On Tuesday, 16 August 2022 at 03:46:58 UTC+2, Ben Bacarisse wrote:
> Why would you do that? Anyway, don't worry; I do like you saying it.
> Say it again and again, please.
At your service...

The expression "lim(x -> 0)" means exactly the same thing as "lim(x-> ∞)" when 0 is bound to ∞
The expression "lim(x -> 1)" means exactly the same thing as "lim(x-> ∞)" when 1 is bound to ∞
The expression "lim(x -> 2)" means exactly the same thing as "lim(x-> ∞)" when 2 is bound to ∞
.... <------------- Once we ignore the discontinuity in the continuum.
The expression "lim(x -> ∞)" means exactly the same thing as "lim(x-> ∞)" when ∞ is bound to ∞

Op 15.aug..2022 om 01:35 schreef wij:
> The vague, no-logic concept of infinity seems dominated people's mind.
> What is infinity? What does "lim(x→∞) f(x)" mean?
>
> If infinity is merely a 'concept', not a number, what does x approach to?
> If x is not getting "closer" to ∞? What does 'approach' mean?
> Therefore, ∞-(x+1) < ∞-x must be valid inequality to mean x+1 is closer than x to infinity ∞.
>
> But valid what? Most people agree ∀n∈ℕ, n<∞.
>
> Is x+1 not closer than x to infinity?
> So, infinity ∞ must have arithmetic meaning. Here is one:
> The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
>
> All in all, that is the definition of infinity (the symbol '∞') proposed.
> All is that simple, the usage treating ∞ as if it is a unique number is
> safe-guaranteed, what left is interpretation. Though I think I figured this
> part (merely means a procedure never terminate), there may be lots more
> instances to test its interpretation in various scenario.

Nice idea. Let's forget what was told us by our teachers and follow our
intuition. ∞ can be used in equations just as any other number.
1/∞ = 0 and when 1/y=x it follows that 1/x=y, so 1/0=∞.
Similarly, 1/-∞ = 0, so 1/0 = -∞ , therefore -∞ = 1/0 = ∞ .
exp(∞) = ∞ and exp(-∞) = 0, so, because ∞ = -∞, 0 = ∞ .
exp(0) = 1 and exp(-∞) = 0, so 1 = 0 = ∞ = -∞ .
1+0=1 and 1+1=2, so 2 = 1 = 0 = ∞ = -∞.
We can continue and show that all numbers are equal.
This leads to what our intuition already expected: a better society,
where numbers are no longer discriminated, but all numbers are equal.
Wouldn't this mathematical model solve a lot of problems in our world?
It is suspected that capitalistic mathematical teachers try to hide this
truth from us to let us believe that people with more money are richer
and can pay more than people with less money, but there is no real
difference.

On Tuesday, 16 August 2022 at 01:15:55 UTC+2, richar...@gmail.com wrote:
> If you don't follow the rules of the Math you are using, you get
> unreliable results.
If you can't make "The Rules of the the Math" explicit, and enforceable by a compiler - it's nobody's fault that the rules can be interpreted as anyone chooses.

On 8/16/22 2:54 AM, Skep Dick wrote:
> On Tuesday, 16 August 2022 at 03:46:58 UTC+2, Ben Bacarisse wrote:
>> Skep Dick <skepd...@gmail.com> writes:
>>> On Monday, 15 August 2022 at 18:25:25 UTC+2, Ben Bacarisse wrote:
>>>> Skep Dick <skepd...@gmail.com> writes:
>>>>
>>>>> On Monday, 15 August 2022 at 17:45:59 UTC+2, Ben Bacarisse wrote:
>>>>>> Obviously if you don't understand the basics of real analysis, you will
>>>>>> doubt anyone who points them out.
>>>>>
>>>>> Obviously, if you don't understand the basics of syntax, semantics,
>>>>> bound and unbound variables you will doubt anyone who points out that
>>>>> "lim(x -> ∞)" is a syntax error IF "∞ is not a number"
>>>> You don't get to say what the syntax means. lim(x->a) and lim(x->oo)
>>>> are well-established form used to mean two quite different kinds of
>>>> limits, despite the similarity in the syntax.
>>>>
>>> Lets try that again... without the errors.
>>>
>>> Yes, I do get to say it. And if you don't like me saying it - I will
>>> repeat it louder.
>> Why would you do that? Anyway, don't worry; I do like you saying it.
>> Say it again and again, please.
>>
>>> lim(x -> ∞) means EXACTLY THE SAME THING AS
>>>
>>> let a = ∞
>>> lim(x -> a)
>> The two kinds of limit are defined differently for rather obvious
>> reasons.
> For rather obvious reaons the two different notations are semantically identical.

Nope, they have different meanings so are semantically different.

>
> Either "∞" is a bound symbol; or "∞" is an ubound symbol in the expression "lim(x -> ∞)".
>
> This is undergraduate computer science stuff. Compiler theory.
>
> https://en.wikipedia.org/wiki/Name_binding
>

Oh, "∞" is a "bound" symbol, it just isn't bound to a "value" but a concept.

You don't seem to understand the difference.

The limit operator is overloaded on the "type" of its operator.

If the limit term has a finite value, it means one thing, if it is an
infinite value it means another.

The two meanings have a lot of similarity, but are subtlety different,
because the definition for finite values isn't applicable for infinite
values.

On 8/16/22 4:42 AM, Skep Dick wrote:
> On Tuesday, 16 August 2022 at 03:46:58 UTC+2, Ben Bacarisse wrote:
>> Why would you do that? Anyway, don't worry; I do like you saying it.
>> Say it again and again, please.
> At your service...
>
> The expression "lim(x -> 0)" means exactly the same thing as "lim(x-> ∞)" when 0 is bound to ∞
> The expression "lim(x -> 1)" means exactly the same thing as "lim(x-> ∞)" when 1 is bound to ∞
> The expression "lim(x -> 2)" means exactly the same thing as "lim(x-> ∞)" when 2 is bound to ∞
> ... <------------- Once we ignore the discontinuity in the continuum.
> The expression "lim(x -> ∞)" means exactly the same thing as "lim(x-> ∞)" when ∞ is bound to ∞
>
>

Nope, because you CAN'T just ignore that discontinuity, because the
universe radically changes when you do.

On 8/16/22 6:31 AM, Skep Dick wrote:
> On Tuesday, 16 August 2022 at 01:15:55 UTC+2, richar...@gmail.com wrote:
>> If you don't follow the rules of the Math you are using, you get
>> unreliable results.
> If you can't make "The Rules of the the Math" explicit, and enforceable by a compiler - it's nobody's fault that the rules can be interpreted as anyone chooses.
>
>

Lets see your complier enforce a speed limit.

The rules are well established, and well known.

And it IS possible to express at least most of them (not sure if all) as
a set of rules that you can put into an appropriate rule system.

Perhaps one issue is that some are semantic, so can't always be detected
at compile time depending on your type system.

For instance, we don't know if 1/x is a valid operation unless we know
that x doesn't have the value 0.

And, it IS your fault for not following them, as YOU are the one
claiming to be working inside them when you make a statement about how
the system work.

You need to actually know a system before you can make actual statements
about it.

On Tuesday, 16 August 2022 at 13:23:41 UTC+2, richar...@gmail.com wrote:
> Nope, because you CAN'T just ignore that discontinuity, because the
> universe radically changes when you do.
Noooo! Waaaaay! Are you serious!?!

Here is the continuum for you: |------------------------------->∞

Could you please draw me a line (a discontinuity) which splits the continuum into "finite" and "infinite" parts ?

On Tuesday, 16 August 2022 at 13:22:14 UTC+2, richar...@gmail.com wrote:
> Oh, "∞" is a "bound" symbol, it just isn't bound to a "value" but a concept.
Shut the fuck up, sophist.

1 is symbol bound to a concept.
2 is symbol bound to a concept.
3 is symbol bound to a concept.
∞ is symbol bound to a concept.

> You don't seem to understand the difference.
I think it's time you stopped lying about my understanding.

Either symbols is a pointer to a valid object, or it isn't.

And you know what happens when you de-reference null-pointers...
> The limit operator is overloaded on the "type" of its operator.
You are running for the hills of polymorphism and type theory. It's not going to help you.
Yes the lim() operator is overloaded. So lets pretend we are talking about limits on Real numbers.

lim( x:ℝ -> a) with a bound to ∞ means the EXACT SAME THING as lim( x:ℝ -> ∞ )

Here is your number line ℝ: 0 |-----------------------------------------> ∞

Could you please draw me a line where a finite number in ℝ begins approaching ∞?

> If the limit term has a finite value, it means one thing, if it is an
> infinite value it means another.
Bullshit. I don't care about the value of the limit.

I care about the meaning of lim( x:ℝ -> ∞ )
if ALL numbers in ℝ are finite, and NO finite number is closer to infinity than any other then what the fuck does it mean for a real number to approach infinity?

> The two meanings have a lot of similarity, but are subtlety different,
> because the definition for finite values isn't applicable for infinite
> values.

Here is your number line ℝ: 0 |-----------------------------------------> ∞

Could you please put a mark on it (adiscontinuity!) where the "finite values" stop and the "infinite values" start?

On Tuesday, 16 August 2022 at 13:29:22 UTC+2, richar...@gmail.com wrote:
> Lets see your complier enforce a speed limit.
Ooooh. You think I am talking about a concrete compiler?
I am talking about the abstract compiler in your head.

> The rules are well established, and well known.
Bullshit!

Stop talking about "The Rules" and show them to me already.
Stop talking about your God and show him to me.

> And it IS possible to express at least most of them (not sure if all) as
> a set of rules that you can put into an appropriate rule system.
You haven't expressed ANY of them yet.

> Perhaps one issue is that some are semantic, so can't always be detected
> at compile time depending on your type system.
Nonsense. Encode your semantics into your type system then!

What is the domain of the lim(x -> y) type?

> For instance, we don't know if 1/x is a valid operation unless we know
> that x doesn't have the value 0.
Whether X is allowed to be bound to 0 in 1/x is an arbitrary choice.

You can forbid it. And mandate that 1/0 is undefined.
You can allow it. And mandate that 1/0 is defined.

Both arguments (arbitrary choices) are an appeal to authority.

> And, it IS your fault for not following them, as YOU are the one
> claiming to be working inside them when you make a statement about how
> the system work.
> You need to actually know a system before you can make actual statements
> about it.

Bullshit. I have repeated it over and over. That I am NOT working INSIDE your system.
I have told you that I am working OUTSIDE of your system.

I am not making POSITIVE statements about how your system works.
I am making NEGATIVE statements about your system DOESN'T work.

Skep Dick <skepdick22@gmail.com> wrote:
> On Monday, 15 August 2022 at 18:39:33 UTC+2, Ben Bacarisse wrote:
>> Cranks don't want to be correct in some "other" system (though,
>> as it happens, 0.999... = 1 in *R as well as in R), they want
>> conventional wisdom to be wrong.
> You have no idea what "they" actually want! Because you are a tone-deaf
> zealot.

Ben is an educated and refined person who happens to care about the
truth.

> What "cranks" actually want is for you to stop assaulting their
> intuitions with nonsense like 0.999... = 1. Because "cranks" actually
> have a killer intuition about Mathematics developed empirically, not
> through the usual academic indoctrination.

There's no need for the scare quotes on cranks. We all know what cranks
are - wij is one. You are another. Neither of you care about the truth.

"Killer intuitions" are usually simply mistakes due to lack of education.
Mathematics graduates have followed a rigorous course of study for
several years, and have thus developed a level of mathematical
understanding far beyond that of cranks.

A good question to you is why are you so disrespectful of education?

> What "cranks" want is NOT the theorem 0.999... = 1.
> What "cranks" want is the theorem 0.999... = 1 - ε

Similarly to what cranks want is a flat Earth. Reality and truth,
however, surpass the restricted horizons of cranks. Theorems are proven
propositions. Only the first of what you've called theorems actually is
one.

--
Alan Mackenzie (Nuremberg, Germany).

On Tuesday, 16 August 2022 at 14:35:22 UTC+2, Alan Mackenzie wrote:
> Ben is an educated and refined person who happens to care about the
> truth.
Any well educated and refined person knows the truth.

Mathematics is relative.
> There's no need for the scare quotes on cranks. We all know what cranks
> are - wij is one. You are another. Neither of you care about the truth.
Anybody who thinks Mathematics pursues "truth" is a crank.

> Mathematics graduates have followed a rigorous course of study for
> several years, and have thus developed a level of mathematical
> understanding far beyond that of cranks.
What an idiotic idea. What is it that they are studying?
What did the first Mathematics graduate study?

> A good question to you is why are you so disrespectful of education?
I am not disrespectful of education. I am disrespectful of indoctrination.

> > What "cranks" want is NOT the theorem 0.999... = 1.
> > What "cranks" want is the theorem 0.999... = 1 - ε

> Similarly to what cranks want is a flat Earth. Reality and truth,
There is no truth in Mathematics. It's invented, not discovered.

Quit peddling your crank-bullshit.

> however, surpass the restricted horizons of cranks.
I guess, you could say that this entire discourse is about decidability.

If there is a crank amongst us - lets figure out a decision procedure to identify that crank.

My decision procedure says you are the crank.

On Tuesday, 16 August 2022 at 07:15:55 UTC+8, richar...@gmail.com wrote:
> On 8/15/22 10:52 AM, wij wrote:
> > On Monday, 15 August 2022 at 20:02:58 UTC+8, richar...@gmail.com wrote:
> >> On 8/15/22 5:38 AM, wij wrote:
> >>> On Monday, 15 August 2022 at 08:34:39 UTC+8, richar...@gmail.com wrote:
> >>>> On 8/14/22 7:35 PM, wij wrote:
> >>>>> The vague, no-logic concept of infinity seems dominated people's mind.
> >>>>> What is infinity? What does "lim(x→∞) f(x)" mean?
> >>>>>
> >>>>> If infinity is merely a 'concept', not a number, what does x approach to?
> >>>>> If x is not getting "closer" to ∞? What does 'approach' mean?
> >>>>> Therefore, ∞-(x+1) < ∞-x must be valid inequality to mean x+1 is closer than x to infinity ∞.
> >>>>>
> >>>>> But valid what? Most people agree ∀n∈ℕ, n<∞.
> >>>>>
> >>>>> Is x+1 not closer than x to infinity?
> >>>>> So, infinity ∞ must have arithmetic meaning. Here is one:
> >>>>> The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
> >>>>>
> >>>>> All in all, that is the definition of infinity (the symbol '∞') proposed.
> >>>>> All is that simple, the usage treating ∞ as if it is a unique number is
> >>>>> safe-guaranteed, what left is interpretation. Though I think I figured this
> >>>>> part (merely means a procedure never terminate), there may be lots more
> >>>>> instances to test its interpretation in various scenario.
> >>>> If we are talking the real number system, as implied by the limit
> >>>> operator, then the definition of what lim(x->inf) f(x) means
> >>>>
> >>>> is there a number L, such that for ANY error e > 0, no matter how small,
> >>>> can we find an X such that for all x > X that |f(x)-L| < e
> >>>>
> >>>> If L exists, then it is the value of lim(x->inf) f(x)
> >>>>
> >>>> Generally, we will find some bounding formula of some X(e) where we can
> >>>> prove that | F(x) - L | < e for all x > X(e),
> >>>
> >>> The issue has been discussed many times. This proposal is primarily about the
> >>> definition of infinity.
> >>>
> >>> Pythagorean's real number is Q, they could use the infinite-approaching argument
> >>> very validly deducing that all numbers are ratio number. Anyone can use Q to
> >>> approach any number and deduce that all real numbers are rational (sure modern
> >>> people won't do this).
> >>>
> >> Actually, the real Pythagorean's eventually realized (c 5th century BC)
> >> that the length of the hypotenuse of a right triangle with the two legs
> >> having length 1 was not a rational number, and this caused them problem.
> >>
> >> Yes, it took them a while, but that is the irrationality of Man when he
> >> sticks to wrong ideas.
> >>> Snippet from https://groups.google.com/g/comp.theory/c/DaybI0JY4Vc
> >>> ...
> >>> To add more material came up to me (not well ordered):
> >>>
> >>> ----------------------------
> >>> There are quite a number of proofs of "repeating decimals are irrational".
> >>> The basic is the correct equation of 1/3 and its decimal form from long
> >>> division (kids understand this 'infinity' with no problem) should be:
> >>>
> >>> 1/3= 0.333... + nonzero_remainder.
> >>>
> >>> ----------------------------
> >>> To translate the 0.999... problem to limit:
> >>>
> >>> Let A= lim(n->∞) 1-1/2^n = 0.999...
> >>> B= lim(n->∞) 1-1/10^n = 0.999...
> >>>
> >>> Assume A=B
> >>> <=> lim(n->∞) 1-1/2^n = lim(n->∞) 1-1/10^n
> >>> <=> lim(n->∞) 1/2^n = lim(n->∞) 1/10^n
> >> And this step is invalid. You either multiplied by a "non-number" or
> >> divided by zero depending on the steps you did to make that transition..
> >>
> >> This is the problem of assuming that "infinity" is a number.
> >>> <=> lim(n->∞) 1 = lim(n->∞) 1/5^n
> >>> <=> 1=0
> >>>
> >>> [Note] I just demonstrate an instance. The limit theory can evolve as it does
> >>> (e.g. one-sided limit... There are many slightly different versions of
> >>> interpretation of limit as it evolves). Readers might find different
> >>> authors use different rules.
> >>> Limit is a technic to find its 'limit', it cannot form a logically
> >>> consistent theory for real number, e.g. the result of limit in general
> >>> must be verified, e.g. numerically, one cannot absolutely trust the
> >>> result of limit arithmetic. And at final, lim(x->c) f(c)= L does not
> >>> 'deduce' f(c)=L (In text book, probably just reads "lim(x->c) f(c)= L, SO
> >>> WRITTEN as f(c)=L"). Limit theory only says the limit of 0.999... is 1,
> >>> the theory does not say 0.999...=1. There is no equality concept in the
> >>> ε-δ theory.
> >>> If one resorts to Dedekind-cut-like theories (I did not really read it),
> >>> from the knowledge that all the combinations of discrete symbols cannot
> >>> represent all the real numbers, I can conclude what those theories
> >>> claim are false, let alone I suspect there should be circular arguments
> >>> there, because many terms there must be well defined as a fundamental
> >>> theory, are undefined (prove me wrong).
> >>>
> >>> The limit example above demonstrated "0.999..." cannot denote a specific number,
> >>> which also means "repeating decimal" cannot specify a unique number (A!=B).
> >>> Using limit is invalid for me (for this question) but the result is correct,
> >>> see the provided reference (I found a typo there).
> >>>
> >>> -----------------------
> >>> Simple arithmetic (this should also be a valid way 2.718... is calculated):
> >>> (0.999....)^n approaches 1/e
> >>> (1.000...1)^n approaches e (or defined as e)
> >>> A possible rebuttal might be that the (1-1/n) in lim(n->∞) (1-1/n)^n is an invalid
> >>> number (approximated like 0.999...), or it is a 'concept' etc...
> >>> But if it is not a number, the whole equation is broken.
> >>>
> >>> -----------------------
> >>> A[0]=0
> >>> A[n]=(A[n-1]+1)/2
> >>>
> >>> The density property says (implicitly) n can enumerate infinitely (otherwise, it
> >>> won't be a rule) and A[∞] never be 1. A[n] infinitely approaches 1 in form
> >>> like 0.999.... This is like in the case of the interval [0,1), infinite numbers
> >>> of 0.999...s are located near the open end of [0,1).
> >>> Can we infinitely refine the scale of a ruler and the last scale never touches
> >>> the scale of 1? I think, yes, something like the √2 story, otherwise all numbers
> >>> can be 'proved' rational.
> >
> > Why do you point to where you seem not addressing.
> > Andy probably just missed a point, I provided a reference.
> > Ben made an error and (assume he saw my reply to Andy) made an error again.
> >
> > Let A= lim(n->∞) 1-1/2^n = 0.999...
> > B= lim(n->∞) 1-1/10^n = 0.999...
> >
> > Assume A=B
> > <=> lim(n->∞) 1-1/2^n = lim(n->∞) 1-1/10^n
> > <=> lim(n->∞) 1/2^n = lim(n->∞) 1/10^n
> > <=> (lim(n->∞) 2^n)*(lim(n->∞) 1/2^n) = (lim(n->∞) 2^n)*(lim(n->∞) 1/10^n)
> As Ben points out lim(n->inf)*2^n) is Not a Number, so the limit doesn't
> exist so you can't multiply by it.
>
> If you don't follow the rules of the Math you are using, you get
> unreliable results.
> > <=> lim(n->∞) 2^n/2^n = lim(n->∞) 2^n/10^n
> > <=> lim(n->∞) 1 = lim(n->∞) 1/5^n
> > <=> 1=0
> >
> > I wonder how much does you guys really understand you are talking?
> I wonder if YOU know what you are talking about.

On Tuesday, 16 August 2022 at 05:36:34 UTC+8, Keith Thompson wrote:
> wij <wyni...@gmail.com> writes:
> > On Tuesday, 16 August 2022 at 02:25:36 UTC+8, Keith Thompson wrote:
> >> wij <wyni...@gmail.com> writes:
> >> > On Monday, 15 August 2022 at 10:37:13 UTC+8, Keith Thompson wrote:
> >> >> wij <wyni...@gmail.com> writes:
> >> >> > The vague, no-logic concept of infinity seems dominated people's mind.
> >> >> > What is infinity? What does "lim(x→∞) f(x)" mean?
> >> >> >
> >> >> > If infinity is merely a 'concept', not a number, what does x approach to?
> >> >> > If x is not getting "closer" to ∞? What does 'approach' mean?
> >> >> > Therefore, ∞-(x+1) < ∞-x must be valid inequality to mean x+1 is closer than x to infinity ∞.
> >> >> >
> >> >> > But valid what? Most people agree ∀n∈ℕ, n<∞.
> >> >> Typically the "<" relationship is defined over the real numbers. Since
> >> >> ∞ is not a real number, n<∞ is no more valid than n<♫.
> >> >>
> >> >> Of course you can define < over other sets. Exactly what set did you
> >> >> have in mind as the domain of the "<" relationship in your statement?
> >> >> > Is x+1 not closer than x to infinity?
> >> >> If it's "closer", can you define how much closer? Is ∞-(x+1) different
> >> >> from ∞-x?
> >> >
> >> > I cannot really figure out what you mean.
> >> > It seems the definition is not properly presented caused your problems, sorry:
> >> >
> >> > '∞' ::=
> >> > 1. ∀n∈ℕ, n<∞
> >> > 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
> >>
> >> ℕ denotes the set of natural numbers, which is either the set of
> >> non-negative integers or the set of positive integers (the difference
> >> doesn't matter here).
> >>
> >> Your "definition" implies that ∞ is not a natural number,
> >
> > No, 2<3.3, 3.3 is not a natural number.
> >
> >> since every natural number is less than ∞.
> >
> > Definition: ∀n∈ℕ, n<∞
> >
> > I still don't quite understand you.
> I'm not sure what it is you don't understand. When I said that ∞ is not
> a natural number, I wasn't trying to point out a flaw.
>
> What your definition does not provide is uniqueness. How do we know
> that there isn't more than one value X such that ∀n∈ℕ, n<∞? There are
> certainly models in which there is more than one such value. I don't
> believe you've specified the model you're using precisely enough.
> >> > Thus, ∞ denotes a unique number. x+1 is 1 closer than x to ∞ (note that it is
> >> > illegal for limit theory to say this way).
> >>
> >> I suggest that your definition isn't a complete definition. It doesn't
> >> imply that ∞ is unique.
> >
> > Yes, it is implied (see below)
> I don't believe it is (see above). If you want ∞ to be unique, that
> should be part of its definition, or an axiom, or a fact derivable from
> axioms. Giving a partial definition and then saying "Oh, by the way,
> it's unique" is not sufficiently precise.
> >> You could have two distinct infinite valuess
> >> say aleph0 and aleph1, that both satisfy your definition.
> >>
> >> ∀n∈ℕ, n<aleph0
> >> ∀n∈ℕ, n<aleph1
> >> aleph0 ≠ aleph1
> >
> > There is only one symbol '∞' denoting infinity. ∞^2, ∞^∞ are also infinite
> > numbers. sin(∞), x^2+∞=0,... are valid numbers (usage is safe-guaranteed, and
> > there is practical meaning).
> > I am not considering aleph0/aleph1. In my understanding, the length of a point
> > is zero, Any_Infinity*0=0, ℝ cannot be stuffed by points (or numbers).
> >
> >> And I still don't see how your definition implies that x+1 is "closer"
> >> than x to ∞.
> >>
> >> 7 is closer than 6 to 10. 10-7 is 3; 10-6 is 4. 3 and 4 are two
> >> distinct values, and comparing them shows us that 7 is closer than 6 to
> >> 10, and how much closer.
> >>
> >> Do you claim that that same reasoning leads to the conclusion that x+1
> >> is closer than x to ∞? Do you claim that ∞-(x+1) is different from ∞-x?
> >> If so, is that claim consistent with your claim that ∞ is a unique
> >> value?
> >
> > closer::= if x-n < y-n, then, x is closer than y to n. (x,y,n>=0)
> >
> > ∞-(x+1) < ∞-x
> > <=> -(x+1) < -x
> > <=> x+1 > x
> > <=> 1>0
> > <=> true
> 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.
> [...]
>
> --
> 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 */

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

wij <wyniijj2@gmail.com> writes:
> On Tuesday, 16 August 2022 at 05:36:34 UTC+8, Keith Thompson wrote:
>> wij <wyni...@gmail.com> writes:
>> > On Tuesday, 16 August 2022 at 02:25:36 UTC+8, Keith Thompson wrote:
>> >> wij <wyni...@gmail.com> writes:
>> >> > On Monday, 15 August 2022 at 10:37:13 UTC+8, Keith Thompson wrote:
>> >> >> wij <wyni...@gmail.com> writes:
>> >> >> > The vague, no-logic concept of infinity seems dominated people's mind.
>> >> >> > What is infinity? What does "lim(x→∞) f(x)" mean?
>> >> >> >
>> >> >> > If infinity is merely a 'concept', not a number, what does x approach to?
>> >> >> > If x is not getting "closer" to ∞? What does 'approach' mean?
>> >> >> > Therefore, ∞-(x+1) < ∞-x must be valid inequality to mean x+1 is closer than x to infinity ∞.
>> >> >> >
>> >> >> > But valid what? Most people agree ∀n∈ℕ, n<∞.
>> >> >> Typically the "<" relationship is defined over the real numbers. Since
>> >> >> ∞ is not a real number, n<∞ is no more valid than n<♫.
>> >> >>
>> >> >> Of course you can define < over other sets. Exactly what set did you
>> >> >> have in mind as the domain of the "<" relationship in your statement?
>> >> >> > Is x+1 not closer than x to infinity?
>> >> >> If it's "closer", can you define how much closer? Is ∞-(x+1) different
>> >> >> from ∞-x?
>> >> >
>> >> > I cannot really figure out what you mean.
>> >> > It seems the definition is not properly presented caused your problems, sorry:
>> >> >
>> >> > '∞' ::=
>> >> > 1. ∀n∈ℕ, n<∞
>> >> > 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
>> >>
>> >> ℕ denotes the set of natural numbers, which is either the set of
>> >> non-negative integers or the set of positive integers (the difference
>> >> doesn't matter here).
>> >>
>> >> Your "definition" implies that ∞ is not a natural number,
>> >
>> > No, 2<3.3, 3.3 is not a natural number.
>> >
>> >> since every natural number is less than ∞.
>> >
>> > Definition: ∀n∈ℕ, n<∞
>> >
>> > I still don't quite understand you.
>> I'm not sure what it is you don't understand. When I said that ∞ is not
>> a natural number, I wasn't trying to point out a flaw.
>>
>> What your definition does not provide is uniqueness. How do we know
>> that there isn't more than one value X such that ∀n∈ℕ, n<∞? There are
>> certainly models in which there is more than one such value. I don't
>> believe you've specified the model you're using precisely enough.
>> >> > Thus, ∞ denotes a unique number. x+1 is 1 closer than x to ∞ (note that it is
>> >> > illegal for limit theory to say this way).
>> >>
>> >> I suggest that your definition isn't a complete definition. It doesn't
>> >> imply that ∞ is unique.
>> >
>> > Yes, it is implied (see below)
>> I don't believe it is (see above). If you want ∞ to be unique, that
>> should be part of its definition, or an axiom, or a fact derivable from
>> axioms. Giving a partial definition and then saying "Oh, by the way,
>> it's unique" is not sufficiently precise.
>> >> You could have two distinct infinite valuess
>> >> say aleph0 and aleph1, that both satisfy your definition.
>> >>
>> >> ∀n∈ℕ, n<aleph0
>> >> ∀n∈ℕ, n<aleph1
>> >> aleph0 ≠ aleph1
>> >
>> > There is only one symbol '∞' denoting infinity. ∞^2, ∞^∞ are also infinite
>> > numbers. sin(∞), x^2+∞=0,... are valid numbers (usage is safe-guaranteed, and
>> > there is practical meaning).
>> > I am not considering aleph0/aleph1. In my understanding, the length of a point
>> > is zero, Any_Infinity*0=0, ℝ cannot be stuffed by points (or numbers).
>> >
>> >> And I still don't see how your definition implies that x+1 is "closer"
>> >> than x to ∞.
>> >>
>> >> 7 is closer than 6 to 10. 10-7 is 3; 10-6 is 4. 3 and 4 are two
>> >> distinct values, and comparing them shows us that 7 is closer than 6 to
>> >> 10, and how much closer.
>> >>
>> >> Do you claim that that same reasoning leads to the conclusion that x+1
>> >> is closer than x to ∞? Do you claim that ∞-(x+1) is different from ∞-x?
>> >> If so, is that claim consistent with your claim that ∞ is a unique
>> >> value?
>> >
>> > closer::= if x-n < y-n, then, x is closer than y to n. (x,y,n>=0)
>> >
>> > ∞-(x+1) < ∞-x
>> > <=> -(x+1) < -x
>> > <=> x+1 > x
>> > <=> 1>0
>> > <=> true
>> 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.
>> [...]
>
> 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 thought what I wrote was clear enough. In any case, I don't think I
can help you.

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

On Wednesday, 17 August 2022 at 03:34:38 UTC+8, wij wrote:
> On Tuesday, 16 August 2022 at 05:36:34 UTC+8, Keith Thompson wrote:
> > wij <wyni...@gmail.com> writes:
> > > On Tuesday, 16 August 2022 at 02:25:36 UTC+8, Keith Thompson wrote:
> > >> wij <wyni...@gmail.com> writes:
> > >> > On Monday, 15 August 2022 at 10:37:13 UTC+8, Keith Thompson wrote:
> > >> >> wij <wyni...@gmail.com> writes:
> > >> >> > The vague, no-logic concept of infinity seems dominated people's mind.
> > >> >> > What is infinity? What does "lim(x→∞) f(x)" mean?
> > >> >> >
> > >> >> > If infinity is merely a 'concept', not a number, what does x approach to?
> > >> >> > If x is not getting "closer" to ∞? What does 'approach' mean?
> > >> >> > Therefore, ∞-(x+1) < ∞-x must be valid inequality to mean x+1 is closer than x to infinity ∞.
> > >> >> >
> > >> >> > But valid what? Most people agree ∀n∈ℕ, n<∞.
> > >> >> Typically the "<" relationship is defined over the real numbers. Since
> > >> >> ∞ is not a real number, n<∞ is no more valid than n<♫.
> > >> >>
> > >> >> Of course you can define < over other sets. Exactly what set did you
> > >> >> have in mind as the domain of the "<" relationship in your statement?
> > >> >> > Is x+1 not closer than x to infinity?
> > >> >> If it's "closer", can you define how much closer? Is ∞-(x+1) different
> > >> >> from ∞-x?
> > >> >
> > >> > I cannot really figure out what you mean.
> > >> > It seems the definition is not properly presented caused your problems, sorry:
> > >> >
> > >> > '∞' ::=
> > >> > 1. ∀n∈ℕ, n<∞
> > >> > 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
> > >>
> > >> ℕ denotes the set of natural numbers, which is either the set of
> > >> non-negative integers or the set of positive integers (the difference
> > >> doesn't matter here).
> > >>
> > >> Your "definition" implies that ∞ is not a natural number,
> > >
> > > No, 2<3.3, 3.3 is not a natural number.
> > >
> > >> since every natural number is less than ∞.
> > >
> > > Definition: ∀n∈ℕ, n<∞
> > >
> > > I still don't quite understand you.
> > I'm not sure what it is you don't understand. When I said that ∞ is not
> > a natural number, I wasn't trying to point out a flaw.
> >
> > What your definition does not provide is uniqueness. How do we know
> > that there isn't more than one value X such that ∀n∈ℕ, n<∞? There are
> > certainly models in which there is more than one such value. I don't
> > believe you've specified the model you're using precisely enough.
> > >> > Thus, ∞ denotes a unique number. x+1 is 1 closer than x to ∞ (note that it is
> > >> > illegal for limit theory to say this way).
> > >>
> > >> I suggest that your definition isn't a complete definition. It doesn't
> > >> imply that ∞ is unique.
> > >
> > > Yes, it is implied (see below)
> > I don't believe it is (see above). If you want ∞ to be unique, that
> > should be part of its definition, or an axiom, or a fact derivable from
> > axioms. Giving a partial definition and then saying "Oh, by the way,
> > it's unique" is not sufficiently precise.
> > >> You could have two distinct infinite valuess
> > >> say aleph0 and aleph1, that both satisfy your definition.
> > >>
> > >> ∀n∈ℕ, n<aleph0
> > >> ∀n∈ℕ, n<aleph1
> > >> aleph0 ≠ aleph1
> > >
> > > There is only one symbol '∞' denoting infinity. ∞^2, ∞^∞ are also infinite
> > > numbers. sin(∞), x^2+∞=0,... are valid numbers (usage is safe-guaranteed, and
> > > there is practical meaning).
> > > I am not considering aleph0/aleph1. In my understanding, the length of a point
> > > is zero, Any_Infinity*0=0, ℝ cannot be stuffed by points (or numbers).
> > >
> > >> And I still don't see how your definition implies that x+1 is "closer"
> > >> than x to ∞.
> > >>
> > >> 7 is closer than 6 to 10. 10-7 is 3; 10-6 is 4. 3 and 4 are two
> > >> distinct values, and comparing them shows us that 7 is closer than 6 to
> > >> 10, and how much closer.
> > >>
> > >> Do you claim that that same reasoning leads to the conclusion that x+1
> > >> is closer than x to ∞? Do you claim that ∞-(x+1) is different from ∞-x?
> > >> If so, is that claim consistent with your claim that ∞ is a unique
> > >> value?
> > >
> > > closer::= if x-n < y-n, then, x is closer than y to n. (x,y,n>=0)
> > >
> > > ∞-(x+1) < ∞-x
> > > <=> -(x+1) < -x
> > > <=> x+1 > x
> > > <=> 1>0
> > > <=> true
> > 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.
> > [...]
> >
> > --
> > 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 */
> I read your reply many times and still not sure what it means.
> IIUC, the concern of naming issue depends on how things develop.

Numbers eventually can be analyzed as natural number + operation. From the
definition of natural number (Peano axioms), it also relies on "operation".
This roughly says number is an operation -> procedure/expression.
Constants like 1,2,3... are symbol names for some operation. Σ(n=0,∞) (-1/n)^2
is an unnamed expression(procedure), a number, be it divergent or convergent.
IIUC, real number is doomed connected to reality. It has to be verified by
reality, number itself becomes description of reality, no more merely 'concept'
or abstract entity. Can we find 'the reality' of 1/∞, sin(∞) or of any other
irrational number? I think, yes, by LOGIC. The former (1/∞) is easy: the
execution time of an algorithm or math. expression.

If academic type people insists that ℝ has to be what the 'research-level'
text-books says, they should also make it clear their ℝ is abstract, not meant to be real.
I think the 'abstract' option is difficult, because we know it will change.

On Wednesday, 17 August 2022 at 04:21:13 UTC+8, wij wrote:
> On Wednesday, 17 August 2022 at 03:34:38 UTC+8, wij wrote:
> > On Tuesday, 16 August 2022 at 05:36:34 UTC+8, Keith Thompson wrote:
> > > wij <wyni...@gmail.com> writes:
> > > > On Tuesday, 16 August 2022 at 02:25:36 UTC+8, Keith Thompson wrote:
> > > >> wij <wyni...@gmail.com> writes:
> > > >> > On Monday, 15 August 2022 at 10:37:13 UTC+8, Keith Thompson wrote:
> > > >> >> wij <wyni...@gmail.com> writes:
> > > >> >> > The vague, no-logic concept of infinity seems dominated people's mind.
> > > >> >> > What is infinity? What does "lim(x→∞) f(x)" mean?
> > > >> >> >
> > > >> >> > If infinity is merely a 'concept', not a number, what does x approach to?
> > > >> >> > If x is not getting "closer" to ∞? What does 'approach' mean?
> > > >> >> > Therefore, ∞-(x+1) < ∞-x must be valid inequality to mean x+1 is closer than x to infinity ∞.
> > > >> >> >
> > > >> >> > But valid what? Most people agree ∀n∈ℕ, n<∞.
> > > >> >> Typically the "<" relationship is defined over the real numbers.. Since
> > > >> >> ∞ is not a real number, n<∞ is no more valid than n<♫.
> > > >> >>
> > > >> >> Of course you can define < over other sets. Exactly what set did you
> > > >> >> have in mind as the domain of the "<" relationship in your statement?
> > > >> >> > Is x+1 not closer than x to infinity?
> > > >> >> If it's "closer", can you define how much closer? Is ∞-(x+1) different
> > > >> >> from ∞-x?
> > > >> >
> > > >> > I cannot really figure out what you mean.
> > > >> > It seems the definition is not properly presented caused your problems, sorry:
> > > >> >
> > > >> > '∞' ::=
> > > >> > 1. ∀n∈ℕ, n<∞
> > > >> > 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
> > > >>
> > > >> ℕ denotes the set of natural numbers, which is either the set of
> > > >> non-negative integers or the set of positive integers (the difference
> > > >> doesn't matter here).
> > > >>
> > > >> Your "definition" implies that ∞ is not a natural number,
> > > >
> > > > No, 2<3.3, 3.3 is not a natural number.
> > > >
> > > >> since every natural number is less than ∞.
> > > >
> > > > Definition: ∀n∈ℕ, n<∞
> > > >
> > > > I still don't quite understand you.
> > > I'm not sure what it is you don't understand. When I said that ∞ is not
> > > a natural number, I wasn't trying to point out a flaw.
> > >
> > > What your definition does not provide is uniqueness. How do we know
> > > that there isn't more than one value X such that ∀n∈ℕ, n<∞? There are
> > > certainly models in which there is more than one such value. I don't
> > > believe you've specified the model you're using precisely enough.
> > > >> > Thus, ∞ denotes a unique number. x+1 is 1 closer than x to ∞ (note that it is
> > > >> > illegal for limit theory to say this way).
> > > >>
> > > >> I suggest that your definition isn't a complete definition. It doesn't
> > > >> imply that ∞ is unique.
> > > >
> > > > Yes, it is implied (see below)
> > > I don't believe it is (see above). If you want ∞ to be unique, that
> > > should be part of its definition, or an axiom, or a fact derivable from
> > > axioms. Giving a partial definition and then saying "Oh, by the way,
> > > it's unique" is not sufficiently precise.
> > > >> You could have two distinct infinite valuess
> > > >> say aleph0 and aleph1, that both satisfy your definition.
> > > >>
> > > >> ∀n∈ℕ, n<aleph0
> > > >> ∀n∈ℕ, n<aleph1
> > > >> aleph0 ≠ aleph1
> > > >
> > > > There is only one symbol '∞' denoting infinity. ∞^2, ∞^∞ are also infinite
> > > > numbers. sin(∞), x^2+∞=0,... are valid numbers (usage is safe-guaranteed, and
> > > > there is practical meaning).
> > > > I am not considering aleph0/aleph1. In my understanding, the length of a point
> > > > is zero, Any_Infinity*0=0, ℝ cannot be stuffed by points (or numbers).
> > > >
> > > >> And I still don't see how your definition implies that x+1 is "closer"
> > > >> than x to ∞.
> > > >>
> > > >> 7 is closer than 6 to 10. 10-7 is 3; 10-6 is 4. 3 and 4 are two
> > > >> distinct values, and comparing them shows us that 7 is closer than 6 to
> > > >> 10, and how much closer.
> > > >>
> > > >> Do you claim that that same reasoning leads to the conclusion that x+1
> > > >> is closer than x to ∞? Do you claim that ∞-(x+1) is different from ∞-x?
> > > >> If so, is that claim consistent with your claim that ∞ is a unique
> > > >> value?
> > > >
> > > > closer::= if x-n < y-n, then, x is closer than y to n. (x,y,n>=0)
> > > >
> > > > ∞-(x+1) < ∞-x
> > > > <=> -(x+1) < -x
> > > > <=> x+1 > x
> > > > <=> 1>0
> > > > <=> true
> > > 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.
> > > [...]
> > >
> > > --
> > > 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 */
> > I read your reply many times and still not sure what it means.
> > IIUC, the concern of naming issue depends on how things develop.
> Numbers eventually can be analyzed as natural number + operation. From the
> definition of natural number (Peano axioms), it also relies on "operation".
> This roughly says number is an operation -> procedure/expression.
> Constants like 1,2,3... are symbol names for some operation. Σ(n=0,∞) (-1/n)^2
> is an unnamed expression(procedure), a number, be it divergent or convergent.
> IIUC, real number is doomed connected to reality. It has to be verified by
> reality, number itself becomes description of reality, no more merely 'concept'
> or abstract entity. Can we find 'the reality' of 1/∞, sin(∞) or of any other
> irrational number? I think, yes, by LOGIC. The former (1/∞) is easy: the
> execution time of an algorithm or math. expression.
>
> If academic type people insists that ℝ has to be what the 'research-level'
> text-books says, they should also make it clear their ℝ is abstract, not meant to be real.
> I think the 'abstract' option is difficult, because we know it will change.

typo: Σ(n=1,∞) (-1/n)^2 (dividing zero is undefined)

On 8/16/22 8:08 AM, Skep Dick wrote:
> On Tuesday, 16 August 2022 at 13:22:14 UTC+2, richar...@gmail.com wrote:
>> Oh, "∞" is a "bound" symbol, it just isn't bound to a "value" but a concept.
> Shut the fuck up, sophist.
>
> 1 is symbol bound to a concept.
> 2 is symbol bound to a concept.
> 3 is symbol bound to a concept.
> ∞ is symbol bound to a concept.

And 1, 2, 3, ... are bound to a value concept, and ∞ is bound to a
non-value concept.

>
>> You don't seem to understand the difference.
> I think it's time you stopped lying about my understanding.
>
> Either symbols is a pointer to a valid object, or it isn't.
>
> And you know what happens when you de-reference null-pointers...

And they are pointers of objects of different types.

>
>> The limit operator is overloaded on the "type" of its operator.
> You are running for the hills of polymorphism and type theory. It's not going to help you.
> Yes the lim() operator is overloaded. So lets pretend we are talking about limits on Real numbers.
>
> lim( x:ℝ -> a) with a bound to ∞ means the EXACT SAME THING as lim( x:ℝ -> ∞ )

Nope because ∞ isn't a Real Number, but the concept of Infinity.

>
> Here is your number line ℝ: 0 |-----------------------------------------> ∞
>
> Could you please draw me a line where a finite number in ℝ begins approaching ∞?

Here is the number line ℝ:
<-----------------| 0 |---------------->

"∞" isn't ON the Real Number Line, since it isn't a Real Number

I suppose you could also notate it as: ( -∞, ∞ )

Since the open interval doesn't include the end points.

>
>> If the limit term has a finite value, it means one thing, if it is an
>> infinite value it means another.
> Bullshit. I don't care about the value of the limit.

Then why are you talking about it.

>
> I care about the meaning of lim( x:ℝ -> ∞ )

lim( x:ℝ -> ∞ ) F(x) means what number L, if it exists, statisfies the
property that for any positve number e, we can find a bound X such that
for ALL x > X, | f(x) - L | < e

> if ALL numbers in ℝ are finite, and NO finite number is closer to infinity than any other then what the fuck does it mean for a real number to approach infinity?

We don't talk about NUMBERS approaching infinity, but of the parameter
to the function approaching infinity.

>
>> The two meanings have a lot of similarity, but are subtlety different,
>> because the definition for finite values isn't applicable for infinite
>> values.
>
> Here is your number line ℝ: 0 |-----------------------------------------> ∞
>
> Could you please put a mark on it (adiscontinuity!) where the "finite values" stop and the "infinite values" start?
>

That ISN'T the number line R, the number line R is:

<-------| 0 |-------->

As ∞ isn't on it. ∞ is NOT a "value", only a "limit"

It is just being clear that you don't understand what "The Real Numbers
R" actually are, or maybe even the concept of "Rules".

That means you don't ACTUALLY know how to properly do things, because if
you don't know the rules, you don't know if you are obeying them.

On 8/16/22 7:54 AM, Skep Dick wrote:
> On Tuesday, 16 August 2022 at 13:23:41 UTC+2, richar...@gmail.com wrote:
>> Nope, because you CAN'T just ignore that discontinuity, because the
>> universe radically changes when you do.
> Noooo! Waaaaay! Are you serious!?!
>
> Here is the continuum for you: |------------------------------->∞
>
> Could you please draw me a line (a discontinuity) which splits the continuum into "finite" and "infinite" parts ?
>

|-------------------------------> | ∞

Does that work for you?

On Wednesday, 17 August 2022 at 00:45:53 UTC+2, richar...@gmail.com wrote:
> On 8/16/22 7:54 AM, Skep Dick wrote:
> > On Tuesday, 16 August 2022 at 13:23:41 UTC+2, richar...@gmail.com wrote:
> >> Nope, because you CAN'T just ignore that discontinuity, because the
> >> universe radically changes when you do.
> > Noooo! Waaaaay! Are you serious!?!
> >
> > Here is the continuum for you: |------------------------------->∞
> >
> > Could you please draw me a line (a discontinuity) which splits the continuum into "finite" and "infinite" parts ?
> >
> |-------------------------------> | ∞
>
>
> Does that work for you?
I don't know. Does it "work" ?

Which finite numbers "approach infinity" in the exression lim(x:ℝ ->∞) ?

On 8/16/22 8:56 AM, Skep Dick wrote:
> On Tuesday, 16 August 2022 at 14:35:22 UTC+2, Alan Mackenzie wrote:
>> Ben is an educated and refined person who happens to care about the
>> truth.
> Any well educated and refined person knows the truth.
>
> Mathematics is relative.

Nope, a given Mathematics tends to be fairly absolute.

You can chose different variations for some purposes, but a given
Mathematics is fairly rigid.

>
>> There's no need for the scare quotes on cranks. We all know what cranks
>> are - wij is one. You are another. Neither of you care about the truth.
> Anybody who thinks Mathematics pursues "truth" is a crank.

Why?
>

>
>> Mathematics graduates have followed a rigorous course of study for
>> several years, and have thus developed a level of mathematical
>> understanding far beyond that of cranks.
> What an idiotic idea. What is it that they are studying?
> What did the first Mathematics graduate study?

Mathemmatics, and Truth.

>
>
>> A good question to you is why are you so disrespectful of education?
> I am not disrespectful of education. I am disrespectful of indoctrination.

No, yoru are disrespectful of Rules and the Tradiition of understanding.

You forget where the stuff you think you know came from.
>
>>> What "cranks" want is NOT the theorem 0.999... = 1.
>>> What "cranks" want is the theorem 0.999... = 1 - ε
>
>
>> Similarly to what cranks want is a flat Earth. Reality and truth,
> There is no truth in Mathematics. It's invented, not discovered.

Nope, much of Mathematics is actually discovered.

>
> Quit peddling your crank-bullshit.

You need to quite peddling your lies.

>
>> however, surpass the restricted horizons of cranks.
> I guess, you could say that this entire discourse is about decidability.
>
> If there is a crank amongst us - lets figure out a decision procedure to identify that crank.
>
> My decision procedure says you are the crank.

And the TRUE decision procedure says YOU are the crank.

Since YOU think everyone gets to decide on their own system, so do I,
and thus I get to decide what it the true system.

You lose.

On Wednesday, 17 August 2022 at 00:44:38 UTC+2, richar...@gmail.com wrote:
> And 1, 2, 3, ... are bound to a value concept, and ∞ is bound to a
> non-value concept.
Distinction without a difference.

> And they are pointers of objects of different types.
You don't understand dependent types.

> Nope because ∞ isn't a Real Number, but the concept of Infinity.
Distinction without a difference. It's a parameter to a function.

It does some work. Like every other symbol does some work.

> >
> > Here is your number line ℝ: 0 |-----------------------------------------> ∞
> >
> > Could you please draw me a line where a finite number in ℝ begins approaching ∞?
> Here is the number line ℝ:
> <-----------------| 0 |---------------->
>
> "∞" isn't ON the Real Number Line, since it isn't a Real Number
>
> I suppose you could also notate it as: ( -∞, ∞ )
>
> Since the open interval doesn't include the end points.
Idiot. You forgot to address the actual question.

Where on ℝ do the finite numbers begin approaching ∞?
You know damn well I meant ℝ+ you twat. I even drew you a picture.

But just as well as you want to play dumb - draw me two lines instead.
Where on ℝ do the finite numbers begin approaching -∞ and +?

> > Bullshit. I don't care about the value of the limit.
> Then why are you talking about it.
I am talking about infinity. You said it's not a value.

Have you changed your mind?

> lim( x:ℝ -> ∞ ) F(x) means what number L, if it exists, statisfies the
> property that for any positve number e, we can find a bound X such that
> for ALL x > X, | f(x) - L | < e
Why are you blabbering about stuff that nobody is asking you about?

I asked you what "lim( x:ℝ -> ∞ )" means, not what "lim( x:ℝ -> ∞ ) F(x)" means.

> > if ALL numbers in ℝ are finite, and NO finite number is closer to infinity than any other then what the fuck does it mean for a real number to approach infinity?
> We don't talk about NUMBERS approaching infinity, but of the parameter
> to the function approaching infinity.
Idiot. The parameter to the function is a number.

"lim( x:ℝ -> ∞ )" means "as the parameter to the function approaches infinity".

But ALL x in ℝ are finite. NO number in ℝ is any closer to infinity than any other number in ℝ.

> That means you don't ACTUALLY know how to properly do things, because if
> you don't know the rules, you don't know if you are obeying them.
What rules? Show them to me already!

You like your definitions so much lets drop ℝ altogether. Lets start with the continuum.
( -∞, ∞ )

-∞ <-------------------------------------> +∞

Give me an algorithm to find 0.

On Wednesday, 17 August 2022 at 01:04:38 UTC+2, richar...@gmail.com wrote:
> Nope, a given Mathematics tends to be fairly absolute.
Really? Which property of a given Mathematics is its "absoluteness"?

> You can chose different variations for some purposes, but a given
> Mathematics is fairly rigid.
It's no more or less "rigid" than the arbitrary choices made to construct it.

> > Anybody who thinks Mathematics pursues "truth" is a crank.
> Why?
Because Mathematics is invented and truth isn't.

> > What an idiotic idea. What is it that they are studying?
> > What did the first Mathematics graduate study?
> Mathemmatics, and Truth.
So how did the first mathematician study Mathematics when nobody had invented any Mathematics yet?

> No, yoru are disrespectful of Rules and the Tradiition of understanding.
I am? It seems to me I have acquired far more understanding than you have without following the "Rules and Traditions".

> You forget where the stuff you think you know came from.
It came from the people who invented it. There are no rules to invention.

> > There is no truth in Mathematics. It's invented, not discovered.
> Nope, much of Mathematics is actually discovered.
Oh really? Where do you look for Mathematics?

> You need to quite peddling your lies.
You need to stop lying about me lying.

> And the TRUE decision procedure says YOU are the crank.
The TRUE decision procedure said that your decision procedure is broken.

> Since YOU think everyone gets to decide on their own system, so do I,
> and thus I get to decide what it the true system.
>
> You lose.
I lose? You have literally conceded my point!

Mathematics is relative! Made up. Constructed.

Welcome to the science of Distributed Systems. Where the most important problems are consesus algorithms.
https://en.wikipedia.org/wiki/Consensus_(computer_science)

On Sunday, August 14, 2022 at 7:35:27 PM UTC-4, wyni...@gmail.com wrote:
> The vague, no-logic concept of infinity seems dominated people's mind.
> What is infinity? What does "lim(x→∞) f(x)" mean?
>
> If infinity is merely a 'concept', not a number, what does x approach to?
> If x is not getting "closer" to ∞? What does 'approach' mean?
> Therefore, ∞-(x+1) < ∞-x must be valid inequality to mean x+1 is closer than x to infinity ∞.
>
> But valid what? Most people agree ∀n∈ℕ, n<∞.
>
> Is x+1 not closer than x to infinity?
> So, infinity ∞ must have arithmetic meaning. Here is one:
> The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞
>
> All in all, that is the definition of infinity (the symbol '∞') proposed.
> All is that simple, the usage treating ∞ as if it is a unique number is
> safe-guaranteed, what left is interpretation. Though I think I figured this
> part (merely means a procedure never terminate), there may be lots more
> instances to test its interpretation in various scenario.

class bignum; set < bignum* > bignumstack; bignum* getbignum(); joy putbignum(bignum* big);
class bignum { /* model counting answer datatype with one line foundation of correctness */
public: nums n; num log(){return n.size();} bignum& operator++() { addone(zero); return *this; } /*logs again and again in major logic theory*/
joy addone(num k){when(k==n.y)n.add(zero); when(++n[k]>>one){n[k]=zero; addone(k+one);}}/*zero to infinity*/
bignum& operator+=(bignum& j){for(num k=zero;k<j.log();k++){when(k==log())n.add(zero);when(j.n[k])addone(k);}return*this;}
bignum& operator*=(num j){bignum*sum=getbignum();for(num g=zero;g<fifthtau+one;g++)when(j&ones[g]){
bignum*times=getbignum();for(num h=zero;h<g;h++)(*times).n.add(zero);
for(num h=zero;h<log();h++)(*times).n.add(n[h]);(*sum)+=(*times);putbignum(times);}
n.clear();for(num g =zero;g<(*sum).log();g++){n.add((*sum).n[g]);}putbignum(sum);return *this;}
bignum& operator*=(bignum& j){bignum*sum=getbignum();for(num g=zero;g<j.log();g++)when(j.n[g]){
bignum*times=getbignum();for(num h=zero;h<g;h++)(*times).n.add(zero);
for(num h=zero;h<log();h++)(*times).n.add(n[h]);(*sum)+=(*times);putbignum(times);}
n.clear();for(num g=zero;g<(*sum).log();g++){n.add((*sum).n[g]);}putbignum(sum);return *this;}
etc}

Skep Dick <skepdick22@gmail.com> wrote:
> On Wednesday, 17 August 2022 at 01:04:38 UTC+2, richar...@gmail.com wrote:
>> Nope, a given Mathematics tends to be fairly absolute.
> Really? Which property of a given Mathematics is its "absoluteness"?

That 1 + 1 = 2, for example. That in a field, there is a multiplicative
inverse for every element except 0, for another example.

>> You can chose different variations for some purposes, but a given
>> Mathematics is fairly rigid.
> It's no more or less "rigid" than the arbitrary choices made to
> construct it.

There is no such choice. If there were, there would be radically
different mathematics in different cultures, much as there are radically
differnt religions.

>> > Anybody who thinks Mathematics pursues "truth" is a crank.
>> Why?
> Because Mathematics is invented and truth isn't.

Again, there is one system of maths and it is universal. If you were
right, there would be several distinct systems.

>> > What an idiotic idea. What is it that they are studying?
>> > What did the first Mathematics graduate study?
>> Mathemmatics, and Truth.
> So how did the first mathematician study Mathematics when nobody had
> invented any Mathematics yet?

The first mathematician likely grasped that 1 + 1 = 2 was an abstract
truth. Or something like that.

>> No, you are disrespectful of Rules and the Tradiition of understanding.
> I am? It seems to me I have acquired far more understanding than you
> have without following the "Rules and Traditions".

Your mathematical understanding is at a low level. It is at such a low
level, you have no concept of what it is you are ignorant of.

>> You forget where the stuff you think you know came from.
> It came from the people who invented it. There are no rules to
> invention.

Yet, somehow, all the "inventors" of mathematics come up with the same
thing.

>> > There is no truth in Mathematics. It's invented, not discovered.
>> Nope, much of Mathematics is actually discovered.
> Oh really? Where do you look for Mathematics?

Reality, both concrete and abstract.

>> You need to quite peddling your lies.
> You need to stop lying about me lying.

I'm not convinced you're lying. Most likely you're just ignorant,
_very_ ignorant.

>> And the TRUE decision procedure says YOU are the crank.
> The TRUE decision procedure said that your decision procedure is broken.

>> Since YOU think everyone gets to decide on their own system, so do I,
>> and thus I get to decide what it the true system.
>>
>> You lose.
> I lose? You have literally conceded my point!

> Mathematics is relative! Made up. Constructed.

No, it is absolute, as demonstrated above. Bear in mind, I've studied
it, you haven't.

[ .... ]

--
Alan Mackenzie (Nuremberg, Germany).

Skep Dick <skepdick22@gmail.com> wrote:
> On Wednesday, 17 August 2022 at 00:45:53 UTC+2, richar...@gmail.com wrote:
>> On 8/16/22 7:54 AM, Skep Dick wrote:
>> > On Tuesday, 16 August 2022 at 13:23:41 UTC+2, richar...@gmail.com wrote:
>> >> Nope, because you CAN'T just ignore that discontinuity, because the
>> >> universe radically changes when you do.
>> > Noooo! Waaaaay! Are you serious!?!

>> > Here is the continuum for you: |------------------------------->∞

>> > Could you please draw me a line (a discontinuity) which splits the
>> > continuum into "finite" and "infinite" parts ?

>> |-------------------------------> | ∞

>> Does that work for you?
> I don't know. Does it "work" ?

The question was does it work _for_ _you_? Well, does it? It seems, not
very well, if at all.

> Which finite numbers "approach infinity" in the exression lim(x:ℝ ->∞) ?

None do. That's the sort of question which indicates you need to study
mathematical analysis at a first year undergraduate level, or even at
school level. That's if you're genuinely interested rather than just
playing a troll game.

--
Alan Mackenzie (Nuremberg, Germany).