Expressing the idea of "sum of even number"?

[wij]

Intuitively, SumOfEvenNum= Σ(n=0,∞) 2*n
=Σ(n=0,∞) n+n
=Σ(n=0,∞) n + Σ(n=0,∞) n
=> Sum of even number is twice the sum of natural numbers (part > whole) !!!
(What if n starts from 1?)

Should the idea "sum of even numbers" be formalized as Σ(n=0,∞/2) 2*n, or in C
language as "for(n=0;;n+=2) n;"?
According to the theory that number is a procedure/expression of operation.
The latter expression has a respective TM, the former is valid in algebra (if
'∞' is defined), but not sure it is TM representable, yet.

I think the C "for" expression is closer, more precise to the interpretation
of "sum of even number". Because "Σ(n=0,∞) 2*n" really says "twice the sum of
natural number". We should mean to select the even numbers from ℕ, not
multiplying all the natural numbers by 2 and count the sum. This is a tricky
part of infinity.

Let a=∞/2, 2*a=∞. Another question arises:
If ∞ is some kind of whole number, Is ∞ an even number?
Or is |ℕ| an even number? Can ℕ be bi-parted?
Does the sentence "sum of even number" make sense?

[Skep Dick]

On Tuesday, 23 August 2022 at 07:27:20 UTC+2, wyni...@gmail.com wrote:
> Intuitively, SumOfEvenNum= Σ(n=0,∞) 2*n
> =Σ(n=0,∞) n+n
> =Σ(n=0,∞) n + Σ(n=0,∞) n
> => Sum of even number is twice the sum of natural numbers (part > whole) !!!
> (What if n starts from 1?)
>
> Should the idea "sum of even numbers" be formalized as Σ(n=0,∞/2) 2*n, or in C
> language as "for(n=0;;n+=2) n;"?

You are right inso far as Σ is just a for loop. But when you remove the conditional check; or when your conditional never matches it becomes an infinite loop. Non-halting program.

So given these two non-halting programs...

for(n=0, sum=0;;n+=1) { sum = sum + n }; return sum;
for(n=0, sum=0;;n+=2) { sum = sum + n }; return sum;

What is their value and how do you compare it? Are you sampling the value of sum; even though the function never returns it?

[Richard Damon]

This just shows that fallacy of trying to treat "Infinity" as just an
ordinary number.

[Skep Dick]

On Tuesday, 23 August 2022 at 13:22:13 UTC+2, richar...@gmail.com wrote:
> This just shows that fallacy of trying to treat "Infinity" as just an
> ordinary number.

Which is precisely the fallacy you are comitting every time you parametrize any functor with ∞

[Richard Damon]

If you mean the functor Limit, that is DEFINED to handle the case of an
Infinity.

If yours isn't, then YOUR functor is wrong, not the definition of Limit.

Maybe your logic just doesn't handle this form of overloading, and is
thus too limited.

[Skep Dick]

On Tuesday, 23 August 2022 at 13:41:25 UTC+2, richar...@gmail.com wrote:
> If you mean the functor Limit, that is DEFINED to handle the case of an
> Infinity.
>
> If yours isn't, then YOUR functor is wrong, not the definition of Limit.
>
> Maybe your logic just doesn't handle this form of overloading, and is
> thus too limited.

😂😂😂😂😂😂😂😂

Handling cases?!? you mean like case expressions/pattern matching?

Where the the functor is defined to handle the symbols 0, 1, 2, 3 (which you call "numbers")

let f = function
| 0 as x -> a x
| 1 as x -> b x
| 2 as x -> c x
| ∞ as x -> d x

You still can't define "wrong". You fucking idiot!

[Richard Damon]

So what it the problem?

YOU were the one that said it was a fallacy to give your functor an
infinity value.

If it can handle it, what is the fallacy.

If it can't handle it, the above apparently doesn't apply.

You are just proving you don't really understand what you are saying or
are just being an intentional Dick, and should be silenced.

[Skep Dick]

On Tuesday, 23 August 2022 at 14:03:16 UTC+2, richar...@gmail.com wrote:
> So what it the problem?

The problem is YOUR inconsistency.

The symbol "0" is a unique identifier. When you bind it (x=0) that is ACTUAL zero, not POTENTIAL zero.
The symbol "∞" is a unique identifier. When you bind it (x=∞) that is ACTUAL infinity, not POTENTIAL infinity.

Binding IS reification/concretization.

> YOU were the one that said it was a fallacy to give your functor an
> infinity value.

I have said no such thing, you fucking liar!

> If it can handle it, what is the fallacy.
> If it can't handle it, the above apparently doesn't apply.

What do you mean by "handle" it?!? f(x) = x can handle ANY symbol!

f(+) = +
f(<) = <
f(∞) = ∞
f(🖕) = 🖕

> You are just proving you don't really understand what you are saying or
> are just being an intentional Dick, and should be silenced.

You dumb fucking idiot! Proofs are formal (Mathematical) objects!

You are abusing the concept of "proof" by applying the concept outside of Mathematics and onto human behavior.

[Skep Dick]

On Tuesday, 23 August 2022 at 14:03:16 UTC+2, richar...@gmail.com wrote:

> You are just proving you don't really understand what you are saying or
> are just being an intentional Dick, and should be silenced.

To demonstrate how fucking stupid you are to shift the context of the conversation away from Mathematical objects and onto humans....

Do you understand what you are saying when you use the symbol "∞"? Prove it!
Also prove that the nonsense you are about to spew in proving the above constitutes a "proof".

[wij]

Average number is a simplified, restricted polynomial, e.g. Σ(r=a,b) d*B^r ....
.... d represents digit, B represents base, r represents position index.
Or, one can say the average number is a constant, name of standardized
procedure/operation, so we don't need to calculate further, Because most
properties are well known.

Such infinite numbers like the sum of even number (or irrational number if no
name is given) are defined never terminate in 'normal expression'. You cannot
ask it to equal to anything except itself, because it is unnamed.
Adding '∞' cannot express all of them. It just a symbol denoting a kind of
infinity.

For the two expressions:
(1) for(n=0, sum=0;;n+=1) { sum = sum + n }; return sum;
(2) for(n=0, sum=0;;n+=2) { sum = sum + n }; return sum;

This is the good thing using C-like expression to express 'sum'. I have not
really inspected the how part of this. This is the QUESTION of this post for
idea. However, I try to explain what I thought:

(3) for(n=0, sum1=sum2=0;;) {
sum0+=n; ++n;
sum1+=n; ++n;
};

Expression (3) may be considered equal to expression (1) in that sum=sum0+sum1.
(3) is not supposed to 'return', because the process 'never terminate', this
is the definition of infinity. (3) is computed by 'another TM'. As to how we
know 'the value', simply to say, unnamed, 'the value' is constantly changing.
If one really wants a 'value', the whole expression is the value, return the
whole expression. The corresponding reality might be that we take a snapshot
of the 'another TM' to know the sum (and in 'standardized' form).

(3)==(1), so the number expressed by (2) is smaller than that by expression (1), i.e. (2)<(1)
As said, I did not really inspect C-like expression. Such conclusion need
axiomatization.

However, I classify such questions to my 'infinite series' theory, I have a
few theorems about infinite series.

Snippet from https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download
The article is for ME, added some material for 'communication'. Suggestions of improvement are welcome.
+-----------------+
| 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).
---

[Richard Damon]

Yes, I do know what it means in the context that I have been using it.

No, I will not "prove" any of this, because you have convinced me that
you don't understand things well enough to be worth the effort. After
all, I am sure you won't understand it because it involves some abstract
concepts.

[Skep Dick]

On Wednesday, 24 August 2022 at 03:00:09 UTC+2, richar...@gmail.com wrote:
> Yes, I do know what it means in the context that I have been using it.

You don't even understand lexical scoping, but you keep talking "in context".

> No, I will not "prove" any of this, because you have convinced me that
> you don't understand things well enough to be worth the effort. After
> all, I am sure you won't understand it because it involves some abstract
> concepts.

The irony of this statement exceeds infinity. I am literally the one talking about Categorical logic. a.k.a "abstract nonsense".

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

[Richard Damon]

Yet, you still don't understand true abstractions, as you need to make
them into something concrete to think of them.

Your "Abstract Nonsense" is removing the meaning from things as you make
a concrete model of an abstraction.

So, you may talk about the abstract, but you show you don't actually
understand it in the abstract.

[Skep Dick]

On Wednesday, 24 August 2022 at 13:28:23 UTC+2, richar...@gmail.com wrote:
> Yet, you still don't understand true abstractions, as you need to make
> them into something concrete to think of them.

What a fucking idiot! I thought of the abstract idea of infinity before I turned it into a concrete symbol: ∞

> Your "Abstract Nonsense" is removing the meaning from things as you make
> a concrete model of an abstraction.

Fucking Moron. That is literally what YOU are doing with all of your "conventional" definitions! You are inventing denotation where there is none!
You are evicting true meaning from its original home (one's mind) and relocating it into symbols.

The symbol ∀ doesn't denote; or mean anything! It's just an upside-down A!
The symbol Ǝ doesn't denote; or mean anything! It's just a backwards E!

> So, you may talk about the abstract, but you show you don't actually understand it in the abstract.

SOOOO... you want me to make my abstract understanding concrete?!?!?
Right after you have insisted that making things concrete removes meaning?

What a fucking idiot!

[wij]

Don't bother. RD believe everything has to be in his Rule-Definition.
What is 'RD'? Nothing is really in there, just himself. Probably, he is still
believe this is a proof.

x=0.999...
10x= 9.999...
10x= 9+x
9x=9
x=1

Repeating decimals are rational !!!

[Richard Damon]

Which IS a valid proof when the domain of the numbers is the Finites,
aka Reals and their subsets.

Care to point out a rule of the Reals that it breaks?

[wij]

As guessed, you still believe it is a proof.
As said in the above snippet. The "..." in representing repeating decimal has tobe explained before further processing.

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)))
...

This example extends to 'repeating decimal'. Such kind of 'proof' is not a
proof but the RESULT of SELF-EXPLANATION what the "..." mean.

>
> Care to point out a rule of the Reals that it breaks?

One question a time. You don't realize you are asking for the rule of the universe.

[Richard Damon]

What do you mean "explained", you stated what it meant, that the decimal
digits just keep repeating forever.

>
> 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 "...")

I never said that 10*x = 9+x,

If your original x was defined as the sum of 9/(10^n) for all n, in the
Natural numbers >= 1.

THen 10 x would be the sum of 90/(10^n) for all n, N in the Natural Numbers.

> Ex2: Let x=√(2+√(2+√(2+...))). Then, possible interpretation of x are:
> x=√(2+x)
> x=√(2+√(2+x))
> x=√(2+√(2+√(2+x)))
> ...
>
> This example extends to 'repeating decimal'. Such kind of 'proof' is not a
> proof but the RESULT of SELF-EXPLANATION what the "..." mean.

So.

>
>>
>> Care to point out a rule of the Reals that it breaks?
>
> One question a time. You don't realize you are asking for the rule of the universe.

The Reals (aka The Real Numbers) are NOT "The Universe".

[Skep Dick]

> > One question a time. You don't realize you are asking for the rule of the universe.
> The Reals (aka The Real Numbers) are NOT "The Universe".

Shit for brains. The Reals are IN a Universe.

Your Universe has objects. Some of those objects are Rules. And one particular object must be your Rule Precedence Rule: THE rule which determines rule precedence.

https://en.wikipedia.org/wiki/Universe_(mathematics)

[Ben Bacarisse]

Yes. We have one explanation: it denotes a sum whose value is the limit
of the sequence of partial sums.

> Ex1: Let x=0.999...

so, with the only detailed explanation we have so far, x = 1.

What's missing is what /you/ think the ... really means. No one doubts
that it can be given an alternative meaning. We just don't know exactly
what you intend the meaning to be.

Do you know what you mean when you write "..."?

--
Ben.

[Skep Dick]

Of course I know what I mean when I write "...".
I mean the exact thing that everyone means - an infinite sequence off the symbol preceding "...".

x = 0.999...
999... * x = 999.... * 0.999...

Now what?

[Richard Damon]

But, at least in general, the Part is not the Whole.

Also, in general, Rules don't need "Precedence", that only needs to come
if they have special interactions.

For instance, the Rules:

x + 0 == x

and

x * 1 == x


Have no need to be ordered. Yes, we have a rule that says if not
otherwise indicated (typeically by parenthesis) and we have a multply
operation and an addition operation, like x * y + z, we interprete that
as wanting to do the multiply first, but that is in a different type of
rule.

[Ben Bacarisse]

The infinite sequence of symbols is not in doubt. Even wij has said as
much. The question is why everyone with an oh-so-clever alternative
meaning won't say more than this trivial remark about the symbols. What
are the rules the govern arithmetic with these sequences? Or, if you
can't do arithmetic with them, what /can/ be done with them?

> x = 0.999...
> 999... * x = 999.... * 0.999...
>
> Now what?

Indeed. Why won't you say? I think wij is hoping that someone else
will come up with the meaning he imagines, but it seems even you won't
help him or her out.

--
Ben.

[Skep Dick]

On Thursday, 25 August 2022 at 13:51:27 UTC+2, Ben Bacarisse wrote:
> The infinite sequence of symbols is not in doubt. Even wij has said as
> much. The question is why everyone with an oh-so-clever alternative
> meaning won't say more than this trivial remark about the symbols. What
> are the rules the govern arithmetic with these sequences? Or, if you
> can't do arithmetic with them, what /can/ be done with them?

I don't understand your question - it's too vague/imprecise.

Can you formalize it?

> Indeed. Why won't you say? I think wij is hoping that someone else
> will come up with the meaning he imagines, but it seems even you won't
> help him or her out.

Ditto.

[Skep Dick]

On Thursday, 25 August 2022 at 13:51:27 UTC+2, Ben Bacarisse wrote:

On the other hand... maybe I was being way too uncharitable.

Your question certainly doesn't make sense to me, but it must absolutely make sense to you.

Perhaps you could tell us what the rules which govern computation with the finite sequence "∞" in the expression "lim(x->∞)" are?

Maybe you can be of some help?

[Malcolm McLean]

On Thursday, 25 August 2022 at 13:17:05 UTC+1, skepd...@gmail.com wrote:
> On Thursday, 25 August 2022 at 13:51:27 UTC+2, Ben Bacarisse wrote:
> > The infinite sequence of symbols is not in doubt. Even wij has said as
> > much. The question is why everyone with an oh-so-clever alternative
> > meaning won't say more than this trivial remark about the symbols. What
> > are the rules the govern arithmetic with these sequences? Or, if you
> > can't do arithmetic with them, what /can/ be done with them?
> I don't understand your question - it's too vague/imprecise.
>

We've got the sequence 0.999... with the nines going on forever.
Now we double it. So obviously we obtain
1.999... followed by an 8? Or do we say that since the 9s go on forever,
the 8 never exists?
There's not necessarily a "right" answer. It depends which relationships you
wish to preseve with your rules for arithetic, and which you are prepared to
sacrifice.

[Skep Dick]

On Thursday, 25 August 2022 at 13:41:35 UTC+2, richar...@gmail.com wrote:
> > https://en.wikipedia.org/wiki/Universe_(mathematics)
> But, at least in general, the Part is not the Whole.

No idea what that implies.

> Also, in general, Rules don't need "Precedence", that only needs to come
> if they have special interactions.
>
> For instance, the Rules:
>
> x + 0 == x

That's not a rule. That's a proposition. And I have no idea whether you mean x + (0 == x) OR (x + 0) == (x)

> x * 1 == x

That's also a proposition. And I have no idea whether you mean x * (1 == x) OR (x * 1) == (x)

> Have no need to be ordered. Yes, we have a rule that says if not
> otherwise indicated (typeically by parenthesis) and we have a multply
> operation and an addition operation, like x * y + z, we interprete that
> as wanting to do the multiply first, but that is in a different type of
> rule.

Fucking shit-for-brains idiot.

BODMAS doesn't tell you the precedence of ==

Is it EBODMAS ?
Is it BEODMAS ?
Is it BOEDMAS ?
Is it BODMEAS ?
Is it BODMAES ?
Is it BODMASE ?

[Skep Dick]

On Thursday, 25 August 2022 at 13:41:35 UTC+2, richar...@gmail.com wrote:

Here you go, shit-for-brains...

def values(x):
return {
'x + (0 == x)': x + (0 == x),
'(x + 0) == (x)' : (x + 0) == (x),
'x * (1 == 0)' : x * (1 == 0),
'(x * 1) == 0': (x * 1) == 0
}

In [2]: values(0)
Out[2]:
{'x + (0 == x)': 1,
'(x + 0) == (x)': True,
'x * (1 == 0)': 0,
'(x * 1) == 0': True}

In [3]: values(1)
Out[3]:
{'x + (0 == x)': 1,
'(x + 0) == (x)': True,
'x * (1 == 0)': 0,
'(x * 1) == 0': False}

In [4]: values(2)
Out[4]:
{'x + (0 == x)': 2,
'(x + 0) == (x)': True,
'x * (1 == 0)': 0,
'(x * 1) == 0': False}

In [5]: values(3)
Out[5]:
{'x + (0 == x)': 3,
'(x + 0) == (x)': True,
'x * (1 == 0)': 0,
'(x * 1) == 0': False}

[Ben Bacarisse]

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

> On Thursday, 25 August 2022 at 13:51:27 UTC+2, Ben Bacarisse wrote:
>> The infinite sequence of symbols is not in doubt. Even wij has said as
>> much. The question is why everyone with an oh-so-clever alternative
>> meaning won't say more than this trivial remark about the symbols. What
>> are the rules the govern arithmetic with these sequences? Or, if you
>> can't do arithmetic with them, what /can/ be done with them?
>
> I don't understand your question - it's too vague/imprecise.

Can you are least answer the question you asked me? Or was that just
for show?

> Can you formalize it?
>
>> Indeed. Why won't you say? I think wij is hoping that someone else
>> will come up with the meaning he imagines, but it seems even you won't
>> help him or her out.
> Ditto.

Despite your deceptive editing, most readers will remember that this was
your question, not mine.

--
Ben.

[Skep Dick]

Despite your accusation of deception, most readers might bother to actually read back and see that it wasn't my question...

Here is my post:

> Of course I know what I mean when I write "...".
>I mean the exact thing that everyone means - an infinite sequence off the symbol preceding "...".

> x = 0.999...
> 999... * x = 999.... * 0.999...
> Now what?

It contains TWO questions (for those who can't tell the difference between judgmental and propositional equality)

First QUESTION: 999... * x = 999.... * 0.999...
Second QUESTION: Now what?

But perhaps it shall become clearer to you... Is the TYPE x = (999.... * 0.999...)/999... " inhabited ?

[Ben Bacarisse]

English has marker to distinguish between the two.

> First QUESTION: 999... * x = 999.... * 0.999...
> Second QUESTION: Now what?

So, will you answer it? And if not, why not?

> But perhaps it shall become clearer to you... Is the TYPE x =
> (999.... * 0.999...)/999... " inhabited ?

I don't understand your question - it's too vague/imprecise. Can you

formalize it?

--
Ben.

[Skep Dick]

On Thursday, 25 August 2022 at 18:21:52 UTC+2, Ben Bacarisse wrote:
> Skep Dick <skepd...@gmail.com> writes:
> English has marker to distinguish between the two.

Indeed! But Mathematics doesn't. A total faux-pas on behalf of Mathematicians.

> So, will you answer it? And if not, why not?

Because it's undecidable! Exactly like the question x = x.

The value of that expression is whatever the decision-procedure decides it is.

> I don't understand your question - it's too vague/imprecise. Can you
> formalize it?

At this point I am not really sure formalization will help us out of this mess...

You think "x=x" is a statement, not a question.

[Julio Di Egidio]

On Tuesday, 23 August 2022 at 07:27:20 UTC+2, wyni...@gmail.com wrote:

> Intuitively, SumOfEvenNum = Σ(n=0,∞) 2*n

As others have pointed out, (in standard mathematics)
'oo' is *not* a number, and the above is not "intuitive" but
rather an abbreviation for the mathematical definition
which is (using ASCII):

Sum_{n=0}^oo f(n) := Lim_{N->oo} Sum_{n=0}^N f(n).

The same goes with the notation for infinite unions
and intersections, and similar: where you see 'oo' that
is an abbreviation for some limit.

> If ∞ is some kind of whole number, Is ∞ an even number?

As said, standardly 'oo' indeed is not a number, just a
short-hand for taking a limit, or the result of taking a
limit where the limit diverges, as in lim_{n->oo} n = oo.

HTH,

Julio

[Jeff Barnett]

I think that if you use a standard epsilon/delta definition of limit,
you will find that, in the case you discuss above, there isn't a limit.
Note that I said using the standard definition. Do you have some
alternate definition of limit and/or the notation you used?
--
Jeff Barnett

[Julio Di Egidio]

You and your gang of retarded cunts just talk nonsense.

Stop spamming the Usenet, go to the fucking pub...

*Plonk*

Julio

[Ben Bacarisse]

Julio Di Egidio <ju...@diegidio.name> writes:

> You and your gang of retarded cunts just talk nonsense.
>
> Stop spamming the Usenet, go to the fucking pub...

Ah, there it is! I thought for a moment someone was impersonating you.

--
Ben.

[Julio Di Egidio]

Your retarded bullshit and lies instead will never change.

ESAD, you and the whole indecent band-wagon.

*Plonk*

Julio

[Julio Di Egidio]

Up. Fuckers...

Julio

[Jeff Barnett]

I think we may be talking to a badly written script; certainly not a
rational being. Think of Eliza with "plonk" and so on randomly inserted
into the response patterns. There's nothing in the responses that shows
any grasp of what it's responding to. Eliza had a built in move the the
"Julio" agent was developed without: in addition to forming an immediate
response such as "how does the rest of the family feel" keyed off a word
like aunt, it could later take a response such as "how does your aunt
have to say" and randomly toss it out. It gave the eerie feeling of
sentience in the agent and that's what's missing here.
--
Jeff Barnett

[Ben Bacarisse]

Jeff Barnett <j...@notatt.com> writes:

> On 8/25/2022 2:36 PM, Ben Bacarisse wrote:
>> Julio Di Egidio <ju...@diegidio.name> writes:
>>
>>> You and your gang of retarded cunts just talk nonsense.
>>>
>>> Stop spamming the Usenet, go to the fucking pub...
>> Ah, there it is! I thought for a moment someone was impersonating you.
>
> I think we may be talking to a badly written script; certainly not a
> rational being. Think of Eliza with "plonk" and so on randomly
> inserted into the response patterns. There's nothing in the responses
> that shows any grasp of what it's responding to.

Actually, I have had interesting technical exchanges with JDE. He is
knowledgeable about (at least) EcmaScript and Prolog, but even then it
only took a message or two for the expletives and the sadly rhetorical
plonk to surface.

--
Ben.

[Keith Thompson]

Julio Di Egidio <ju...@diegidio.name> writes:
> On Thursday, 25 August 2022 at 22:36:34 UTC+2, Ben Bacarisse wrote:
>> Julio Di Egidio <ju...@diegidio.name> writes:
>>
>> > You and your gang of retarded cunts just talk nonsense.
>> >
>> > Stop spamming the Usenet, go to the fucking pub...
>>
>> Ah, there it is! I thought for a moment someone was impersonating you.
>
> Your retarded bullshit and lies instead will never change.
>
> ESAD, you and the whole indecent band-wagon.
>
> *Plonk*
>
> Julio

Thanking you for making it clear so quickly and emphatically that I
never need to pay attention to you.

--
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 */

[Skep Dick]

On Friday, 26 August 2022 at 00:04:21 UTC+2, Keith Thompson wrote:
> Julio Di Egidio <ju...@diegidio.name> writes:
> > On Thursday, 25 August 2022 at 22:36:34 UTC+2, Ben Bacarisse wrote:
> >> Julio Di Egidio <ju...@diegidio.name> writes:
> >>
> >> > You and your gang of retarded cunts just talk nonsense.
> >> >
> >> > Stop spamming the Usenet, go to the fucking pub...
> >>
> >> Ah, there it is! I thought for a moment someone was impersonating you.
> >
> > Your retarded bullshit and lies instead will never change.
> >
> > ESAD, you and the whole indecent band-wagon.
> >
> > *Plonk*
> >
> > Julio
> Thanking you for making it clear so quickly and emphatically that I
> never need to pay attention to you.

Look, you aren't worth much paying-attention-to either, but you never seem to indicate that overtly.

If anything - you are an ass for making other people having to spend time figuring it out.

[Julio Di Egidio]

On Thursday, 25 August 2022 at 19:49:56 UTC+2, Julio Di Egidio wrote:

Up. Fuckers...

Julio

[Ben Bacarisse]

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

> At this point I am not really sure formalization will help us out of
> this mess...

It's your mess. I am quite content with the conventional interpretation
of 0.999... You, I think, don't want to agree that 0.999... = 1 but you
can't say anything useful about what else it might mean. All you've
done in fire questions at me as if I should be able to tell you what you
mean.

--
Ben.

[Skep Dick]

What is it that you don't understand about expressions of the form "x = y".

In the most pedestrian understanding (Clasical logic) such expressions could mean true; or they could mean false.

And so it goes for "0.999... = 1". It could mean true; or it could mean false.

Obviously, I've made it clear that when I say "0.999... = 1" I mean false and when I say "0.999... + ε = 1" I mean true.

Where have I ever asked you to tell me what I mean? I am asking you to tell me what you mean by ""0.999... = 1" !

[Ben Bacarisse]

Maybe wait until sober before posting?

--
Ben.

[Skep Dick]

I am pretty sure that when I sober up I'll ask you exactly the same thing...

Probably because I see "0.999... = 1" as a proposition with an unspecified semantic value and you don't.

[Ben Bacarisse]

Maybe tomorrow you will remember why I declined to explain this entirely
conventional piece of notation to you.

--
Ben.

[wij]

Read again what you wrote "lim_{n->oo} n = oo"
You said: ...'oo' indeed is not a number, just a short-hand for taking a

limit, or the result of taking a limit where the limit diverges,

That is convenient excuse to refute "oo".

Can you read it backward: "oo = lim_{n->oo} n"? What does this mean?

n approaches c infinitely but never equals to c (this is probably in an older?
definition of limit, people keeps re-phrasing in different ways, but the
essence is no different).
=> the limit of n is c // no problem
=> n=c // very problematic (invalid deduction, actually)

https://groups.google.com/g/comp.theory/c/Px5i7DAd0Vs talked about the
illogic of limit theory and ambiguous use of infinity.

[wij]

As said. 0.999... or "Repeating decimal" suffer from pattern interpretation problems:
(1) 0.999...= 0.(9) = 0.(99)= 0.(999)
(2) 0.999...= 0.((9)(99)) // Andy Walker provided such interpretation
(3) 1-1/∞= 0.999...
1-2/∞= 0.999...
1-3/2^∞= 0.999...
1-4/10^∞= 0.999...
(4) 0.999...= 0.9 + 0.999 + 0.99 + 0.9999 +...
= 0.4+0.5 + 0.444+0.555 +...
= Σ(n=1,∞) f(n) // this f(n) can be nearly anything and yields different result.

See the snippet [Infinite series] in previous post about re-grouping/
re-arrange issues of infinite series.
(5) More interpretations are possible

Which one does "0.999..." really mean? Note that these expressions eventually
will be translated to procedure/operation of natural numbers and then, physical
entities.

As to possible point of current discussion, a consensus that "Repeating decimal
does not specify a unique entity" should be established.
However, such expressions "1-1/∞" or "1-1/10^∞" or "∑(n=1,∞) 9/10^n" can specify
a unique entity, because elements in these expressions can be defined.

As to whether "1-1/2^∞" and "1-1/10^∞" are equal or not, from the snippet:
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...

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

Conclusion: A and B denote different numbers. (a physical device computing
the truth value of A=B is a deterministic process, yields false, no way 'equal').

[wij]

--- Correction:
(4) 0.999...= 0.9 + 0.09+ 0.009 + 0.0009+ ...
= 0.09 + 0.9 + 0.0009 + 0.009+ ...
= 0.04+0.05 + 0.4+0.5+ 0.0004+0.0005+ 0.004+0.005+....
= 0.44+0.55 + 0.0044+0.0055+...

[Skep Dick]

Well, it's tomorrow, and my drunk-questions were spot on - I have no idea what was confusing you.

I am only asking the question precisely because I am aware of at least two conventions. Maybe you aren't as well informed as I am?

Perhaps what you were trying to say all along is that your convention is more conventional? OK, have it! But you still havent told me WHICH convention (that is more conventional) is YOUR convention.

Are you talking about the convention (0.999... = 1) ↔ ⊤ OR the convention (0.999... = 1) ↔ ⊥ ?

[Ben Bacarisse]

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

> On Friday, 26 August 2022 at 08:16:18 UTC+8, Ben Bacarisse wrote:
>> Skep Dick <skepd...@gmail.com> writes:
>>
>> > At this point I am not really sure formalization will help us out of
>> > this mess...
>> It's your mess. I am quite content with the conventional interpretation
>> of 0.999... You, I think, don't want to agree that 0.999... = 1 but you
>> can't say anything useful about what else it might mean. All you've
>> done in fire questions at me as if I should be able to tell you what you
>> mean.
>>
>

> As said. 0.999... or "Repeating decimal" suffer from pattern
> interpretation problems:

Only here. The meaning of ... after digits is almost universally
agreed. So much so that other notations should be used for any other
meaning.

> (1) 0.999...= 0.(9) = 0.(99)= 0.(999)

Yes.

> (2) 0.999...= 0.((9)(99)) // Andy Walker provided such
> interpretation

No. Just write 0.((9)(99)) if that's what you mean (and explain the
concept either in terms of games or order relations).

> (3) 1-1/∞= 0.999...

Introducing a new number called ∞ by which one can divide other numbers
does not produce any clarity. It produces even more ambiguity. What I
would like to see from a "alterntivist" is clarity. If that requires a
new number, so be it, but the set of numbers needs to be defined along
with the operations on those.

> 1-2/∞= 0.999...
> 1-3/2^∞= 0.999...
> 1-4/10^∞= 0.999...

Without further explanation I will conclude that n/∞ = 0 and that n/k^∞
= 0. That does not sound very useful, but what else can I conclude from
your lack of explanation?

| (4) 0.999...= 0.9 + 0.09+ 0.009 + 0.0009+ ...
| = 0.09 + 0.9 + 0.0009 + 0.009+ ...
| = 0.04+0.05 + 0.4+0.5+ 0.0004+0.0005+ 0.004+0.005+....
| = 0.44+0.55 + 0.0044+0.0055+...

= 1.

| = Σ(n=1,∞) f(n) // this f(n) can be nearly anything and yields
| different result.
|
| See the snippet [Infinite series] in previous post about

| re-grouping/re-arrange issues of infinite series.

No. See any good textbook on how to calculate these limits.

> (5) More interpretations are possible

*Sigh* I'd like to see just /one/ alternative interpretation. All
anyone posts is bad algebra and one-line hints. Do the work. Define
your extension to the reals and explain how algebra is to be done in
this new set.

> Which one does "0.999..." really mean?

0.999... means lim(k->oo) Sum(k=1,n) 9/10^k. That limit is 1. It will
mean that even if you stop being lazy and define what you want it to
mean. All you will have done is add confusion. You should define (for
example) 0.999___ and then you can prove that 0.999... =/= 0.999___.

> Note that these expressions eventually will be translated to
> procedure/operation of natural numbers and then, physical entities.
>
> As to possible point of current discussion, a consensus that
> "Repeating decimal does not specify a unique entity" should be
> established.

You are not a dictator. You can't take away the meaning I (and vast
numbers of boring old mathematicians) give to the symbols 0.999... Even
when your great work is published it will (for a century or so at least)
still mean lim(k->oo) Sum(k=1,n) 9/10^k.

You can introduce ambiguity by not using the symbols like the rest of
us, or you can be a proper mathematician and introduce an unambiguous
way to say what you want. Of course, not only is that real work, you
also have the up-hill struggle to persuade anyone to be interested.

> However, such expressions "1-1/∞" or "1-1/10^∞" or
> "∑(n=1,∞) 9/10^n" can specify a unique entity, because elements in
> these expressions can be defined.

If only! Yes they /can/ be defined. I just wish someone would get off
their... er, I mean, stop prevaricating and do the work to define them
and their operations.

> As to whether "1-1/2^∞" and "1-1/10^∞" are equal or not, from the snippet:
> 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...
>
> Assume A=B

Or, more simply, prove that A = B = 1.

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

Not with the conventional meanings of any of these symbols.

> Conclusion: A and B denote different numbers. (a physical device
> computing the truth value of A=B is a deterministic process, yields
> false, no way 'equal').

Conclusion: you don't know what the symbols mean and you don't dare
define new ones with the meaning you;d like them to have.

--
Ben.

[Skep Dick]

On Friday, 26 August 2022 at 12:33:52 UTC+2, Ben Bacarisse wrote:
> Conclusion: you don't know what the symbols mean and you don't dare
> define new ones with the meaning you;d like them to have.

Yeeep. Drunk me was 100% right. We have a priest/zealot in the house!

An ardent believer in denotational semantics. The sort of typee who thinks symbols like an upside-down A's, backwards Es and "0" denote something meaningful.
A hypocrite who demands "clarity", yet cannot even begin to tell you what "clarity" denotes.

Nor could he even tell you what ∞ and "->" denote in the expression lim(k -> ∞)

[Richard Damon]

The problem is that you aren't clear on what domain you are working in.

If it is "The Reals" then 0.999... IS Equal to 1 and your numbers in (3)
just don't exist, because ∞ is NOT a "Number" that can participate in
Arithmetic.

If you aren't working in the domain of the Reals, you need to state what
domain you ARE in, and then look to see what rules are applicable in the
domain, and make sure you stay within those rules.

Yes, if you are trying to work in a system based on all the properties
of The Reals, but lets assume that ∞ can be added as an ordinary number,
you DO get all those sorts of problems, becuase that sort of system just
doesn't work.

[Richard Damon]

Gee, I explained that a long time ago, it just didn't fit your concept
of how to reason. In part, because you need the FULL expression of the
lim, not just a piece. Just the part lim (x -> ∞) doesn't have complete
meaning by itself.

lim(x -> ∞) f(x) means

What value L (if it exists) statisfies the rule that for all e > 0 we
can find a X such that for all x > X we have that | f(x) - L | < e.


It can be noted that this definition is slightly different (but related)
to the definition of lim(x -> y) f(x). when y is a finite number.

[Skep Dick]

On Friday, 26 August 2022 at 13:15:30 UTC+2, richar...@gmail.com wrote:
> The problem is that you aren't clear on what domain you are working in.

Shit-for-brains fucking idiot.

What domain are YOU working in when you SAY things like "lim(x -> ∞) f(x)". What is the domain of the lim()?!?

> If you aren't working in the domain of the Reals, you need to state what
> domain you ARE in, and then look to see what rules are applicable in the
> domain, and make sure you stay within those rules.

OK! Go ahead!

STATE YOUR DOMAIN.

What is the domain of the lim()?!?

> Yes, if you are trying to work in a system based on all the properties
> of The Reals, but lets assume that ∞ can be added as an ordinary number,
> you DO get all those sorts of problems, becuase that sort of system just
> doesn't work.

Oh, so it "doesn't work" when I pass ∞ as a parameter to +(x, ∞),
but then it matically works when you pass ∞ as a parameeter to lim(x -> ∞)

Fucking hypocrite!

[Richard Damon]

It depends on which system we are talking about, which you don't seem to
know, or care about, so you don't know anything useful.

[Skep Dick]

On Friday, 26 August 2022 at 13:22:27 UTC+2, richar...@gmail.com wrote:
> Gee, I explained that a long time ago, it just didn't fit your concept
> of how to reason. In part, because you need the FULL expression of the
> lim, not just a piece. Just the part lim (x -> ∞) doesn't have complete
> meaning by itself.

I am not asking for the complete meaning. I am asking for the incomplete meaning.

You underestand partial application, yes?

> lim(x -> ∞) f(x) means
>
> What value L (if it exists) statisfies the rule that for all e > 0 we
> can find a X such that for all x > X we have that | f(x) - L | < e.

What the fuck are "e", "L" and "X" ?!? Where did they come from?!?!

> It can be noted that this definition is slightly different (but related)
> to the definition of lim(x -> y) f(x). when y is a finite number.

Oh, I see!!!! So lim(x -> y) f(x) is polymorphic then? And it deepends on the TYPE of y?

Could you please show me the logic for the dispatcher? In particular - could you show me the implementation for the decider which determines whether y is a finite; or infinite type?

[Skep Dick]

On Friday, 26 August 2022 at 13:25:11 UTC+2, richar...@gmail.com wrote:
> > Are you talking about the convention (0.999... = 1) ↔ ⊤ OR the convention (0.999... = 1) ↔ ⊥ ?
> >
> It depends on which system we are talking about, which you don't seem to
> know, or care about, so you don't know anything useful.

Fucking shit-for-brains idiot!

That is PRECISELY what I am asking with the QUESTION: (0.999... = 1) ↔ ⊤ OR (0.999... = 1) ↔ ⊥

Are you so incompetent that you are even unable to identify your own damn axiom?!?

[Richard Damon]

Who says I use a "decider" to determine that.

This seems to be YOUR problem of trying to reduce everything to a
'computation', even when they aren't defined that way.

That becomes YOUR task to show you can do it that way.

Maybe that it a key flaw in your thought system.

[Richard Damon]

What Hypocracy?

∞ just isn't a full member of The Real Number System, with only limited
application.

If you don't acknowledge the limits of what your definitions can handle,
you break your system.

I think this just shows that you don't understand how broken you system
might be.

[Skep Dick]

On Friday, 26 August 2022 at 13:38:14 UTC+2, richar...@gmail.com wrote:
> Who says I use a "decider" to determine that.

Oh, OK! So you are openly admitting that you don't have a mechanism for deciding?

So then how do you handle dispatch?

> This seems to be YOUR problem of trying to reduce everything to a
> 'computation', even when they aren't defined that way.

Shit-for-brains fucking idiot. This has nothing to do with reduction or computation.

Either YOU (not your computer) can determine that an object is infinite, or you can't. If you can - tell me how!

> That becomes YOUR task to show you can do it that way.

That seems a bit like burden tennis, don't you think?

YOU are alraedy doing it. YOU are admitting to DOING it.

Tell us how.

> Maybe that it a key flaw in your thought system.

My system?!? What system?

I am just a debugger...

[Skep Dick]

On Friday, 26 August 2022 at 13:41:32 UTC+2, richar...@gmail.com wrote:
> ∞ just isn't a full member of The Real Number System, with only limited
> application.

😂😂😂🤣🤣🤣🤣

So it's a non-full member of The Real Number System then. Is that like being half-pregnant?

> If you don't acknowledge the limits of what your definitions can handle,
> you break your system.

Yes, I know!

So what are the limits of what lim(x->y)f(x) can handle?!?

> I think this just shows that you don't understand how broken you system
> might be.

I think you don't understand that my system is the debugger of YOUR broken system.

[Richard Damon]

So, if you don't know the system your working in, how do you know what
to do?

This is precisely the problem that wij is having, they want to be
working in The Real Number System, but don't want to need to follow the
rules of it, so get non-sense.

Your question is putting the cart before the horse, we don't DEFINIE
whether 0.999... == 1 or not, that comes out of the actual definitions
we use for the more basic things.

Remember, the axioms that are available come out of the choice of the
system we choose.

Yes, you create a system by choosing the axioms, but once created, that
choice is fixed for that system.

[Skep Dick]

On Friday, 26 August 2022 at 13:53:49 UTC+2, richar...@gmail.com wrote:
> On 8/26/22 7:37 AM, Skep Dick wrote:
> > On Friday, 26 August 2022 at 13:25:11 UTC+2, richar...@gmail.com wrote:
> >>> Are you talking about the convention (0.999... = 1) ↔ ⊤ OR the convention (0.999... = 1) ↔ ⊥ ?
> >>>
> >> It depends on which system we are talking about, which you don't seem to
> >> know, or care about, so you don't know anything useful.
> > Fucking shit-for-brains idiot!
> >
> > That is PRECISELY what I am asking with the QUESTION: (0.999... = 1) ↔ ⊤ OR (0.999... = 1) ↔ ⊥
> >
> > Are you so incompetent that you are even unable to identify your own damn axiom?!?
> >
> So, if you don't know the system your working in, how do you know what
> to do?

Well if you "do know the systeme you are working in" why can't you tell me its rules?

> This is precisely the problem that wij is having, they want to be
> working in The Real Number System, but don't want to need to follow the
> rules of it, so get non-sense.

You've been talking about The Rules for weeks. Can you show us those rules already or... ?

>
> Your question is putting the cart before the horse, we don't DEFINIE
> whether 0.999... == 1 or not, that comes out of the actual definitions
> we use for the more basic things.

Jesus fucking christ. I am asking you if 0.999... == 1 is a theorem, or if the negation of 0.999... == 1 is a theorem in your system.

Why can't you tell me?

> Remember, the axioms that are available come out of the choice of the
> system we choose.

Well that's a backwards way of going about it!

I always start with theorems first.

> Yes, you create a system by choosing the axioms, but once created, that
> choice is fixed for that system.

Which has what to do with anything?

[Richard Damon]

What is the problem with your system? I can just inspect a value and
tell if it is the value that is infinity or a Real Number, because that
is just how they are defined. ∞ is infinite, and regular numbers are
regular numbers.

I don't use "Objects".

If you want to computify things, then YOU need to figure out how to
define things so you can do that.

You are the one that seems to be having the problem figuring out what
you are doing.

[Richard Damon]

No, you system appears not to understand how to handle my system.

I can just say if y is ∞ and it works.

[Skep Dick]

On Friday, 26 August 2022 at 14:05:29 UTC+2, richar...@gmail.com wrote:
> What is the problem with your system? I can just inspect a value and
> tell if it is the value that is infinity

Ah! So infinity IS a value! So you were lying all along?

> I don't use "Objects".

HAHAHAHAHAHAHAHAHA.

So you don't have (Mathematical) object in your system?

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

Fucking idiot.

> If you want to computify things, then YOU need to figure out how to
> define things so you can do that.

I understand that perfectly.

I am just wondering why YOU think YOU are exept from that rule when it comes to computing with Real Numbers.

> You are the one that seems to be having the problem figuring out what
> you are doing.

Idiot. I know precisely what I am doing.

What I am DOING is trying to figure out HOW you are doing arithmetic with "Real numbers" which are NOT computationally defined.

[Skep Dick]

On Friday, 26 August 2022 at 14:11:59 UTC+2, richar...@gmail.com wrote:
> I can just say if y is ∞ and it works.

HAHAHAHAHAHAHAHAHAHAHA.

So you have a decider? is(x,y) -> {True, False}

Could you explain to me how your decider is any diffeerent from (x == ∞) -> {True, False} ?

[Richard Damon]

On 8/26/22 8:00 AM, Skep Dick wrote:
> On Friday, 26 August 2022 at 13:53:49 UTC+2, richar...@gmail.com wrote:
>> On 8/26/22 7:37 AM, Skep Dick wrote:
>>> On Friday, 26 August 2022 at 13:25:11 UTC+2, richar...@gmail.com wrote:
>>>>> Are you talking about the convention (0.999... = 1) ↔ ⊤ OR the convention (0.999... = 1) ↔ ⊥ ?
>>>>>
>>>> It depends on which system we are talking about, which you don't seem to
>>>> know, or care about, so you don't know anything useful.
>>> Fucking shit-for-brains idiot!
>>>
>>> That is PRECISELY what I am asking with the QUESTION: (0.999... = 1) ↔ ⊤ OR (0.999... = 1) ↔ ⊥
>>>
>>> Are you so incompetent that you are even unable to identify your own damn axiom?!?
>>>
>> So, if you don't know the system your working in, how do you know what
>> to do?
> Well if you "do know the systeme you are working in" why can't you tell me its rules?

I tend to work in the Reals, but the question didn't start with me.

You seem to have forgotten the context of the thread.

>
>
>> This is precisely the problem that wij is having, they want to be
>> working in The Real Number System, but don't want to need to follow the
>> rules of it, so get non-sense.
> You've been talking about The Rules for weeks. Can you show us those rules already or... ?
>
>>
>> Your question is putting the cart before the horse, we don't DEFINIE
>> whether 0.999... == 1 or not, that comes out of the actual definitions
>> we use for the more basic things.
> Jesus fucking christ. I am asking you if 0.999... == 1 is a theorem, or if the negation of 0.999... == 1 is a theorem in your system.
>
> Why can't you tell me?

In The Real Number System, it would be a theorem. The question comes
will wij admit they are working in The Real Number System, and thus need
to accept that fact, and that other things they are trying to do aren't
allowed, or will the admit they aren't working in The Real Number System.

>
>> Remember, the axioms that are available come out of the choice of the
>> system we choose.
> Well that's a backwards way of going about it!
>
> I always start with theorems first.

Hard to work with a Theorem if you don't know the system it is supposed
to be part of.

>
>> Yes, you create a system by choosing the axioms, but once created, that
>> choice is fixed for that system.
> Which has what to do with anything?

Truth.

[Richard Damon]

On 8/26/22 8:15 AM, Skep Dick wrote:
> On Friday, 26 August 2022 at 14:05:29 UTC+2, richar...@gmail.com wrote:
>> What is the problem with your system? I can just inspect a value and
>> tell if it is the value that is infinity
> Ah! So infinity IS a value! So you were lying all along?
>
>> I don't use "Objects".
> HAHAHAHAHAHAHAHAHA.
>
> So you don't have (Mathematical) object in your system?
>
> https://en.wikipedia.org/wiki/Mathematical_object
>
> Fucking idiot.

So, WHICH definition of "Object" are you using. You seem to keep
changing it.

You are now the one being inconsistent.

Mathematical Objects are not the exact same thing as the objects of a
programming language.

>
>> If you want to computify things, then YOU need to figure out how to
>> define things so you can do that.
> I understand that perfectly.
>
> I am just wondering why YOU think YOU are exept from that rule when it comes to computing with Real Numbers.

But you don't seem to be able to test the "type" of a object to use the
appropriate definition to go with it.

5.25 can be a Real Number

∞ is the symbol that represents infinity, which ISN'T a 'value' in the
Real Number System, but can be use in a few special situations.

>
>> You are the one that seems to be having the problem figuring out what
>> you are doing.
> Idiot. I know precisely what I am doing.
>
> What I am DOING is trying to figure out HOW you are doing arithmetic with "Real numbers" which are NOT computationally defined.

Who says they aren't computationally defined?

The definition of how you do arithmetic with the Reals is all in the
body of knowledge that is out there, and is too vast to just post in an
usenet posting.

[Richard Damon]

I have said what I have said and you have shown yourself too stupid to
understand.

You seem to not understand how context works.

To know if y is ∞, I need to know what y actually IS.

If I don't then I need to express results conditionally based on what it
might be.

somehow, you don't seem to understand this.

is(x,y) needs to know what x and y are to be evaluated.

[Skep Dick]

On Friday, 26 August 2022 at 14:27:52 UTC+2, richar...@gmail.com wrote:
> > Jesus fucking christ. I am asking you if 0.999... == 1 is a theorem, or if the negation of 0.999... == 1 is a theorem in your system.
> >
> > Why can't you tell me?
> In The Real Number System, it would be a theorem.

Holy fucking shitballs you imprecise moron! They are BOTH theorems! I am asking you WHICH "it" is YOUR theorem.

is "IT" the theorem ( 0.999... == 1); or is is "IT" the theorem not(0.999... == 1)

Can you bind "IT" to a value!?!

> The question comes
> will wij admit they are working in The Real Number System, and thus need
> to accept that fact, and that other things they are trying to do aren't
> allowed, or will the admit they aren't working in The Real Number System.

Shit-for-brains fucking idiot. You are constantly doing things which are "not allowed" IN The Real Number System.

You are doing arithmetic in R. Which is "not allowed".

Does that mean you are lying about working in The Real Number System?

> > I always start with theorems first.
> Hard to work with a Theorem if you don't know the system it is supposed
> to be part of.

Nothing hard about it! It's exactly how the Real World works!

I don't need to know which universe I am in, in order to accept gravity as a theorem.

> > Which has what to do with anything?
> Truth.

Truth? Is that like God?

Are we in alt.religion or comp.theory ?!?!

[Ben Bacarisse]

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

> On Friday, 26 August 2022 at 03:12:28 UTC+2, Ben Bacarisse wrote:
>> Skep Dick <skepd...@gmail.com> writes:

>> > Probably because I see "0.999... = 1" as a proposition with an
>> > unspecified semantic value and you don't.
>>
>> Maybe tomorrow you will remember why I declined to explain this entirely
>> conventional piece of notation to you.

> Well, it's tomorrow, and my drunk-questions were spot on - I have no
> idea what was confusing you.

So you still want me to waste time typing out standard material from
goodness know how many sources? No. Go read them.

> I am only asking the question precisely because I am aware of at least
> two conventions. Maybe you aren't as well informed as I am?
>
> Perhaps what you were trying to say all along is that your convention
> is more conventional? OK, have it! But you still havent told me WHICH
> convention (that is more conventional) is YOUR convention.
>
> Are you talking about the convention (0.999... = 1) ↔ ⊤ OR the
> convention (0.999... = 1) ↔ ⊥ ?

If you don't know what 0.999... = 1 means, you know where you can find
out.

The question is what /you/ mean. That will remain a mystery because you
won't say more than report, as stated, silly uncontested facts like
"0.999... + epsilon = 1" is true.

--
Ben.

[Skep Dick]

On Friday, 26 August 2022 at 14:39:36 UTC+2, richar...@gmail.com wrote:
> > So you don't have (Mathematical) object in your system?
> >
> > https://en.wikipedia.org/wiki/Mathematical_object
> >
> > Fucking idiot.
> So, WHICH definition of "Object" are you using. You seem to keep
> changing it.

I am not using any "definition" of object. Moron. I am just USING the concept of an object.

There's absoluteley no need to define it! Every child understands the condept intuitivley.

An object is an object is an object. Everything is an object.

> You are now the one being inconsistent.

I have told you this a dozen times already!!! I am consistently inconsistent!

It's not really my fault that you can't decide whether that amounts to consistency; or inconsistency.

> Mathematical Objects are not the exact same thing as the objects of a
> programming language.

In so far as they are conceptual - there is absolutely NO difference.

But if you want to - you are welcome to treat ALL objects as first class citizens in some programming language.

> But you don't seem to be able to test the "type" of a object to use the
> appropriate definition to go with it.

The fuck? You said the context/domain is defined.

+(x:ℝ, x:ℝ) -> y:ℝ

> 5.25 can be a Real Number

So you don't know what you are representing with with those symbols?

> ∞ is the symbol that represents infinity, which ISN'T a 'value' in the
> Real Number System, but can be use in a few special situations.

So it's half-pregnant when it needs to be?

> > What I am DOING is trying to figure out HOW you are doing arithmetic with "Real numbers" which are NOT computationally defined.
> Who says they aren't computationally defined?

Who says they are?

> The definition of how you do arithmetic with the Reals is all in the
> body of knowledge that is out there, and is too vast to just post in an
> usenet posting.

Body of knowledge or body of bullshit?

If arithmetic with infinite-precision (why does this even need saying?) Real Numbers is computable then you've solved the halting problem.

[Richard Damon]

On 8/26/22 8:43 AM, Skep Dick wrote:
> On Friday, 26 August 2022 at 14:27:52 UTC+2, richar...@gmail.com wrote:
>>> Jesus fucking christ. I am asking you if 0.999... == 1 is a theorem, or if the negation of 0.999... == 1 is a theorem in your system.
>>>
>>> Why can't you tell me?
>> In The Real Number System, it would be a theorem.
> Holy fucking shitballs you imprecise moron! They are BOTH theorems! I am asking you WHICH "it" is YOUR theorem.

Your context evaluator is obviously broken, as BY DEFINITON in the real
Number system 0.999... == 1 is the Theorem.

>
> is "IT" the theorem ( 0.999... == 1); or is is "IT" the theorem not(0.999... == 1)
>
> Can you bind "IT" to a value!?!
>
>
>> The question comes
>> will wij admit they are working in The Real Number System, and thus need
>> to accept that fact, and that other things they are trying to do aren't
>> allowed, or will the admit they aren't working in The Real Number System.
> Shit-for-brains fucking idiot. You are constantly doing things which are "not allowed" IN The Real Number System.
>
> You are doing arithmetic in R. Which is "not allowed".

Doing arithmetic on ∞ is not allowed in R, since ∞ is not an element in R,

[Skep Dick]

On Friday, 26 August 2022 at 14:44:08 UTC+2, Ben Bacarisse wrote:
> So you still want me to waste time typing out standard material from
> goodness know how many sources? No. Go read them.

Fucking idiot. Nobody is asking you to post "standard material" or ANY material.

I am asking you to identify WHICH standard you think is THE standard.

Because I've read material, and I've read material - and there's no fucking standard! There are just echo chambers.

> If you don't know what 0.999... = 1 means, you know where you can find
> out.

Fucking idiot! Of courseI know what it means in gneral! It means a Boolean!

But I am not interested in general answers, I am looking for the particular and precis answer. I am looking for the Boolean value YOU have assigned to it,
and of course I know where I can find that out. The location of that information is in YOUR head. That is PRECISELY why I am asking YOU.

> The question is what /you/ mean. That will remain a mystery because you
> won't say more than report, as stated, silly uncontested facts like
> "0.999... + epsilon = 1" is true.

What the fuck?! I have told you over and over what /I/ mean!

I have assigned the Boolean value "True" to the type "0.999... + epsilon = 1". That's precisly what it means!

"0.999... + epsilon = 1" means True:Boolean

Do you even understand what you are asking?!?

[Richard Damon]

On 8/26/22 8:53 AM, Skep Dick wrote:
> On Friday, 26 August 2022 at 14:39:36 UTC+2, richar...@gmail.com wrote:
>>> So you don't have (Mathematical) object in your system?
>>>
>>> https://en.wikipedia.org/wiki/Mathematical_object
>>>
>>> Fucking idiot.
>> So, WHICH definition of "Object" are you using. You seem to keep
>> changing it.
> I am not using any "definition" of object. Moron. I am just USING the concept of an object.
>
> There's absoluteley no need to define it! Every child understands the condept intuitivley.
>
> An object is an object is an object. Everything is an object.

So, you can't define it?

>
>> You are now the one being inconsistent.
> I have told you this a dozen times already!!! I am consistently inconsistent!
>
> It's not really my fault that you can't decide whether that amounts to consistency; or inconsistency.

Nope, you are just inconsistent so your logic system is worthless.

PERIOD.

>
>> Mathematical Objects are not the exact same thing as the objects of a
>> programming language.
> In so far as they are conceptual - there is absolutely NO difference.
>
> But if you want to - you are welcome to treat ALL objects as first class citizens in some programming language.

And what language is that?

I think you will find you just defined your system to blow up.

I guess is has in it the set of all sets that don't contain themselves.

And thus has "blown itself up"

FAIL.

>
>> But you don't seem to be able to test the "type" of a object to use the
>> appropriate definition to go with it.
> The fuck? You said the context/domain is defined.
>
> +(x:ℝ, x:ℝ) -> y:ℝ

Right, and ∞ is NOT an element of the Reals, so that doesn't define what
you meant by +(x, ∞)

[Skep Dick]

On Friday, 26 August 2022 at 14:56:46 UTC+2, richar...@gmail.com wrote:
> On 8/26/22 8:43 AM, Skep Dick wrote:
> > On Friday, 26 August 2022 at 14:27:52 UTC+2, richar...@gmail.com wrote:
> >>> Jesus fucking christ. I am asking you if 0.999... == 1 is a theorem, or if the negation of 0.999... == 1 is a theorem in your system.
> >>>
> >>> Why can't you tell me?
> >> In The Real Number System, it would be a theorem.
> > Holy fucking shitballs you imprecise moron! They are BOTH theorems! I am asking you WHICH "it" is YOUR theorem.
> Your context evaluator is obviously broken, as BY DEFINITON in the real
> Number system 0.999... == 1 is the Theorem.
> >
> > is "IT" the theorem ( 0.999... == 1); or is is "IT" the theorem not(0.999... == 1)
> >
> > Can you bind "IT" to a value!?!
> >
> >
> >> The question comes
> >> will wij admit they are working in The Real Number System, and thus need
> >> to accept that fact, and that other things they are trying to do aren't
> >> allowed, or will the admit they aren't working in The Real Number System.
> > Shit-for-brains fucking idiot. You are constantly doing things which are "not allowed" IN The Real Number System.
> >
> > You are doing arithmetic in R. Which is "not allowed".
> Doing arithmetic on ∞ is not allowed in R, since ∞ is not an element in R,

Doing ANY operations on infinite precision numbers is also not allowed in R. Because you can't perform infinite number of steps in finite time.

This is not a limit of Turing machines.
This is alimit of humans in a time-bound Universe.

[Skep Dick]

On Friday, 26 August 2022 at 15:09:47 UTC+2, richar...@gmail.com wrote:
> > An object is an object is an object. Everything is an object.
> So, you can't define it?

I did define it! Recursively! Which part of my definition are you struggling with?

> > It's not really my fault that you can't decide whether that amounts to consistency; or inconsistency.
> Nope, you are just inconsistent so your logic system is worthless.

Your logic can't tell THAT I am consistently inconsisteent.

Now THAT's worthless.

> > But if you want to - you are welcome to treat ALL objects as first class citizens in some programming language.
> And what language is that?

The programming language in which ALL mathematical objects are first class citizens.

> I think you will find you just defined your system to blow up.

Not possible. My system doesn't admit the principle of explosion.

Alrady explained it to you. A bunch of times.

> I guess is has in it the set of all sets that don't contain themselves.
> And thus has "blown itself up"
> FAIL.

You intentionally admit principles which blow up your system on trivialities !?!

If you pulled that stunt when designing industrial systems you'd be thrown in prison!

Oh wait.. you DO admit principlees which blow up your system. You already admitted that your system does "if x is ∞ then..."

https://en.wikipedia.org/wiki/Curry's_paradox

It's a shame that you have accepted sucha stupid, explosive principle as a foundation of a system...
Not smart. Not smart at all..

> > +(x:ℝ, x:ℝ) -> y:ℝ
> Right, and ∞ is NOT an element of the Reals, so that doesn't define what
> you meant by +(x, ∞)

Hypocrite.

If x is an element of the Reals you still haven't defined what you mean by lim(x -> ∞)f(x)

[Malcolm McLean]

acos(sqrt(2)) = PI/4
You can prove that geometrically (draw a right angled triangle with unit small sides. It's an isosceles triangle because the
small sides are of equal length, and the angle between them is a right angle. So the other two angles must be PI/4).

The numbers are irrational (the sides of the triangle are not ratios of each other, and the angle isn't a ratio of the angle an
arc subtends with two radii of its own length). But we can specify them exactly.

[Skep Dick]

On Friday, 26 August 2022 at 15:39:16 UTC+2, malcolm.ar...@gmail.com wrote:
> acos(sqrt(2)) = PI/4
> You can prove that geometrically (draw a right angled triangle with unit small sides. It's an isosceles triangle because the
> small sides are of equal length, and the angle between them is a right angle. So the other two angles must be PI/4).
>
> The numbers are irrational (the sides of the triangle are not ratios of each other, and the angle isn't a ratio of the angle an
> arc subtends with two radii of its own length). But we can specify them exactly.

Which has what to do with arithmetic on real numbers?

[Richard Damon]

Nope.

Nothing says that operations on the Reals need to give the full results
in finite time/steps.

That is just a limitation of YOUR domain.

[Richard Damon]

So, since "epsilon" is NOT part of the "Real Number System", you are
clearly not working with that system, even though that was the system
that wij claimed to have been working on wheen he started.

He, of course, was making a similar mistake.

[Richard Damon]

On 8/26/22 9:22 AM, Skep Dick wrote:
> On Friday, 26 August 2022 at 15:09:47 UTC+2, richar...@gmail.com wrote:
>>> An object is an object is an object. Everything is an object.
>> So, you can't define it?
> I did define it! Recursively! Which part of my definition are you struggling with?
>
>>> It's not really my fault that you can't decide whether that amounts to consistency; or inconsistency.
>> Nope, you are just inconsistent so your logic system is worthless.
> Your logic can't tell THAT I am consistently inconsisteent.
>
> Now THAT's worthless.
>
>>> But if you want to - you are welcome to treat ALL objects as first class citizens in some programming language.
>> And what language is that?
> The programming language in which ALL mathematical objects are first class citizens.
>
>> I think you will find you just defined your system to blow up.
> Not possible. My system doesn't admit the principle of explosion.

Then it doesn't support the principle of LOGIC.

>
> Alrady explained it to you. A bunch of times.

As you have amply demonstrated.

You system is consistently WORTHLESS because it doesn't have a
foundation to stand on.

>
>> I guess is has in it the set of all sets that don't contain themselves.
>> And thus has "blown itself up"
>> FAIL.
> You intentionally admit principles which blow up your system on trivialities !?!
>
> If you pulled that stunt when designing industrial systems you'd be thrown in prison!
>
> Oh wait.. you DO admit principlees which blow up your system. You already admitted that your system does "if x is ∞ then..."
>
> https://en.wikipedia.org/wiki/Curry's_paradox

So, you don't understand Curry's paradox.

I never said that x was defined as if x then ... so t doesn't fit the
definition.

>
> It's a shame that you have accepted sucha stupid, explosive principle as a foundation of a system...
> Not smart. Not smart at all..
>
>>> +(x:ℝ, x:ℝ) -> y:ℝ
>> Right, and ∞ is NOT an element of the Reals, so that doesn't define what
>> you meant by +(x, ∞)
> Hypocrite.
>
> If x is an element of the Reals you still haven't defined what you mean by lim(x -> ∞)f(x)

Obviously you lie, because I have. To repeat it:


lim(x -> ∞) f(x) means L if and only if there exists an L in the Reals
that for any e in the Reals > 0 we can find an X in the Real such that:

| f(x) - L | < e for all x > X

IF that is true, then L is the value of lim(x->∞) f(x)

If there is no such L, then the limit doesn't exist.

[wij]

On Friday, 26 August 2022 at 18:33:52 UTC+8, Ben Bacarisse wrote:
> wij <wyni...@gmail.com> writes:

>
> > On Friday, 26 August 2022 at 08:16:18 UTC+8, Ben Bacarisse wrote:
> >> Skep Dick <skepd...@gmail.com> writes:
> >>

> >> > At this point I am not really sure formalization will help us out of
> >> > this mess...
> >> It's your mess. I am quite content with the conventional interpretation
> >> of 0.999... You, I think, don't want to agree that 0.999... = 1 but you
> >> can't say anything useful about what else it might mean. All you've
> >> done in fire questions at me as if I should be able to tell you what you
> >> mean.
> >>
> >
> > As said. 0.999... or "Repeating decimal" suffer from pattern
> > interpretation problems:
> Only here. The meaning of ... after digits is almost universally
> agreed. So much so that other notations should be used for any other
> meaning.

Do you agree that derivation like this is not a proof?
(high schools use such deductions to 'prove' repeating decimals are rational).

Proof: 1=0.999...
x=0.999...
10x= 9.999...
10x= 9+x
9x=9
x=1

> > (1) 0.999...= 0.(9) = 0.(99)= 0.(999)
> Yes.

You should say no. They are not what "∑(n=1,∞) 9/10^n)" mean.

> > (2) 0.999...= 0.((9)(99)) // Andy Walker provided such
> > interpretation
> No. Just write 0.((9)(99)) if that's what you mean (and explain the
> concept either in terms of games or order relations).

The example is given by Andy. It could be valid to mean something.

> > (3) 1-1/∞= 0.999...
>
> Introducing a new number called ∞ by which one can divide other numbers
> does not produce any clarity. It produces even more ambiguity. What I
> would like to see from a "alterntivist" is clarity. If that requires a
> new number, so be it, but the set of numbers needs to be defined along
> with the operations on those.

The symbol ∞ I use is perfectly defined, not like yours.

> > 1-2/∞= 0.999...
> > 1-3/2^∞= 0.999...
> > 1-4/10^∞= 0.999...
> Without further explanation I will conclude that n/∞ = 0 and that n/k^∞
> = 0. That does not sound very useful, but what else can I conclude from
> your lack of explanation?

You are using your ambiguous ∞ again. You have no unambiguous base to say
"n/∞ = 0" or "n/k^∞= 0" (details had been discussed in another post). In this
case, n/∞ = 0 <=> n= ∞*0 <=> n=0. n has to be 0, not any number. In the
examples above, like 1-1/2^∞= 0.999..., n is not 0.

> | (4) 0.999...= 0.9 + 0.09+ 0.009 + 0.0009+ ...
> | = 0.09 + 0.9 + 0.0009 + 0.009+ ...
> | = 0.04+0.05 + 0.4+0.5+ 0.0004+0.0005+ 0.004+0.005+....
> | = 0.44+0.55 + 0.0044+0.0055+...
> = 1.
> | = Σ(n=1,∞) f(n) // this f(n) can be nearly anything and yields
> | different result.
> |
> | See the snippet [Infinite series] in previous post about
> | re-grouping/re-arrange issues of infinite series.
> No. See any good textbook on how to calculate these limits.

The limit theory perfectly says THE LIMIT of lim(x->c) f(x) is L, why you
keep pretending I don't know limit better than you do? I am questioning f(c)=L (EQUAL), like
1/∞= 2/∞= 3/∞. Do all graph of n/x join at any remote point at infinite remote
point? There is no value x in ℝ such that 1/x=2/x=3/x, not even ∞ (not even valid
in your math.). I don't think you can find any valid logic except brainlessly reciting the
limit theory, as so far exhibited.

> > (5) More interpretations are possible
> *Sigh* I'd like to see just /one/ alternative interpretation. All
> anyone posts is bad algebra and one-line hints. Do the work. Define
> your extension to the reals and explain how algebra is to be done in
> this new set.

I don't see you really understand algebra.

> > Which one does "0.999..." really mean?
> 0.999... means lim(k->oo) Sum(k=1,n) 9/10^k. That limit is 1. It will
> mean that even if you stop being lazy and define what you want it to
> mean. All you will have done is add confusion. You should define (for
> example) 0.999___ and then you can prove that 0.999... =/= 0.999___.

It is you who are lazy, just reciting textbook to pass exam. for ???.

> > Note that these expressions eventually will be translated to
> > procedure/operation of natural numbers and then, physical entities.
> >
> > As to possible point of current discussion, a consensus that
> > "Repeating decimal does not specify a unique entity" should be
> > established.
> You are not a dictator. You can't take away the meaning I (and vast
> numbers of boring old mathematicians) give to the symbols 0.999... Even
> when your great work is published it will (for a century or so at least)
> still mean lim(k->oo) Sum(k=1,n) 9/10^k.
>
> You can introduce ambiguity by not using the symbols like the rest of
> us, or you can be a proper mathematician and introduce an unambiguous
> way to say what you want. Of course, not only is that real work, you
> also have the up-hill struggle to persuade anyone to be interested.

Let's see. I estimate it won't take long. Because I try to be reasonable.

> > However, such expressions "1-1/∞" or "1-1/10^∞" or
> > "∑(n=1,∞) 9/10^n" can specify a unique entity, because elements in
> > these expressions can be defined.
> If only! Yes they /can/ be defined. I just wish someone would get off
> their... er, I mean, stop prevaricating and do the work to define them
> and their operations.

A= Σ(n=1,∞) 1/2^n = 0.999... Agree?
If you really understand algebra, A!=B is logically unavoidable.

> > As to whether "1-1/2^∞" and "1-1/10^∞" are equal or not, from the snippet:
> > 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...
> >
> > Assume A=B
> Or, more simply, prove that A = B = 1.
> > <=> 1-1/2^∞= 1-1/10^∞ // converted from the formula of geometric series
> > <=> 1/2^∞= 1/10^∞
> > <=> 5^∞=1
> > <=> false
> Not with the conventional meanings of any of these symbols.

I just use a perfectly defined symbol '∞', nothing more.
If you really understand algebra, the definition of '∞' is sufficient:
1. ∀n∈ℕ, n<∞
2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞

> > Conclusion: A and B denote different numbers. (a physical device
> > computing the truth value of A=B is a deterministic process, yields
> > false, no way 'equal').
> Conclusion: you don't know what the symbols mean and you don't dare
> define new ones with the meaning you;d like them to have.
>
> --
> Ben.

The meaning the symbol ∞ used by you (and in textbook) is vague, all you
can say is "it is a concept", "it does not exist"..., nothing very significant.

[Malcolm McLean]

You said that no "operations" could be performed on irrationals of infinite precision.
So I gave an example of an operation. We've got one irrational (root 2) and one operation
(arc cosine) and, if we're working in radians, we obtain another irrational (PI/4). These are
exact.

[wij]

Just comment. I don't get the point what you two are talking about.
The definition of pi is defined as the ratio of the circumference and diameter
of a circle. No other definition can 'exactly' define the value of pi.
Many irrationals can only be expressed like e, pi, sqrt(2), atan(1),...
My point is that if '∞' not defined, many existing irrationals are difficult to
talk about. e.g. many infinite series whose limit is pi but the 'value' of those
expressions are themselves, another irrational (endless process), not exactly pi.

[Richard Damon]

What you seem to miss is tha "Infinite Sums" are defined by limit
theory. Thus, while at any finite stage, there total sum will not be
exactly pi, the final limit will be.

This is in fact, the nature of all the irrationals, we may not be able
to get an exact value in finite time, but their definition converges in
the limit to the exact value.

Some, like sqrt(2) have a "finite represention" in that computational
form, and we can perhaps do exact algerbra based on that form, but to
get a numeric "digits" representation is only available in the limit.

There are many series/sums that can be show DO exactly represent pi "in
the limit" which is the DEFINITION (in the Reals) of those series/sums.

[Malcolm McLean]

e to the PI i equals minus one (Euler's identity). So if we can agree on e, i, and minus
one, then that gives us another definition of PI.
Whilst normally e is defined in terms of infinite series, it can also be defined as the
value for which the differential of e^x (taking tiny steps in x) is e^x.

However there is no finite sum of rationals which yields an irrational. So if you're
teaching mathematics, you need to make sure that there's at least a rudimentary
understanding of infinity before you introduce the concept of irrational numbers.

[wij]

For those who doubt the result of calculus is refuted, not exactly.
The derivative result involving power function is checked exact, exponential and
trigonometric functions are still questionable (but the error is infinitesimal,
most engineering problems don't need to worry much).

[Malcolm McLean]

Euler's number is the number you obtain when something grows in proportion to its
size, or to say the same thing in a different manner, changes rate of change by its
rate of change.

[Richard Damon]

Nope, there derivative results of the trigonometric functions IS exact
(at least if you are talking in the domain of R, or possible R^N, like
the complex numbers).

The key is that if you are seeing infinitesimals, you have left the
concept of the Reals, as infinitesimals are not part of the Real Number
System.

Note, that as normally used, "Calculus" is a math field defined on the
Real Number Field (or with simple extension, the Complex Numbers or
Vectors in R^N)

Since it works by "defining" operations with a specific class of
infinitesimals that do not leave its operations, there are difficulties
with trying to combine it with other extensions to the Real Number
System that also add infinities and infinitesimals.

[Ben Bacarisse]

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

> On Friday, 26 August 2022 at 18:33:52 UTC+8, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>> > On Friday, 26 August 2022 at 08:16:18 UTC+8, Ben Bacarisse wrote:
>> >> Skep Dick <skepd...@gmail.com> writes:
>> >>
>> >> > At this point I am not really sure formalization will help us out of
>> >> > this mess...
>> >> It's your mess. I am quite content with the conventional interpretation
>> >> of 0.999... You, I think, don't want to agree that 0.999... = 1 but you
>> >> can't say anything useful about what else it might mean. All you've
>> >> done in fire questions at me as if I should be able to tell you what you
>> >> mean.
>> >>
>> >
>> > As said. 0.999... or "Repeating decimal" suffer from pattern
>> > interpretation problems:
>> Only here. The meaning of ... after digits is almost universally
>> agreed. So much so that other notations should be used for any other
>> meaning.
>
> Do you agree that derivation like this is not a proof?
> (high schools use such deductions to 'prove' repeating decimals are
> rational).

What a shame. But it's irrelevant to the discussion here.

> Proof: 1=0.999...
> x=0.999...
> 10x= 9.999...
> 10x= 9+x
> 9x=9
> x=1

Eh? The fact that 0.999... = 1 comes from (a) defining field in
question (the reals), (b) defining what the ... means (an infinite sum),
(c) defining what an infinite sum is (the limit of the sequence on
partial sums), and finally, showing that the limit of the sum in
question is 1.

>> > (1) 0.999...= 0.(9) = 0.(99)= 0.(999)
>> Yes.
>
> You should say no. They are not what "∑(n=1,∞) 9/10^n)" mean.

They all mean the same. The ()s denote repetition of digits. They have
no effect on how the sum of digits is determined.

>> > (2) 0.999...= 0.((9)(99)) // Andy Walker provided such
>> > interpretation
>> No. Just write 0.((9)(99)) if that's what you mean (and explain the
>> concept either in terms of games or order relations).
>
> The example is given by Andy. It could be valid to mean something.

Not my concern. Cranks often misrepresent what others have said and I
am not going to defend something I never said.

>> > (3) 1-1/∞= 0.999...
>>
>> Introducing a new number called ∞ by which one can divide other numbers
>> does not produce any clarity. It produces even more ambiguity. What I
>> would like to see from a "alterntivist" is clarity. If that requires a
>> new number, so be it, but the set of numbers needs to be defined along
>> with the operations on those.
>
> The symbol ∞ I use is perfectly defined, not like yours.

Yet I have not see a definition of it from you. Why is that?

>> > 1-2/∞= 0.999...
>> > 1-3/2^∞= 0.999...
>> > 1-4/10^∞= 0.999...
>> Without further explanation I will conclude that n/∞ = 0 and that n/k^∞
>> = 0. That does not sound very useful, but what else can I conclude from
>> your lack of explanation?
>
> You are using your ambiguous ∞ again.

Don't be silly. It's not "my" ∞. You used it in a context in which I
can determine that the sub-expression (e.g. 3/∞) is zero. You give no
meaning for 3/∞ other than it must be zero, since you now know that
0.999... = 1. I will make any deductions I like based on the fact that
0.999... = 1 so you really should start using some new notation what
whatever it is you want to be "not quite 1".

>> | (4) 0.999...= 0.9 + 0.09+ 0.009 + 0.0009+ ...
>> | = 0.09 + 0.9 + 0.0009 + 0.009+ ...
>> | = 0.04+0.05 + 0.4+0.5+ 0.0004+0.0005+ 0.004+0.005+....
>> | = 0.44+0.55 + 0.0044+0.0055+...
>> = 1.
>> | = Σ(n=1,∞) f(n) // this f(n) can be nearly anything and yields
>> | different result.
>> |
>> | See the snippet [Infinite series] in previous post about
>> | re-grouping/re-arrange issues of infinite series.
>> No. See any good textbook on how to calculate these limits.
>
> The limit theory perfectly says THE LIMIT of lim(x->c) f(x) is L, why you
> keep pretending I don't know limit better than you do?

You don't know how to define the other sort of limit. It's staggering
that you will accuse several people who appear to have degrees in
mathematics of not knowing how limits are defined simply because you
think there is only one.

> I am questioning f(c)=L (EQUAL),

No one is talking about such a limit. Pick up a book. Learn about the
asymptotic limit and then start being more respectful of the years some
people have put into learning about this material.

> like 1/∞= 2/∞= 3/∞. Do all graph of
> n/x join at any remote point at infinite remote point? There is no
> value x in ℝ such that 1/x=2/x=3/x, not even ∞ (not even valid in your
> math.).

Indeed. So whatever is you meant by ∞ when you wrote that 1-2/∞=
0.999... was not in ℝ. Will you ever say what it is? I doubt it. That
would involve doing some real work.

> I don't think you can find any valid logic except brainlessly
> reciting the limit theory, as so far exhibited.

Yes, I am stuck having to report on the result of four centuries of
mathematical investigation into the reals, limits and calculus.

>> > (5) More interpretations are possible
>> *Sigh* I'd like to see just /one/ alternative interpretation. All
>> anyone posts is bad algebra and one-line hints. Do the work. Define
>> your extension to the reals and explain how algebra is to be done in
>> this new set.
>
> I don't see you really understand algebra.

Of course you don't.

>> > Which one does "0.999..." really mean?
>> 0.999... means lim(k->oo) Sum(k=1,n) 9/10^k. That limit is 1. It will
>> mean that even if you stop being lazy and define what you want it to
>> mean. All you will have done is add confusion. You should define (for
>> example) 0.999___ and then you can prove that 0.999... =/= 0.999___.
>
> It is you who are lazy, just reciting textbook to pass exam. for ???.

But I am not claiming anything new or interesting. Should I join the
massed ranks of cranks that post here and make something up, just so you
can't call me lazy? No, that would be daft. So, yes, I will be lazy.
I will continue to recite the accumulated understanding I got from my
many years of education.

>> > Note that these expressions eventually will be translated to
>> > procedure/operation of natural numbers and then, physical entities.
>> >
>> > As to possible point of current discussion, a consensus that
>> > "Repeating decimal does not specify a unique entity" should be
>> > established.
>> You are not a dictator. You can't take away the meaning I (and vast
>> numbers of boring old mathematicians) give to the symbols 0.999... Even
>> when your great work is published it will (for a century or so at least)
>> still mean lim(k->oo) Sum(k=1,n) 9/10^k.
>>
>> You can introduce ambiguity by not using the symbols like the rest of
>> us, or you can be a proper mathematician and introduce an unambiguous
>> way to say what you want. Of course, not only is that real work, you
>> also have the up-hill struggle to persuade anyone to be interested.
>
> Let's see. I estimate it won't take long. Because I try to be
> reasonable.

No, refusing to say what you mean is not reasonable.

>> > However, such expressions "1-1/∞" or "1-1/10^∞" or
>> > "∑(n=1,∞) 9/10^n" can specify a unique entity, because elements in
>> > these expressions can be defined.
>> If only! Yes they /can/ be defined. I just wish someone would get off
>> their... er, I mean, stop prevaricating and do the work to define them
>> and their operations.

Still no definition of what you mean of course.

> A= Σ(n=1,∞) 1/2^n = 0.999... Agree?

Yes, because 0.999... = Σ(n=1,∞) 9/10^n = 1.

> If you really understand algebra, A!=B is logically unavoidable.

The B you defined before: B= Σ(n=1,∞) 9/10^n, has the same sum.

Ultimately, there is no getting around this basic disagreement. And if
you knew how to find the sum (i.e. the limit of the sequence of partial
sums) you would know that the sums are equal. This is an argument on
the level of 2+2 != 1+3. The truth of the matter follows from the
conventional meaning of the symbols.

>> > As to whether "1-1/2^∞" and "1-1/10^∞" are equal or not, from the snippet:
>> > 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...
>> >
>> > Assume A=B
>> Or, more simply, prove that A = B = 1.
>> > <=> 1-1/2^∞= 1-1/10^∞ // converted from the formula of geometric series
>> > <=> 1/2^∞= 1/10^∞
>> > <=> 5^∞=1
>> > <=> false
>> Not with the conventional meanings of any of these symbols.
>
> I just use a perfectly defined symbol '∞', nothing more.

Ah. You think that's a definition. I see the problem. It isn't.

> If you really understand algebra, the definition of '∞' is sufficient:
> 1. ∀n∈ℕ, n<∞
> 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is
> -∞

Even this is not a definition. Have you read anything about extensions
to the reals? That would show you how this sort of thing needs to be
defined. (And why are you talking about ℕ?)

>> > Conclusion: A and B denote different numbers. (a physical device
>> > computing the truth value of A=B is a deterministic process, yields
>> > false, no way 'equal').
>> Conclusion: you don't know what the symbols mean and you don't dare
>> define new ones with the meaning you;d like them to have.
>

> The meaning the symbol ∞ used by you (and in textbook) is vague, all you
> can say is "it is a concept", "it does not exist"..., nothing very
> significant.

Unfortunately some notations use ∞ metaphorically (the case that's
confusing you is lim{x->∞}), but the proper definition of the limit is
precise and does not refer to ∞ at all. But it seems you've never seen
such a definition because you still think there is only one kind of
limit.

--
Ben.

[Keith Thompson]

Ben Bacarisse <ben.u...@bsb.me.uk> writes:
> wij <wyni...@gmail.com> writes:
[...]

>> I just use a perfectly defined symbol '∞', nothing more.
>
> Ah. You think that's a definition. I see the problem. It isn't.
>
>> If you really understand algebra, the definition of '∞' is sufficient:
>> 1. ∀n∈ℕ, n<∞
>> 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is
>> -∞
>
> Even this is not a definition. Have you read anything about extensions
> to the reals? That would show you how this sort of thing needs to be
> defined. (And why are you talking about ℕ?)

[...]

A couple of weeks ago, I asked wij whether they were talking about the
hyperreals or not. They weren't even able to answer that. That's about
where I gave up.

Part of the discussion:

I wrote:
>> If ∞-(x+1) < ∞-x, then you have multiple infinite values, even if you
>> only use the term "∞" to refer to one of them.
>>
>> More concretely, you seem to be saying that ∞-1 and ∞-2 are distinct
>> values. They're clearly both infinite, right? But neither of them is
>> equal to ∞?
>>
>> Certainly there are systems in which that's all true -- but I don't know
>> what system you're working with.
>>
>> If you're talking about hyperreals, you can save a lot of time and
>> effort by saying so. Likewise if you're talking about some other well
>> defined system in which ∞ is treated as a unique number. There are a
>> number of such systems.
>> [...]

wig replied:
> I read your reply many times and still not sure what it means.
> IIUC, the concern of naming issue depends on how things develop.

I would have though that "Are you talking about hypperreals or not?"
(which is a *slight* simplification of what I wrote) would be an easy
question to answer.

--
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 */

[Richard Damon]

I think that wij ultimate doesn't understand the various Trans-Finite
number systems, but just wants to work in some system call "The Reals"
that happens to have infinites added to them.

[Ben Bacarisse]

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

> On Friday, 26 August 2022 at 14:44:08 UTC+2, Ben Bacarisse wrote:
>> So you still want me to waste time typing out standard material from
>> goodness know how many sources? No. Go read them.
>
> Fucking idiot. Nobody is asking you to post "standard material" or ANY
> material.

That's good.

> I am asking you to identify WHICH standard you think is THE standard.

Limits in real analysis. Do you mean to say you've been commenting on
an infinite sum without knowing what such things "usually" mean? Why
would you say anything about a topic when you don't know the standard
meanings?

>> If you don't know what 0.999... = 1 means, you know where you can find
>> out.
>
> Fucking idiot! Of courseI know what it means in gneral! It means a
> Boolean!

No. I seriously doubt you don't know what it means, but if you don't
why did you ever think it was reasonable to talk about the subject?

> But I am not interested in general answers, I am looking for the
> particular and precis answer. I am looking for the Boolean value YOU
> have assigned to it, and of course I know where I can find that
> out. The location of that information is in YOUR head. That is
> PRECISELY why I am asking YOU.

OK. But I don't have to play that game do I?

The big picture here is that I am not trying to persuade you of
anything. You can find out how an infinite sum of digit values is
defined if you really don't know (which I doubt). If you never get to
know what I mean, well, fine. Why should I care?

>> The question is what /you/ mean. That will remain a mystery because you
>> won't say more than report, as stated, silly uncontested facts like
>> "0.999... + epsilon = 1" is true.
>
> What the fuck?! I have told you over and over what /I/ mean!
>
> I have assigned the Boolean value "True" to the type "0.999... +
> epsilon = 1". That's precisly what it means!
>
> "0.999... + epsilon = 1" means True:Boolean

So what? Since 0.999... = 1 of course 0.999... + epsilon = 1 is true
(technically satisfiable). Of course that's not what you mean, but you
dare not say more since you fear being too clearly wrong.

> Do you even understand what you are asking?!?

More to the point, do you even know what you are saying?

--
Ben.

[Ben Bacarisse]

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

> On Friday, 26 August 2022 at 12:33:52 UTC+2, Ben Bacarisse wrote:
>> Conclusion: you don't know what the symbols mean and you don't dare
>> define new ones with the meaning you;d like them to have.
>

> Yeeep. Drunk me was 100% right. We have a priest/zealot in the house!

More wine? Attempts to insult me don't later the facts. If you want to
know what I mean the information is readily available on the web. Any
one who would like to know what /you/ mean gets called a priest. So be
it, but we still don't know what you mean. And I suspect we never will.

> Nor could he even tell you what ∞ and "->" denote in the expression
> lim(k -> ∞)

I could, but I won't, because it's standard and available in any course
on real analysis. It's a troll move to get me to waste time telling you
things you should have learnt before commenting on this topic. But I
think the tread is working out just fine for you, no?

--
Ben.

[Skep Dick]

On Saturday, 27 August 2022 at 01:59:19 UTC+2, Ben Bacarisse wrote:
> More wine? Attempts to insult me don't later the facts.

I don't have to insult you. You insult yourself by appealing to "facts" in Mathematics.

There are cultural trends and fads, sure - but facts? Heh!

>If you want to know what I mean the information is readily available on the web.

That is soooo peculiar to me, the location of YOUR meaning is NOT in your own head, but on the web?!?

Wow! Bourbaki really were right - you morons have totally debased meaning in Mathematics.

>Any one who would like to know what /you/ mean gets called a priest. So be
> it, but we still don't know what you mean. And I suspect we never will.

The confusion is all yours. There are multiple possible theories of formal semantics you see.

Or maybe you don't see.

> I could, but I won't, because it's standard and available in any course on real analysis.

OH! You mean the coursework that has been bullshiting us about epsilon-delta calculus for centuries?

No thanks. Liebnitz and Newton (actual Mathematicians) did it much better than the abstract academic monkeys.

>It's a troll move to get me to waste time telling you things you should have learnt before commenting on this topic.

I did learn them. And right after my Mathematic class I went to Physics class. And guess what? Apparently Physicists don't care about epsilon-delta. They never left the universe of infinitesimals - physicists just differentiate anywhere, everywhere, whenever they like.

>But I think the tread is working out just fine for you, no?

If by "fine" you mean your utmost refusal to tell me which theory of meaning you subscribe to... then sure.