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?

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?

On 8/23/22 1:27 AM, wij 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;"?
> 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?

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

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 ∞

On 8/23/22 7:34 AM, Skep Dick wrote:
> 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 ∞
>

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.

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!

On 8/23/22 7:58 AM, Skep Dick wrote:
> 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!
>

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.

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.

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".

On Tuesday, 23 August 2022 at 15:24:09 UTC+8, Skep Dick wrote:
> 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?

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'.

On 8/23/22 8:16 AM, Skep Dick wrote:
> 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".
>

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.

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

On 8/24/22 4:58 AM, Skep Dick wrote:
> 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
>
>
>

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.

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!

On Wednesday, 24 August 2022 at 20:55:56 UTC+8, Skep Dick wrote:
> 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!

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 !!!

On 8/24/22 7:57 PM, wij wrote:
> On Wednesday, 24 August 2022 at 20:55:56 UTC+8, Skep Dick wrote:
>> 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!
>
> 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 !!!

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?

On Thursday, 25 August 2022 at 08:09:53 UTC+8, richar...@gmail.com wrote:
> On 8/24/22 7:57 PM, wij wrote:
> > On Wednesday, 24 August 2022 at 20:55:56 UTC+8, Skep Dick wrote:
> >> 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!
> >
> > 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 !!!
> Which IS a valid proof when the domain of the numbers is the Finites,
> aka Reals and their subsets.

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.

On 8/24/22 8:26 PM, wij wrote:
> On Thursday, 25 August 2022 at 08:09:53 UTC+8, richar...@gmail.com wrote:
>> On 8/24/22 7:57 PM, wij wrote:
>>> On Wednesday, 24 August 2022 at 20:55:56 UTC+8, Skep Dick wrote:
>>>> 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!
>>>
>>> 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 !!!
>> Which IS a valid proof when the domain of the numbers is the Finites,
>> aka Reals and their subsets.
>
> 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.

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".

> > 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)

wij <wyniijj2@gmail.com> writes:

> On Thursday, 25 August 2022 at 08:09:53 UTC+8, richar...@gmail.com wrote:
>> On 8/24/22 7:57 PM, wij wrote:
>> > On Wednesday, 24 August 2022 at 20:55:56 UTC+8, Skep Dick wrote:
>> >> 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!
>> >
>> > 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 !!!
>> Which IS a valid proof when the domain of the numbers is the Finites,
>> aka Reals and their subsets.
>
> As guessed, you still believe it is a proof.

> As said in the above snippet. The "..." in representing repeating
> decimal has to be explained before further processing.

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.

On Thursday, 25 August 2022 at 12:37:00 UTC+2, Ben Bacarisse wrote:
> wij <wyni...@gmail.com> writes:
>
> > On Thursday, 25 August 2022 at 08:09:53 UTC+8, richar...@gmail.com wrote:
> >> On 8/24/22 7:57 PM, wij wrote:
> >> > On Wednesday, 24 August 2022 at 20:55:56 UTC+8, Skep Dick wrote:
> >> >> 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!
> >> >
> >> > 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 !!!
> >> Which IS a valid proof when the domain of the numbers is the Finites,
> >> aka Reals and their subsets.
> >
> > As guessed, you still believe it is a proof.
>
> > As said in the above snippet. The "..." in representing repeating
> > decimal has to be explained before further processing.
> 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 "..."?

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?

On 8/25/22 3:39 AM, Skep Dick wrote:
>>> 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)

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.

Skep Dick <skepdick22@gmail.com> writes:

> On Thursday, 25 August 2022 at 12:37:00 UTC+2, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>> > On Thursday, 25 August 2022 at 08:09:53 UTC+8, richar...@gmail.com wrote:
>> >> On 8/24/22 7:57 PM, wij wrote:
>> >> > On Wednesday, 24 August 2022 at 20:55:56 UTC+8, Skep Dick wrote:
>> >> >> 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!
>> >> >
>> >> > 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 !!!
>> >> Which IS a valid proof when the domain of the numbers is the Finites,
>> >> aka Reals and their subsets.
>> >
>> > As guessed, you still believe it is a proof.
>>
>> > As said in the above snippet. The "..." in representing repeating
>> > decimal has to be explained before further processing.
>> 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 "..."?
>
> 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 "...".

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.

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.

On Thursday, 25 August 2022 at 13:51:27 UTC+2, Ben Bacarisse wrote:
> Skep Dick <skepd...@gmail.com> writes:
>
> > On Thursday, 25 August 2022 at 12:37:00 UTC+2, Ben Bacarisse wrote:
> >> wij <wyni...@gmail.com> writes:
> >>
> >> > On Thursday, 25 August 2022 at 08:09:53 UTC+8, richar...@gmail.com wrote:
> >> >> On 8/24/22 7:57 PM, wij wrote:
> >> >> > On Wednesday, 24 August 2022 at 20:55:56 UTC+8, Skep Dick wrote:
> >> >> >> 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!
> >> >> >
> >> >> > 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 !!!
> >> >> Which IS a valid proof when the domain of the numbers is the Finites,
> >> >> aka Reals and their subsets.
> >> >
> >> > As guessed, you still believe it is a proof.
> >>
> >> > As said in the above snippet. The "..." in representing repeating
> >> > decimal has to be explained before further processing.
> >> 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 "..."?
> >
> > 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 "...".
> 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.
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?