you need to say what numbers you /are/ talking about. And
yo need to lay out the rules or arithmetic that enable someone to use
your set whilst as the same time distinguishing between lim_{n->oo}
Sum_{k=1,n} 9/10^k" and "1". It's a lot of work. Are you up for it?

--
Ben.

[Mike Terry]

On 01/09/2022 12:27, Ben Bacarisse wrote:
> wij <wyni...@gmail.com> writes:
>
>> On Thursday, 1 September 2022 at 03:05:16 UTC+8, Keith Thompson wrote:
> <cut>
>>> No, lim(x->c) f(x)=L does not imply f(c)=L.
>>
>> Exactly what I mean.
>> This is my point: Limit cannot yield an equal conclusion.
>
> It can if someone wants it to. The reals are closed under taking the
> least upper bounds of sets of rationals. That's how they are defined.
> It's the whole point of the set. There are other ways to express the
> same definition -- Cauchy sequences converge in the reals, every Dedkind
> cut is a real, and so on -- but it's the basic definition of the set we
> use to do calculus.
>
> (There's a mass of very interesting history here with the motivation
> that numbers like pi and e should be actual numbers. They are not
> algebraic, so the set we want needs to be closed under some operation
> other than taking roots of polynomials.)
>
>> 'lim' should always stick to its expression, 'lim' cannot be
>> removed.
>
> In the reals, that just gives us lots of ways to write numbers. The
> rules of real arithmetic can not distinguish between
>
> lim_{n->oo} 1/n!

I'm sure you mean

lim_{n->oo} [sum_{k=0 to n} 1/k!]
or equivalently
sum_{n=0 to oo} 1/n!

>
> and
>
> lim_{n->oo) (1+1/n)^n
>
> and so we say they are equal. Do you say they are equal? (Please
> answer this question. You are not very diligent when it comes to
> answering clarifying questions.)

Maybe not a great example, since wij (or anyone lacking the appropriate maths background) could be
excused for having no idea that those two quite different expressions have the same limit.

>
> Similarly, the rules of real arithmetic don't let us distinguish between
> lim_{n->oo} Sum_{k=1,n} 9/10^k and 1. That's why we write 0.999... = 1.
>
> No one objects to "leaving the lim there" except on grounds of
> convenience. After all, people have told you repeatedly that
> 0.999... is just another way to write a limit. We could ditch the
> ... and always write the limit, but there would still be no way to tell
> the difference between that long form number "lim_{n->oo} Sum_{k=1,n}
> 9/10^k" and "1". People will just write "1" because it's shorter.
>
> You need to say, explicitly, that you are not talking about the real
> numbers. And you need to say what numbers you /are/ talking about. And
> yo need to lay out the rules or arithmetic that enable someone to use
> your set whilst as the same time distinguishing between lim_{n->oo}
> Sum_{k=1,n} 9/10^k" and "1". It's a lot of work. Are you up for it?
>

For me, a bigger problem with wij is that he repeatedly misrepresents the claims of "limit theory"
in an attempt to make it appear silly. If this were deliberate, I'd have no hesitation calling him
an intellectually dishonest tosser. However, I suspect he does it simply through complete
ignorance/misunderstanding of what "limit theory" actually says! Since it's been explained to him
by several people why those claims are simply wrong (not at all consequences of "limit theory") and
since he still repeats those claims /on behalf of limit theory/ , some other description applies
(less harsh than intellectual dishonesty).

[Example:

WIJ:
The limit theory perfectly says THE LIMIT of lim(x->c) f(x) is L, and
jumps to (logically invalid) conclusion f(c)=L (EQUAL).

Limit theory does NOT say or jump to any conclusion f(c)=L, as everyone keeps pointing out to wij.
Keith explained it clearly upthread - but wij will be repeating this claim verbatim in a few weeks
time...

Similarly wij should stop making claims re the beliefs of Pythagoreans - there's no reason to
believe he has any understanding of what they actually believed, and they are not around to answer
back. His putting silly words into their long dead mouths is just crass.]


Mike.

[wij]

On Thursday, 1 September 2022 at 19:27:51 UTC+8, Ben Bacarisse wrote:
> wij <wyni...@gmail.com> writes:
>
> > On Thursday, 1 September 2022 at 03:05:16 UTC+8, Keith Thompson wrote:
> <cut>
> >> No, lim(x->c) f(x)=L does not imply f(c)=L.
> >
> > Exactly what I mean.
> > This is my point: Limit cannot yield an equal conclusion.
> It can if someone wants it to. The reals are closed under taking the
> least upper bounds of sets of rationals. That's how they are defined.
> It's the whole point of the set. There are other ways to express the
> same definition -- Cauchy sequences converge in the reals, every Dedkind
> cut is a real, and so on -- but it's the basic definition of the set we
> use to do calculus.

The real is not closed (lots of irrationals cannot be expressed with finite
symbols). Maybe you are talking sub-set of ℝ.
I don't use Dedkind-cut, Cauchy sequence, theories. 1. They cannot construct
all real numbers as claimed. 2.These theories, as definition of real number,
should presume no real numbers exist (they seem to do circular argument).

> (There's a mass of very interesting history here with the motivation
> that numbers like pi and e should be actual numbers. They are not
> algebraic, so the set we want needs to be closed under some operation
> other than taking roots of polynomials.)

Number is operation/algorithm/expression (my number view). So e can be a
number, pi should be 'unreachable' by its definition. Like my definition of ∞,
assigning a symbol for it should work. e can be defined (of course, different
definition here). I don't use (high-level) set theory.
√2 is essentially an expression, so a number. One can see it as a 'name'.

> > 'lim' should always stick to its expression, 'lim' cannot be
> > removed.
> In the reals, that just gives us lots of ways to write numbers. The
> rules of real arithmetic can not distinguish between
>
> lim_{n->oo} 1/n!
>
> and
>
> lim_{n->oo) (1+1/n)^n
>
> and so we say they are equal. Do you say they are equal? (Please
> answer this question. You are not very diligent when it comes to
> answering clarifying questions.)
>
> Similarly, the rules of real arithmetic don't let us distinguish between
> lim_{n->oo} Sum_{k=1,n} 9/10^k and 1. That's why we write 0.999... = 1.
>
> No one objects to "leaving the lim there" except on grounds of
> convenience. After all, people have told you repeatedly that
> 0.999... is just another way to write a limit. We could ditch the
> ... and always write the limit, but there would still be no way to tell
> the difference between that long form number "lim_{n->oo} Sum_{k=1,n}
> 9/10^k" and "1". People will just write "1" because it's shorter.

I guess a typo in the reply above. From the idea of information theory
A= lim(x->∞) 1-1/n = 0.999...
B= lim(x->∞) 1-1/2^n = 0.999...
C= lim(x->∞) 1-1/10^n= 0.999...
D= lim(x->∞) 1-2/10^n= 0.999...

Before one determine A=B=C=D=0.999..., All such numbers(expressions) are
distinguishable, SO WE CAN DISCUSS them. (We don't invent rules to make them
equal for no GOOD and SOUND reason).
Yes, all these could come down to the definition of 'equal'. I use the definition:
A=B ::= The occurrence of A can be replaced by B, vice versa.

To make my idea more clear, starting from infinity should be simpler:
123...
566...
....34
These are all infinities (infinitely many, probably more than the first look).
They are infinity (number) because they each can be valid expression and
TM can be designed to represent them.... I found the simple and good way to
handle infinity is making it unique. Therefore,

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

From this definition, others should be clear and hopefully deterministic
(to save long discussion of 0.999... issues discussed).

> You need to say, explicitly, that you are not talking about the real
> numbers. And you need to say what numbers you /are/ talking about. And
> yo need to lay out the rules or arithmetic that enable someone to use
> your set whilst as the same time distinguishing between lim_{n->oo}
> Sum_{k=1,n} 9/10^k" and "1". It's a lot of work. Are you up for it?
>
> --
> Ben.

I already made my idea clear in https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
and mentioned many times. It was originally written for me to follow, so
something may not be clear to readers. But, I think my idea should be clear
enough so far. Adding these explanation to that file ruins its purpose.

Real numbers are not all constructable. I don't think real numbers can be better
addressed without infinity (think about the vast number of irrationals not
addressable). Some real number 'explode', some 'implode'.
What I am talking about is really noting but from the definition of infinity.
All should be followed, not some new theory, maybe just new recognition.

As you can see, I cannot name what I thought differently. Because we are dealing
the same thing (basically, the measurement of the real world, the distance
between two points in the real space). What is in text-book is inconsistent,
useful when the authority says you must say the same to gain ??? or avoid
punishment (If I were taking an exam. I would answer the same as you would.
Why not? somebody pay for it and I want the prize), or useful when nothing involving infinity.

[Ben Bacarisse]

Yes, of course. I don't take enough time with posts like this.
Thanks. I hope this does derail my explanation...

>> and
>> lim_{n->oo) (1+1/n)^n
>> and so we say they are equal. Do you say they are equal? (Please
>> answer this question. You are not very diligent when it comes to
>> answering clarifying questions.)
>
> Maybe not a great example, since wij (or anyone lacking the
> appropriate maths background) could be excused for having no idea that
> those two quite different expressions have the same limit.

OK, but it's useful to know either way. Yes, no and don't know are all
interesting bits of information.

There's a language issue here. He does not mean that we incorrectly
conclude that f(c)=L which is what Keith (and I) have explained. He
means that we must /never/ conclude that the limit is equal to anything,
especially in the case of lim_{x->oo}. That's what he means by

||| 'lim' should always stick to its expression, 'lim' cannot be
||| removed.

He means you can't "remove" the limit (what we call evaluate the limit)
to get a number. That's what motivated my particular reply -- I think I
grasped what he's been trying to say.

--
Ben.

[Ben Bacarisse]

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

> On Thursday, 1 September 2022 at 19:27:51 UTC+8, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>> > On Thursday, 1 September 2022 at 03:05:16 UTC+8, Keith Thompson wrote:
>> <cut>
>> >> No, lim(x->c) f(x)=L does not imply f(c)=L.
>> >
>> > Exactly what I mean.
>> > This is my point: Limit cannot yield an equal conclusion.
>> It can if someone wants it to. The reals are closed under taking the
>> least upper bounds of sets of rationals. That's how they are defined.
>> It's the whole point of the set. There are other ways to express the
>> same definition -- Cauchy sequences converge in the reals, every Dedkind
>> cut is a real, and so on -- but it's the basic definition of the set we
>> use to do calculus.
>
> The real is not closed (lots of irrationals cannot be expressed with finite
> symbols). Maybe you are talking sub-set of ℝ.

No. I think you need to learn a bit more. "Closed", on it's own is
meaningless in this context. And I meant what I said. R is closed with
respect to taking least upper bounds of sets or rationals. It is
defined so that limits do what I say they do and not what you want them
to do.

> I don't use Dedkind-cut, Cauchy sequence, theories. 1. They cannot construct
> all real numbers as claimed. 2.These theories, as definition of real number,
> should presume no real numbers exist (they seem to do circular
> argument).

They are not circular. I can't type of a whole book on the reals for
you. Read one. The definitions are not circular.

>> (There's a mass of very interesting history here with the motivation
>> that numbers like pi and e should be actual numbers. They are not
>> algebraic, so the set we want needs to be closed under some operation
>> other than taking roots of polynomials.)
>
> Number is operation/algorithm/expression (my number view). So e can be a
> number, pi should be 'unreachable' by its definition. Like my definition of ∞,
> assigning a symbol for it should work. e can be defined (of course, different
> definition here). I don't use (high-level) set theory.
> √2 is essentially an expression, so a number. One can see it as a
> 'name'.

Your mission, should you choose to accept it, is to work out the details
and try to persuade someone that it's worth looking at. (And I read
below you claim to have done the first half of that.)

>> > 'lim' should always stick to its expression, 'lim' cannot be
>> > removed.
>> In the reals, that just gives us lots of ways to write numbers. The
>> rules of real arithmetic can not distinguish between
>>
>> lim_{n->oo} 1/n!
>>
>> and
>>
>> lim_{n->oo) (1+1/n)^n
>>
>> and so we say they are equal. Do you say they are equal? (Please
>> answer this question. You are not very diligent when it comes to
>> answering clarifying questions.)
>>
>> Similarly, the rules of real arithmetic don't let us distinguish between
>> lim_{n->oo} Sum_{k=1,n} 9/10^k and 1. That's why we write 0.999... = 1.
>>
>> No one objects to "leaving the lim there" except on grounds of
>> convenience. After all, people have told you repeatedly that
>> 0.999... is just another way to write a limit. We could ditch the
>> ... and always write the limit, but there would still be no way to tell
>> the difference between that long form number "lim_{n->oo} Sum_{k=1,n}
>> 9/10^k" and "1". People will just write "1" because it's shorter.
>
> I guess a typo in the reply above.

Yes. Mike spotted it as well. But you did not answer the question.
You want to keep the limit and never say that it's actually equal to
something (at least I think that's what you meant). So if I write

lim+{n->oo} sum_{k=1,n} 1/k! and lim_{n->oo) (1+1/n)^n

can you tell is they are the same? Are they equal? Can we not just
give them a name and write e for either of them instead? Please answer.

> From the idea of information theory
> A= lim(x->∞) 1-1/n = 0.999...
> B= lim(x->∞) 1-1/2^n = 0.999...
> C= lim(x->∞) 1-1/10^n= 0.999...
> D= lim(x->∞) 1-2/10^n= 0.999...
>
> Before one determine A=B=C=D=0.999..., All such numbers(expressions) are
> distinguishable, SO WE CAN DISCUSS them.

Provided we have a meaning for the symbols. You have not given one
here. Standard textbooks give a meaning from which we conclude that
A=B=C=D=1 so you would be well-advised to use different symbols.

I can't stress this enough. Unless you are a crank trying to say that
the world is wrong about the reals, you a defining something new, so you
should use new notation.

> (We don't invent rules to make them
> equal for no GOOD and SOUND reason).
> Yes, all these could come down to the definition of 'equal'. I use the definition:
> A=B ::= The occurrence of A can be replaced by B, vice versa.

That's reasonable (except for a few details that are not really
important here). 0.999... can be used in place of 1 since they denote
the same real number.

In Wij-numbers 0.999___ =/= 1. Note the new notation so no one think
you are talking about the reals and limit of partial sums.

> To make my idea more clear, starting from infinity should be simpler:
> 123...
> 566...
> ....34

None of these have a conventional meaning and you have not explained
your number system so I can't comment. What happens if I add them,
divide them, subtract one from another etc.?

> These are all infinities (infinitely many, probably more than the first look).
> They are infinity (number) because they each can be valid expression and
> TM can be designed to represent them.... I found the simple and good way to
> handle infinity is making it unique. Therefore,
>
> '∞' ::=
> 1. ∀n∈ℕ, n<∞
> 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
>
> From this definition, others should be clear and hopefully deterministic
> (to save long discussion of 0.999... issues discussed).

That's a start. But I still don't know the rules for this new number
system. I can lookup other standard systems with infinite numbers, but
you won't say if you mean any of those. I expect not, since you want
this to be your very own invention. What's more, they are usually
extension to the reals, so 0.999... = 1 in them as well.

Maybe you want to extend the reals so can have 0.999... and 0.999___ as
well?

>> You need to say, explicitly, that you are not talking about the real
>> numbers. And you need to say what numbers you /are/ talking about. And
>> yo need to lay out the rules or arithmetic that enable someone to use
>> your set whilst as the same time distinguishing between lim_{n->oo}
>> Sum_{k=1,n} 9/10^k" and "1". It's a lot of work. Are you up for it?
>>
>> --
>> Ben.
>
> I already made my idea clear in
> https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download

Great. I'll assume you have done all the work. Is there any reason I
should read it?

> and mentioned many times. It was originally written for me to follow, so
> something may not be clear to readers. But, I think my idea should be clear
> enough so far. Adding these explanation to that file ruins its
> purpose.

But my posts have been about why you are wrong about the reals. I've
not said you are wrong about your own numbers. I think it's /likely/
that you are wrong about them, but I'd have to have a reason to read
about them first. And I don't see a reason. Why are they interesting?

> Real numbers are not all constructable.

True.

> I don't think real numbers can be better
> addressed without infinity (think about the vast number of irrationals not
> addressable). Some real number 'explode', some 'implode'.
> What I am talking about is really noting but from the definition of infinity.
> All should be followed, not some new theory, maybe just new
> recognition.
>
> As you can see, I cannot name what I thought differently. Because we
> are dealing the same thing (basically, the measurement of the real
> world, the distance between two points in the real space). What is in
> text-book is inconsistent, useful when the authority says you must say
> the same to gain ??? or avoid punishment (If I were taking an exam. I
> would answer the same as you would. Why not? somebody pay for it and
> I want the prize), or useful when nothing involving infinity.

Is there someone who can read over your text before posting? I find
your writing very hard to follow. I don't know what these remarks mean.

--
Ben.

[wij]

I just say something previously missed. Your wording is like always very 'political'.

ExprA: Σ(n->∞) 1/n!
ExprB: lim(n->∞) (1+1/n)^n

Yes, we can give them a name for sure. But, from the point of definition,
ExprA and ExprB are different.

e^k= ∑(n->∞) k/n! should be written as "e^k≒ ∑(n->∞) k/n!" to be more precise,
because this equation as I know come from a long deduction and
it has irrational 'remainder' omitted. ExprB is simple: (1+1/∞)^∞ (my notation)
So, simply put, ExprA and ExprB are not equal (no true equation can link them).

As said, as infinity is defined, the rest should be hopefully deterministic.

[Keith Thompson]

wij <wyni...@gmail.com> writes:
> On Thursday, 1 September 2022 at 03:05:16 UTC+8, Keith Thompson wrote:
>> wij <wyni...@gmail.com> writes:
>> > On Tuesday, 30 August 2022 at 23:30:28 UTC+8, Ben Bacarisse wrote:
>> >> wij <wyni...@gmail.com> writes:
>> >>
>> >> > "1/3= 0.333... + nonzero_remainder" can be proved by contradiction.
>> >> No it can't. With the usual meanings 1/3 = 0.333...
>> >
>> > Your only source of knowledge is the limit theory in the book (if dig deeper,
>> > like most book-worm, lack concrete meaning, just words).
>> > You keep evading the foundamental question and talk irrelevant superstition:
>> > A logical deduction that lim(x->c) f(c)=L yields the conclusion f(c)=L (EQUAL).
>> > Answer this question, everybody will be convinced.
>> > Make sure you really solve it, because all such problems since at least
>> > calculus was invented, will be solved by you !!!
>> Answer what question?
>>
>> If that were a question, the answer would be no.
>> You wrote:
>> lim(x->c) f(c)=L yields the conclusion f(c)=L (EQUAL).
>> I think you meant:
>> lim(x->c) f(x)=L yields the conclusion f(c)=L (EQUAL).
>> And I assume the added "(EQUAL)" was just for emphasis.
>>
>> No, lim(x->c) f(x)=L does not imply f(c)=L.
>
> Exactly what I mean.

Oh? I still don't know what you were asking. You specifically
challenged us to "Answer this question". Please tell us what
question you were referring to.

> This is my point: Limit cannot yield an equal conclusion .'lim' should
> always stick to its expression, 'lim' cannot be removed. Otherwise, I ask for a
> valid logic removing it from the limit expression to form an identity expression.
> Or, I request a CONCLUSION suitable for all kind of further application,
> e.g. lim(x->0) 1-x cannot yield 1-0=1 conclusion (by limit's δ-ϵ definition).
> Same as lim 0.999...=1 cannot (logically) yield the conclusion 0.999...=1.

lim(x->c) f(x)=L does not imply that f(c)=L.
lim(x->c) f(x)=L does not imply that f(c)≠L.
Sometimes it's equal, sometimes it isn't.

The limit is equal to L. The value of f(c) may or may not be equal to L.

I can't tell whether this addresses what you were talking about.

>> Let f(x) be (x^2-x)/(x-1).
>>
>> lim(x->1) f(x) = 1, but f(1) is undefined.
>>
>> If you want an example that doesn't involve division by 0,
>> let f(x) be:
>> 42 if x = 1
>> x if x ≠ 1
>> limit(x->1) f(x) = 1, but f(1) = 42.
>

> A little confusion to read. So, I just say: The first kind of "lim(x-c)" I read
> is "x approaches c, but can never be c".

Not "can never be c", but "never needs to be c". The value of lim(x->c)
f(x) does not depend on the value of f(c). The limit is defined in
terms of values of x that are arbitrarily close to, but not equal to, c.

If you don't understand how limits are conventionally defined, I suggest reading
https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%CE%B4)-definition_of_limit
which explains it better than I probably could

[...]

[Ben Bacarisse]

Since I see no reason to be interested in your "numbers", I'll thank you
for answering that in whatever system you are using Σ(n->∞) 1/n! is not
equal to lim(n->∞) (1+1/n)^n and leave it at that.

--
Ben.

[wij]

Correction: e^k ≒ ∑(n=0,∞) k^n/n!

[wij]

The conclusion from limit.

> > This is my point: Limit cannot yield an equal conclusion .'lim' should
> > always stick to its expression, 'lim' cannot be removed. Otherwise, I ask for a
> > valid logic removing it from the limit expression to form an identity expression.
> > Or, I request a CONCLUSION suitable for all kind of further application,
> > e.g. lim(x->0) 1-x cannot yield 1-0=1 conclusion (by limit's δ-ϵ definition).
> > Same as lim 0.999...=1 cannot (logically) yield the conclusion 0.999...=1.
> lim(x->c) f(x)=L does not imply that f(c)=L.
> lim(x->c) f(x)=L does not imply that f(c)≠L.
> Sometimes it's equal, sometimes it isn't.
>
> The limit is equal to L. The value of f(c) may or may not be equal to L.
>
> I can't tell whether this addresses what you were talking about.

Ok. So, explain why 0.999...=1, based on limit theory?
Nobody questions 0.9, 0.99,...,0.999... approaches 1. But how such limit
argument can lead to conclusion 0.999...=1?

> >> Let f(x) be (x^2-x)/(x-1).
> >>
> >> lim(x->1) f(x) = 1, but f(1) is undefined.
> >>
> >> If you want an example that doesn't involve division by 0,
> >> let f(x) be:
> >> 42 if x = 1
> >> x if x ≠ 1
> >> limit(x->1) f(x) = 1, but f(1) = 42.
> >
> > A little confusion to read. So, I just say: The first kind of "lim(x-c)" I read
> > is "x approaches c, but can never be c".
> Not "can never be c", but "never needs to be c". The value of lim(x->c)
> f(x) does not depend on the value of f(c). The limit is defined in
> terms of values of x that are arbitrarily close to, but not equal to, c.
>
> If you don't understand how limits are conventionally defined, I suggest reading
> https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%CE%B4)-definition_of_limit
> which explains it better than I probably could
>
> [...]
> --
> 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 */

Why you people always like to convince yourself I don't understand limit?
It is you who don't understand what limit really is.

[Ben Bacarisse]

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

> Ok. So, explain why 0.999...=1, based on limit theory?

So, just to be clear, you object to something you don't understand, yes?
You have not read an explanation of how the reals are defined?

The set R is defined (note defined -- we choose this to be true) so that
every bounded monotonic sequence of rationals denotes a real number, a
number we call "the limit". Sometimes that might be another rational,
and sometimes we just accept that the limit is a number expressible only
by that and other limit expressions.

> Nobody questions 0.9, 0.99,...,0.999... approaches 1.

(I think there is a missing , there. If not, you have ended the
sequence with 1.)

Yes. And it's a bounded monotonic sequence of rationals so by
definition it is one way to express some real number. What real number
could that be? The sequence surpasses every real that is less than 1
and yet is bounded above by one. 1 is the least upper bound of the
sequence. By definition, the sequence represents the real number 1.

The definition of lim_{n->oo} S_n formalises this.

> Why you people always like to convince yourself I don't understand
> limit?

Because you ask "explain why 0.999... = 1, based on limit theory". If
you asked why 2+2 = 4, I'd assume you don't understand arithmetic. If
you asked why the derivative of x^2 is 2x, I'd assume you don't
understand differential calculus.

Even if you disagree with what you call "limit theory", you should
understand it before criticising it. You should be able to explain why
0.999... = 1 yourself so that you objections could be taken seriously.
What's more, if you knew why 0.999... = 1 you'd be able to say exactly
what part of the explanation you object to and the discussion would
start from a position not yet reached.

--
Ben.

[Keith Thompson]

"The conclusion from limit." is a sentence fragment, not a question.

>> > This is my point: Limit cannot yield an equal conclusion .'lim' should
>> > always stick to its expression, 'lim' cannot be removed. Otherwise, I ask for a
>> > valid logic removing it from the limit expression to form an identity expression.
>> > Or, I request a CONCLUSION suitable for all kind of further application,
>> > e.g. lim(x->0) 1-x cannot yield 1-0=1 conclusion (by limit's δ-ϵ definition).
>> > Same as lim 0.999...=1 cannot (logically) yield the conclusion 0.999...=1.
>> lim(x->c) f(x)=L does not imply that f(c)=L.
>> lim(x->c) f(x)=L does not imply that f(c)≠L.
>> Sometimes it's equal, sometimes it isn't.
>>
>> The limit is equal to L. The value of f(c) may or may not be equal to L.
>>
>> I can't tell whether this addresses what you were talking about.
>
> Ok. So, explain why 0.999...=1, based on limit theory?
> Nobody questions 0.9, 0.99,...,0.999... approaches 1. But how such limit
> argument can lead to conclusion 0.999...=1?

Is that the question you've been talking about? If so, that's
disappointing, given your earlier claim:

Answer this question, everybody will be convinced.
Make sure you really solve it, because all such problems since at least
calculus was invented, will be solved by you !!!

Explaining why 0.999...=1 is not some intractable problem that has
stymied mathematicians for centuries. It's nearly trivial.

Do we even agree on what "0.999..." means? I say it means the limit of
the sequence 0.9, 0.99, 0.999, 0.9999, etc. as the number of 9s
increases without bound. The limit of that sequence is exactly 1.
(No member of the sequence has the value 1.)

Do you understand the previous paragraph? Do you agree? If not, please
explain exactly what you disagree with.

[...]

>> If you don't understand how limits are conventionally defined, I suggest reading
>> https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%CE%B4)-definition_of_limit
>> which explains it better than I probably could
>

> Why you people always like to convince yourself I don't understand limit?
> It is you who don't understand what limit really is.

OK, what is a limit? If you understand it and we don't, can you explain
it to us?

Have you read the definition I cited above? Do you think it's wrong?
If so, what exactly is wrong with it?

[wij]

On Friday, 2 September 2022 at 20:24:54 UTC+8, Ben Bacarisse wrote:
> wij <wyni...@gmail.com> writes:
>
> > Ok. So, explain why 0.999...=1, based on limit theory?
> So, just to be clear, you object to something you don't understand, yes?
> You have not read an explanation of how the reals are defined?
>
> The set R is defined (note defined -- we choose this to be true) so that
> every bounded monotonic sequence of rationals denotes a real number, a
> number we call "the limit". Sometimes that might be another rational,
> and sometimes we just accept that the limit is a number expressible only
> by that and other limit expressions.

Real numbers are not all constructable, not all are name-able.
Your theory of ℝ is flawed. No need to fabricate any flawed ADVANCED MATH
to 'prove'. As long as infinite long decimals are recognized as real number,
kids can understand why. Not even high-school math is required to prove this claim.

> > Nobody questions 0.9, 0.99,...,0.999... approaches 1.
> (I think there is a missing , there. If not, you have ended the
> sequence with 1.)
>
> Yes. And it's a bounded monotonic sequence of rationals so by
> definition it is one way to express some real number. What real number
> could that be? The sequence surpasses every real that is less than 1
> and yet is bounded above by one. 1 is the least upper bound of the
> sequence. By definition, the sequence represents the real number 1.
>
> The definition of lim_{n->oo} S_n formalises this.

Not interested in your flawed THEORY of real number. Answer my specific questions.

> > Why you people always like to convince yourself I don't understand
> > limit?
> Because you ask "explain why 0.999... = 1, based on limit theory". If
> you asked why 2+2 = 4, I'd assume you don't understand arithmetic. If
> you asked why the derivative of x^2 is 2x, I'd assume you don't
> understand differential calculus.
>
> Even if you disagree with what you call "limit theory", you should
> understand it before criticising it. You should be able to explain why
> 0.999... = 1 yourself so that you objections could be taken seriously.
> What's more, if you knew why 0.999... = 1 you'd be able to say exactly
> what part of the explanation you object to and the discussion would
> start from a position not yet reached.
>
> --
> Ben.

My question is simple, clear and specific: A valid logic to remove 'lim' from
a limit expression to form a conclusion. E.g. "lim 0.999...=1" (make it more
formal the way you like) can yield the conclusion "0.999...=1".

Skep Dick already showed that you cannot even decide whether x-1 or x+1 is
closer to infinity or not in expression "lim(x->∞) 1-1/10^x = 0.999...".
Provide more specific explanations what you mean by the occurrence of infinity
in text-books.

As said, there is no arithmetic rule applicable in limit calculation. If you are so
confident. Please provide arithmetic/math rules in limit calculation that people
can follow without ambiguity.

Above are three KEY questions for you (if you insist the kid-understandable
"0.999...∈ [0,1)" and "1/3= 0.333... + nonzero_remainer" are not correct).
Better HONESTLY answer these three key questions.

-----
I say HONESTLY is because you like playing blind, ignoring facts with your
smart talk you thought. You like to play smart, and had kept even playing smart
before olcott. Even B.H. knew what you really did. The result was 'humiliated' by
olcott. He is, in a way, smarter than you are. People can see.
You just fool yourself by your own smart talk. 'crank', 'troll' is you.
So, HONESTLY answer the questions. If you like to keep replying crap, save it.

[wij]

If there is none, why you point me to the limit theory and indicate every should follow what?

Sorry, your reply read like a new comer to the "0.999...=1" problems.
You don't seem to understand the issues of limit.
You don't realize how broadly and how deeply "0.999...=1" problem involves.
I cannot possibly repeatedly explain these to you.

[Ben Bacarisse]

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

> On Friday, 2 September 2022 at 20:24:54 UTC+8, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>> > Ok. So, explain why 0.999...=1, based on limit theory?
>> So, just to be clear, you object to something you don't understand, yes?
>> You have not read an explanation of how the reals are defined?
>>
>> The set R is defined (note defined -- we choose this to be true) so that
>> every bounded monotonic sequence of rationals denotes a real number, a
>> number we call "the limit". Sometimes that might be another rational,
>> and sometimes we just accept that the limit is a number expressible only
>> by that and other limit expressions.
>
> Real numbers are not all constructable, not all are name-able.

Yes.

> Your theory of ℝ is flawed.

It's not mine. And since every theory is flawed, the theory of the
reals will have flaws as well. But even if it's flawed, you don't get
to say what everyone else means by 0.999....

> No need to fabricate any flawed ADVANCED MATH
> to 'prove'. As long as infinite long decimals are recognized as real number,
> kids can understand why. Not even high-school math is required to
> prove this claim.

I can't understand this. Is your object that the real numbers are not
simple enough. If you have a simpler definition of R, it would be very
well received. If you have defined another simpler set of numbers,
you'd have to say why anyone would be interested in it.

>> > Nobody questions 0.9, 0.99,...,0.999... approaches 1.
>> (I think there is a missing , there. If not, you have ended the
>> sequence with 1.)
>>
>> Yes. And it's a bounded monotonic sequence of rationals so by
>> definition it is one way to express some real number. What real number
>> could that be? The sequence surpasses every real that is less than 1
>> and yet is bounded above by one. 1 is the least upper bound of the
>> sequence. By definition, the sequence represents the real number 1.
>>
>> The definition of lim_{n->oo} S_n formalises this.
>
> Not interested in your flawed THEORY of real number.

Then don't waste my time by asking "So, explain why 0.999...=1, based on
limit theory?".

> Answer my specific questions.

As far as I know I have answered every question you've posed. The only
one I could find is "So, explain why 0.999...=1, based on limit theory?"
and you tell me you are not interested in the answer.

I think it would be simpler for me if I limited myself to correcting
any mistakes I spot.

>> > Why you people always like to convince yourself I don't understand
>> > limit?
>> Because you ask "explain why 0.999... = 1, based on limit theory". If
>> you asked why 2+2 = 4, I'd assume you don't understand arithmetic. If
>> you asked why the derivative of x^2 is 2x, I'd assume you don't
>> understand differential calculus.
>>
>> Even if you disagree with what you call "limit theory", you should
>> understand it before criticising it. You should be able to explain why
>> 0.999... = 1 yourself so that you objections could be taken seriously.
>> What's more, if you knew why 0.999... = 1 you'd be able to say exactly
>> what part of the explanation you object to and the discussion would
>> start from a position not yet reached.
>>
>

> My question is simple, clear and specific: A valid logic to remove 'lim' from
> a limit expression to form a conclusion. E.g. "lim 0.999...=1" (make it more
> formal the way you like) can yield the conclusion "0.999...=1".

There's no question here.

> Skep Dick already showed that you cannot even decide whether x-1 or x+1 is
> closer to infinity or not in expression "lim(x->∞) 1-1/10^x =
> 0.999...".

So what? I'm not going to explain (again) what those symbols mean
because you've just told me you are not interested in the theory of the
reals.

> Provide more specific explanations what you mean by the occurrence of
> infinity in text-books.

You are not interested in the theory of the reals, so why should I write
a book about for you?

> As said, there is no arithmetic rule applicable in limit
> calculation. If you are so confident. Please provide arithmetic/math
> rules in limit calculation that people can follow without ambiguity.

They are all available in standard books. But you've just said you are
not interested in the theory, so why do you care? Just keep telling
everyone they are wrong, and you'll get along just fine. Like PO, SD
and the others, you don't want to get a paper published, so just keep
posting here and be happy.

> Above are three KEY questions for you (if you insist the kid-understandable
> "0.999...∈ [0,1)" and "1/3= 0.333... + nonzero_remainer" are not correct).
> Better HONESTLY answer these three key questions.

Why? The every time I answer a question you either ignore the answer
(look above for an example of a multi-paragraph answer you simply
ignored) or tell me you are not interesting what I have to say.

> -----
> I say HONESTLY is because you like playing blind, ignoring facts with
> your smart talk you thought. You like to play smart, and had kept even
> playing smart before olcott. Even B.H. knew what you really did. The
> result was 'humiliated' by olcott. He is, in a way, smarter than you
> are. People can see. You just fool yourself by your own smart
> talk. 'crank', 'troll' is you. So, HONESTLY answer the questions. If
> you like to keep replying crap, save it.

I generally look to get information from people I think are
knowledgeable and smart. You don't think I am, so why keep asking me to
explain standard things you? You probably don't think Terrence Tao is
knowledgeable and smart because you rejected his book as well. If there
is no one you consider knowledgeable and smart, then there is no one who
can answer your questions in a way that you will pay attention to. You
are doomed to remain in ignorance of how real analysis is done.

--
Ben.

[Andy Walker]

On 03/09/2022 06:31, wij wrote:
[to Ben:]

> Real numbers are not all constructable, not all are name-able.

True; and irrelevant. The theory of ℝ is different from that
of the computable numbers, and from that of the surreal numbers, and
from that of the complex numbers, and ....

> Your theory of ℝ is flawed.

"Different" is not the same as "flawed".

> No need to fabricate any flawed ADVANCED MATH
> to 'prove'. As long as infinite long decimals are recognized as real number,
> kids can understand why.

If it's all so simple, then you need to explain why it took over
2000 years to get from a recognition that some useful numbers are not
rational, to a workable knowledge of the different sorts of infinity and
to theories of computable numbers, etc.

[...]

> My question is simple, clear and specific: A valid logic to remove 'lim' from
> a limit expression to form a conclusion. E.g. "lim 0.999...=1" (make it more
> formal the way you like) can yield the conclusion "0.999...=1".

That's not a question, as others have pointed out. The best
sense I can make of your statement is that you want to know why the
limit of the sequence suggested by 0.9, 0.99, 0.999, ... being 1 means
that the limit of the sequence suggested by 0.9, 0.99, 0.999, ... is 1.
But that is trivial, and requires that you are using "0.999..." in two
different ways. But if you use "0.999..." consistently either to mean
the limit or the sequence, then you either are asking about "lim 1", or
are trying to equate a sequence to a number; neither of those seems to
be what you have in mind.

At the moment, you seem to be asking Ben and others to make
sense of some word salad that you produce. But even word salads need
context. What do /you/ mean by "0.999..."? A limit in ℝ? A sequence?
A rough equivalent to "0.(9)" with which we can follow rules with some
resemblance to Hackenbush [I call it "Hackenstrings"]? Something in
some number system other than ℝ, such as the hyperreals, surreals or
computables? What? At the moment, it's not possible even to guess.

> Skep Dick already showed that you cannot even decide whether x-1 or x+1 is
> closer to infinity or not in expression "lim(x->∞) 1-1/10^x = 0.999...".

Skep is a troll. If you ask meaningless questions, you will
get meaningless answers.

> Provide more specific explanations what you mean by the occurrence of infinity
> in text-books.
> As said, there is no arithmetic rule applicable in limit calculation. If you are so
> confident. Please provide arithmetic/math rules in limit calculation that people
> can follow without ambiguity.

As others have said, if you can explain what problems you are
having with specific wording in textbooks, then we can help. But they
are likely to be talking about ℝ; if you want to know about other sorts
of number, then you need to specify which. It would also help if you
didn't falsely accuse Ben [in particular] of dishonesty; unlike some
in this NG, he tries very hard to be helpful.

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

[wij]

I could not possibly expect your standard things, I was originally seeking for
opinion and help about MY OWN IDEA. What I did is following the trend.

I don't complain Tao (and others), who was likely in 'the system' all his life
devoting what he could.

[wij]

Lots of my response to the above would be duplicated, so I took the liberty to
save it. My purpose is not arguing with your belief.
You have made your stance clear. Mine should also be clear by now.

[Ben Bacarisse]

Andy Walker <a...@cuboid.co.uk> writes:

> At the moment, you seem to be asking Ben and others to make
> sense of some word salad that you produce. But even word salads need
> context. What do /you/ mean by "0.999..."? A limit in ℝ? A sequence?
> A rough equivalent to "0.(9)" with which we can follow rules with some
> resemblance to Hackenbush [I call it "Hackenstrings"]? Something in
> some number system other than ℝ, such as the hyperreals, surreals or
> computables? What? At the moment, it's not possible even to guess.

Well, I have a few guesses... My first guess is that it won't be
anything anyone else has come up with. My second guess is that it's not
a new set of numbers but a critique of everything conventional centred
around 0.999... = 1 denial.

And there's really only one kind of 0.999... = 1 denier these days.
Nowadays pretty much every 'alternative thinker' who posts about maths
is, in fact, a programmer, so 0.999... is a process, maybe even an
actual algorithm. Since it never gets to 1 (in this case), the equality
is obviously wrong as any fule kno. It's so clear to them that they
can't be bothered to find out what anyone else really means.

Limits (some, most, all?) must be left as limits and never resolved to
be equal to anything because the code doesn't get there.

The central lacuna is not seeing that the equality means it's the
process that is equal to 1, not that 1 is reached. The reals /are/ the
Dedkind cuts, or they /are/ the equivalence classes of Cauchy sequence.

All just guesses of course...

PS. Thanks for your kind remarks.

--
Ben.

[olcott]

Dead frogs eating boulders for lunch?

Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer

[Keith Thompson]

OK, let's step back a bit and keep it simple.

You challenged us to "Answer this question".

I don't know what question you're referring to.

Please reply to this with the actual question, phrased as a question.

[...]

[Jeff Barnett]

On 9/3/2022 3:43 PM, olcott wrote:
<snip>

> Dead frogs eating boulders for lunch?

<snip>

That's the first intelligent thing you've said in any message in any of
the dozens of POOP threads that still survive. Maybe there's hope for
you yet! Did you make it up or spot it someplace else? It's so hard to
tell because you so often copy technical wisdom out of context that
doesn't apply where you paste it. But this time you knocked it out of
the park! Congratulations from all of us.

I see that you had the foresight to copyright it, so let's hope that
it's actually yours.
--
Jeff Barnett