On Tuesday, 16 August 2022 at 00:34:41 UTC+8, Ben Bacarisse wrote:
> wij <wyni...@gmail.com> writes:
>
> > On Monday, 15 August 2022 at 23:45:59 UTC+8, Ben Bacarisse wrote:
> >> wij <wyni...@gmail.com> writes:
>
> >> > Ben made an error and (assume he saw my reply to Andy) made an error
> >> > again.
> >> I made no mistake.
> >> > 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)
> >> No. This step is invalid. You can't multiply a limit by anything but a
> >> real number. lim(n->∞) 2^n is not a real number.
> >> > <=> lim(n->∞) 2^n/2^n = lim(n->∞) 2^n/10^n
> >> The product law for limits is only valid when both limits are real.
> >> > <=> lim(n->∞) 1 = lim(n->∞) 1/5^n
> >> > <=> 1=0
> >> >
> >> > I wonder how much does you guys really understand you are talking?
> >> Obviously if you don't understand the basics of real analysis, you will
> >> doubt anyone who points them out.
> >
> > Do you understand this?
> > https://tutorial.math.lamar.edu/classes/calci/limitsproperties.aspx
> Yes of course I do. But what is the point of citing a document that
> does not support your erroneous algebra? In fact that document says
> nothing at all about /any/ of the limits in your so-called proof (though
> you'd have to know a bit of the subject to see that).
>
> --
> Ben.

It is 4 times. I think I am qualified to call you IDIOT and limit responding to you.

On Monday, 15 August 2022 at 18:40:15 UTC+2, wyni...@gmail.com wrote:
> Thank you. I kind of lost, wondering what the world is.
The Mathematics world is full of academics who have never done a minute of engineering in their lives.

They mostly don't unverstand the value of closures/closed sets, mean while that is the essence of control theory/engineering.

> Luckily, computers give me confidence I am not crazy (and I used to play electronics. I think I understand
> how real thing works).
Wonderful! Then you already have the analogue (continuous) intuition of what R/*R is like.
You are a step ahead of most programmers who only have good intuition for the natural numbers.

> I feel ℝ is not closed and incomplete.
Asolutely! This is a well-known fact. The Real numbers are closed under addition, subtraction and multiplication, but not division because it's undefined for x/0.

And a fundamental fact of ALL Mathematics. No number system is closed under equality!
x == x is a Boolean, not a number!

What it seems to me is that you desperately want to be able to do computation with Real numbers (despite the limits of those pesky discrete computers!). And you are not alone.

>But I am a programmer, just learn what I feel need to learn (for time/learning efficiency reason).
Perfect! That's an indispensable intuition for engineers. We do Just-In-Time learning.

It works. Most of the time. Except when you desperately need a number system different to the status quo, and all the idiot-Mathematicians are trying to indoctrinate you instead of help you solve your pragmatic problems.
> You have mentioned Hyperreal several times. After seeing what my idea is,
> should I really learn it? What would I get?
You would get most of the theorems you are looking for and most of the answers you seek in your original post.

You will get to treat infinity as just-another-number.
You will get to understand the meaning of lim(x→∞) f(x) in terms of infinitesimals and infinites (they are complementary).

Like I said, the fndamental theorem of *R is 1/ε = ω/1.
In English: 1 divided by an infinitesimal quantity is an infinite quantity. Infinity multiplied by a really small quantity is 1.

This is a really really nice setting for an engineer, because you can reason about quotients, proportions, logarithmic functions, and all the usual stuff we want out of information theory/signal processing!

On Mon, 15 Aug 2022 10:26:16 -0700 (PDT)
Skep Dick <skepdick22@gmail.com> wrote:

> On Monday, 15 August 2022 at 18:40:15 UTC+2, wyni...@gmail.com wrote:
> > Thank you. I kind of lost, wondering what the world is.
> The Mathematics world is full of academics who have never done a
> minute of engineering in their lives.
>
> They mostly don't unverstand the value of closures/closed sets, mean
> while that is the essence of control theory/engineering.
>
> > Luckily, computers give me confidence I am not crazy (and I used to
> > play electronics. I think I understand how real thing works).
> Wonderful! Then you already have the analogue (continuous) intuition
> of what R/*R is like. You are a step ahead of most programmers who
> only have good intuition for the natural numbers.
>
> > I feel ℝ is not closed and incomplete.
> Asolutely! This is a well-known fact. The Real numbers are closed
> under addition, subtraction and multiplication, but not division
> because it's undefined for x/0.
>
> And a fundamental fact of ALL Mathematics. No number system is closed
> under equality! x == x is a Boolean, not a number!
>
> What it seems to me is that you desperately want to be able to do
> computation with Real numbers (despite the limits of those pesky
> discrete computers!). And you are not alone.
>
> >But I am a programmer, just learn what I feel need to learn (for
> >time/learning efficiency reason).
> Perfect! That's an indispensable intuition for engineers. We do
> Just-In-Time learning.
>
> It works. Most of the time. Except when you desperately need a number
> system different to the status quo, and all the idiot-Mathematicians
> are trying to indoctrinate you instead of help you solve your
> pragmatic problems.
> > You have mentioned Hyperreal several times. After seeing what my
> > idea is, should I really learn it? What would I get?
> You would get most of the theorems you are looking for and most of
> the answers you seek in your original post.
>
> You will get to treat infinity as just-another-number.
> You will get to understand the meaning of lim(x→∞) f(x) in terms of
> infinitesimals and infinites (they are complementary).
>
> Like I said, the fndamental theorem of *R is 1/ε = ω/1.
> In English: 1 divided by an infinitesimal quantity is an infinite
> quantity. Infinity multiplied by a really small quantity is 1.
>
> This is a really really nice setting for an engineer, because you can
> reason about quotients, proportions, logarithmic functions, and all
> the usual stuff we want out of information theory/signal processing!

Infinitesimals don't exist: 1 / infinity = 0

/Flibble

On Monday, 15 August 2022 at 19:39:35 UTC+2, Mr Flibble wrote:
> Infinitesimals don't exist: 1 / infinity = 0
Numbers don't exist!

0 is undefined.
The successor function is undefined.

Now fuck off.

On Mon, 15 Aug 2022 10:52:11 -0700 (PDT)
Skep Dick <skepdick22@gmail.com> wrote:

> On Monday, 15 August 2022 at 19:39:35 UTC+2, Mr Flibble wrote:
> > Infinitesimals don't exist: 1 / infinity = 0
> Numbers don't exist!
>
> 0 is undefined.
> The successor function is undefined.
>
> Now fuck off.

Spicy.

/Flibble

wij <wyniijj2@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, since every
natural number is less than ∞.

> 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. 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

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?

I get the impression that you're trying to use common sense rules that
apply to the integers, and apply them to ∞. That doesn't work. For
example, common sense tells us that x+1 > x (and we can prove it given
the right axioms). But that's not true if x=∞ *and*, as you assert, ∞
is a unique value.

You can certainly define systems in which ∞ is a distinct value, and
with some effort you can define the results of various operations on ∞
and finite numbers and make them work consistently. I don't see that
you've actually done so.

[...]

--
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 Monday, 15 August 2022 at 20:25:36 UTC+2, Keith Thompson wrote:
> You can certainly define systems in which ∞ is a distinct value, and
> with some effort you can define the results of various operations on ∞
> and finite numbers and make them work consistently. I don't see that
> you've actually done so.
Why does every single system need re-defining from first principles?
Why do we constantly have to re-invent the wheel?

Why is the concept of "importing libraries/dependencies" so under-utilised in Mathematics?!?

Skep Dick <skepdick22@gmail.com> writes:
> On Monday, 15 August 2022 at 20:25:36 UTC+2, Keith Thompson wrote:
>> You can certainly define systems in which ∞ is a distinct value, and
>> with some effort you can define the results of various operations on ∞
>> and finite numbers and make them work consistently. I don't see that
>> you've actually done so.
> Why does every single system need re-defining from first principles?
> Why do we constantly have to re-invent the wheel?
>
> Why is the concept of "importing libraries/dependencies" so
> under-utilised in Mathematics?!?

If there are multiple possible libraries you could import, you still
have to specify which one you're using.

--
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 Monday, 15 August 2022 at 20:53:59 UTC+2, Keith Thompson wrote:
> If there are multiple possible libraries you could import, you still
> have to specify which one you're using.
Well yeah! But your search algorithm is going to take significantly longer if you aren't even bothering to eliminate the systems which **definitely** can't satisfy the necessary theorems!

On Monday, August 15, 2022 at 11:38:14 AM UTC-7, Skep Dick wrote:
> On Monday, 15 August 2022 at 20:25:36 UTC+2, Keith Thompson wrote:
> > You can certainly define systems in which ∞ is a distinct value, and
> > with some effort you can define the results of various operations on ∞
> > and finite numbers and make them work consistently. I don't see that
> > you've actually done so.
> Why does every single system need re-defining from first principles?
> Why do we constantly have to re-invent the wheel?
>
> Why is the concept of "importing libraries/dependencies" so under-utilised in Mathematics?!?
>
In most cases mathematicians study specific subject matter and there is
no importing involved. For example: I studied Perturbations of Operators
on Banach Spaces. Nothing extraneous was needed.

wij <wyniijj2@gmail.com> writes:

> On Tuesday, 16 August 2022 at 00:14:19 UTC+8, Skep Dick wrote:
>> On Monday, 15 August 2022 at 16:52:32 UTC+2, wyni...@gmail.com wrote:

>> > 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)
>> > <=> 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?
>>
>> You are 100% correct when using the Hyperreal numbers! That is *R not R.

> You have mentioned Hyperreal several times. After seeing what my idea is,
> should I really learn it? What would I get?

Learning is always a good idea, but the hyperreals won't give you what
you want as far as the limits you presented go. And 0.999... = 1 in *R
as well. In fact, the convergence of such well-behaved series is almost
the "poster boy" case for using *R since *R's infinitesimals formalise
Euler's convergence criterion.

--
Ben.

On Monday, 15 August 2022 at 21:11:31 UTC+2, dklei...@gmail.com wrote:
> In most cases mathematicians study specific subject matter and there is
> no importing involved. For example: I studied Perturbations of Operators
> on Banach Spaces. Nothing extraneous was needed.

When you say "study specific field" do yo mean that you spent time examining other people's no constructions, or did you construct a new field from first principles?

And I am not going to beat around the bush here... By "construct" I mean "invent".

Did you invent anything; or did you marvel at other people's inventions?

wij <wyniijj2@gmail.com> writes:

> On Tuesday, 16 August 2022 at 00:34:41 UTC+8, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>> > On Monday, 15 August 2022 at 23:45:59 UTC+8, Ben Bacarisse wrote:
>> >> wij <wyni...@gmail.com> writes:
>>
>> >> > Ben made an error and (assume he saw my reply to Andy) made an error
>> >> > again.
>> >> I made no mistake.
>> >> > 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)
>> >> No. This step is invalid. You can't multiply a limit by anything but a
>> >> real number. lim(n->∞) 2^n is not a real number.
>> >> > <=> lim(n->∞) 2^n/2^n = lim(n->∞) 2^n/10^n
>> >> The product law for limits is only valid when both limits are real.
>> >> > <=> lim(n->∞) 1 = lim(n->∞) 1/5^n
>> >> > <=> 1=0
>> >> >
>> >> > I wonder how much does you guys really understand you are talking?
>> >> Obviously if you don't understand the basics of real analysis, you will
>> >> doubt anyone who points them out.
>> >
>> > Do you understand this?
>> > https://tutorial.math.lamar.edu/classes/calci/limitsproperties.aspx
>> Yes of course I do. But what is the point of citing a document that
>> does not support your erroneous algebra? In fact that document says
>> nothing at all about /any/ of the limits in your so-called proof (though
>> you'd have to know a bit of the subject to see that).
>>
>
> It is 4 times. I think I am qualified to call you IDIOT and limit
> responding to you.

You can call me what you like. And do please limit your responses
(ideally to zero), but that will not change the fact that the document
you cite does not support your derivation. If you want to know why the
multiplication rule you've seen does no apply, just ask. Someone you
have not insulted might explain it.

--
Ben.

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.

> > 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)

> 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

> I get the impression that you're trying to use common sense rules that
> apply to the integers, and apply them to ∞. That doesn't work. For
> example, common sense tells us that x+1 > x (and we can prove it given
> the right axioms). But that's not true if x=∞ *and*, as you assert, ∞
> is a unique value.
>
> You can certainly define systems in which ∞ is a distinct value, and
> with some effort you can define the results of various operations on ∞
> and finite numbers and make them work consistently. I don't see that
> you've actually done so.
>

'∞' is a symbol like i,e,π,√2. Other arithmetic properties are the same as
usual numbers, implied and sufficient. Defining them will cause ambiguous problems.
Did these answer you?

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

On Tuesday, 16 August 2022 at 01:26:18 UTC+8, Skep Dick wrote:
> On Monday, 15 August 2022 at 18:40:15 UTC+2, wyni...@gmail.com wrote:
> > Thank you. I kind of lost, wondering what the world is.
> The Mathematics world is full of academics who have never done a minute of engineering in their lives.
>
> They mostly don't unverstand the value of closures/closed sets, mean while that is the essence of control theory/engineering.

Yes, many people I know had never get their hand dirty in their life and become
construction/mechanics engineers or doctors. But that alone is not really that
bad, their environment caused that. Their 'abstract' idea may also be very
intelligent and work fine.

> > Luckily, computers give me confidence I am not crazy (and I used to play electronics. I think I understand
> > how real thing works).
> Wonderful! Then you already have the analogue (continuous) intuition of what R/*R is like.
> You are a step ahead of most programmers who only have good intuition for the natural numbers.

1/3= 0.333... + nonzero_remainder is intuition. Kids understand this naturally without question.

> > I feel ℝ is not closed and incomplete.
> Asolutely! This is a well-known fact. The Real numbers are closed under addition, subtraction and multiplication, but not division because it's undefined for x/0.
>
> And a fundamental fact of ALL Mathematics. No number system is closed under equality!
> x == x is a Boolean, not a number!
>
> What it seems to me is that you desperately want to be able to do computation with Real numbers (despite the limits of those pesky discrete computers!). And you are not alone.

Actually, I might be able to not speaking for 10 years. All are in my mind.
Actually, what I want to achieve is a program that can do reasoning
(calculation, proving and reasoning are the same thing, harder to explain)
My hunch is that computer may be able to do math. problems, proving theorems.
I know my math. is not good comparing to MANY (everybody is good at something).
And there are already many symbolic computation systems, but I guess they
should be stuck by inconsistency issues.

> >But I am a programmer, just learn what I feel need to learn (for time/learning efficiency reason).
> Perfect! That's an indispensable intuition for engineers. We do Just-In-Time learning.

Yes, we know what we work for.

> It works. Most of the time. Except when you desperately need a number system different to the status quo, and all the idiot-Mathematicians are trying to indoctrinate you instead of help you solve your pragmatic problems.
> > You have mentioned Hyperreal several times. After seeing what my idea is,
> > should I really learn it? What would I get?
> You would get most of the theorems you are looking for and most of the answers you seek in your original post.
>
> You will get to treat infinity as just-another-number.
> You will get to understand the meaning of lim(x→∞) f(x) in terms of infinitesimals and infinites (they are complementary).
>
> Like I said, the fndamental theorem of *R is 1/ε = ω/1.
> In English: 1 divided by an infinitesimal quantity is an infinite quantity. Infinity multiplied by a really small quantity is 1.
>
> This is a really really nice setting for an engineer, because you can reason about quotients, proportions, logarithmic functions, and all the usual stuff we want out of information theory/signal processing!

Terrific. I got motive.

We may have many idea to exchange. You can email me or my C++ library project
at https://sourceforge.net/projects/cscall/

On Tuesday, 16 August 2022 at 04:50:43 UTC+8, wij wrote:
> On Tuesday, 16 August 2022 at 01:26:18 UTC+8, Skep Dick wrote:
> > On Monday, 15 August 2022 at 18:40:15 UTC+2, wyni...@gmail.com wrote:
> > > Thank you. I kind of lost, wondering what the world is.
> > The Mathematics world is full of academics who have never done a minute of engineering in their lives.
> >
> > They mostly don't unverstand the value of closures/closed sets, mean while that is the essence of control theory/engineering.
> Yes, many people I know had never get their hand dirty in their life and become
> construction/mechanics engineers or doctors. But that alone is not really that
> bad, their environment caused that. Their 'abstract' idea may also be very
> intelligent and work fine.
> > > Luckily, computers give me confidence I am not crazy (and I used to play electronics. I think I understand
> > > how real thing works).
> > Wonderful! Then you already have the analogue (continuous) intuition of what R/*R is like.
> > You are a step ahead of most programmers who only have good intuition for the natural numbers.
> 1/3= 0.333... + nonzero_remainder is intuition. Kids understand this naturally without question.
> > > I feel ℝ is not closed and incomplete.
> > Asolutely! This is a well-known fact. The Real numbers are closed under addition, subtraction and multiplication, but not division because it's undefined for x/0.
> >
> > And a fundamental fact of ALL Mathematics. No number system is closed under equality!
> > x == x is a Boolean, not a number!
> >
> > What it seems to me is that you desperately want to be able to do computation with Real numbers (despite the limits of those pesky discrete computers!). And you are not alone.
> Actually, I might be able to not speaking for 10 years. All are in my mind.
> Actually, what I want to achieve is a program that can do reasoning
> (calculation, proving and reasoning are the same thing, harder to explain)
> My hunch is that computer may be able to do math. problems, proving theorems.
> I know my math. is not good comparing to MANY (everybody is good at something).
> And there are already many symbolic computation systems, but I guess they
> should be stuck by inconsistency issues.
> > >But I am a programmer, just learn what I feel need to learn (for time/learning efficiency reason).
> > Perfect! That's an indispensable intuition for engineers. We do Just-In-Time learning.
> Yes, we know what we work for.
> > It works. Most of the time. Except when you desperately need a number system different to the status quo, and all the idiot-Mathematicians are trying to indoctrinate you instead of help you solve your pragmatic problems.
> > > You have mentioned Hyperreal several times. After seeing what my idea is,
> > > should I really learn it? What would I get?
> > You would get most of the theorems you are looking for and most of the answers you seek in your original post.
> >
> > You will get to treat infinity as just-another-number.
> > You will get to understand the meaning of lim(x→∞) f(x) in terms of infinitesimals and infinites (they are complementary).
> >
> > Like I said, the fndamental theorem of *R is 1/ε = ω/1.
> > In English: 1 divided by an infinitesimal quantity is an infinite quantity. Infinity multiplied by a really small quantity is 1.
> >
> > This is a really really nice setting for an engineer, because you can reason about quotients, proportions, logarithmic functions, and all the usual stuff we want out of information theory/signal processing!
> Terrific. I got motive.
>
> We may have many idea to exchange. You can email me or my C++ library project
> at https://sourceforge.net/projects/cscall/

Forgot to say thanks. Originally, I did not want to talk much. But you have handled
many loud noises (speech was like Laozi https://en.wikipedia.org/wiki/Laozi)
So, I decided to take the opportunity. You did helped a lot.

wij <wyniijj2@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.Thompson+u@gmail.com
Working, but not speaking, for Philips
void Void(void) { Void(); } /* The recursive call of the void */

On 8/15/2022 3:47 AM, win wrote:

> The proposed definition of infinity is super simple and safe-guaranteed, and, it
> SOLVED many infinity related paradoxes: Classes of liar's paradoxes, Zeno's
> paradoxes, Supertask paradox, myth of infinite series,... and can build a
> "one-point slope theory", Your choice. What the standard analysis solves?
> Inconsistencies from limit, exam/thesis/paper/degree/title/money, such things I guess.
>
> People can use the proposed definition of infinity as an 'informal' option to test.
> It is super simple, safe-guaranteed, no need to say more.

What raucous bull shit! You haven't proposed a definition of infinity
nor solved any paradox. Do as I suggested: go read a High School math
book and then express an opinion. But I must admit that you are very
humorous. Thanks for the entertainment. The only improvement is when you
and Peter O. start debating. The sound is like castrated wolves howling
at the full moon. When you are on your own like here, it brings up that
old Buddhist question: what is the sound of one hand clapping? Or is
that howling? Enjoy your self.
--
Jeff Barnett

On 8/15/22 12:02 PM, Skep Dick wrote:
> 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"
>

Nope, because the law of limits has a special definition for limits to
infinity.

Just because you don't understand the definitions doesn't mean they
don't exist.

On 8/15/22 12:27 PM, Skep Dick wrote:
> 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.
> Yes, I do get to say it. And if you don't like me saying it - I will repeat it louder.
>
> lim(x -> a) means EXACTLY
>
> let a = ∞
> lim(x -> a)
>
>

Except there is a SPECIAL rule for limits to the non-finite-number infinity.

Thus you can't use the the rule for limits to a real number.

Definition aren't just syntax, but can refer to semantics

lim(x -> a) has two different definitions.

One for a being a member of the Real Numbers (or a Complex Number), and
another for a being an infinity.

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 Monday, August 15, 2022 at 12:17:57 PM UTC-7, Skep Dick wrote:
> On Monday, 15 August 2022 at 21:11:31 UTC+2, dklei...@gmail.com wrote:
> > In most cases mathematicians study specific subject matter and there is
> > no importing involved. For example: I studied Perturbations of Operators
> > on Banach Spaces. Nothing extraneous was needed.
> When you say "study specific field" do yo mean that you spent time examining other people's no constructions, or did you construct a new field from first principles?
>
> And I am not going to beat around the bush here... By "construct" I mean "invent".
>
> Did you invent anything; or did you marvel at other people's inventions?
>
There were a couple of hundred relevant papers. I read them and tried a
slightly different approach. I uncovered a dozen or more new relationships
and theorems. The most noteworthy is still used from time to time (called
the Kleinecke-Shirokov Theorem). I am a Platonist in the sense I believe
mathematical entities exist and are discovered. Not invented.

If you want to wonder at mathematics admire the monster - the largest
sporatic simple group. Found not invented.

Skep Dick <skepdick22@gmail.com> writes:

> On Monday, 15 August 2022 at 18:39:33 UTC+2, Ben Bacarisse wrote:
>> Skep Dick <skepd...@gmail.com> writes:
>>
>> > On Monday, 15 August 2022 at 16:52:32 UTC+2, wyni...@gmail.com wrote:
>>
>> >> 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)
>> >> <=> 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?
>> >
>> > You are 100% correct when using the Hyperreal numbers! That is *R not
>> > R.
>> Wij is not working in *R. He cites standard rules about limits in R
>> (rules he or she does not properly understand) to support this bogus
>> proof.
> He has openly told you what theorems he is interested in!
> None of the theorems he wants hold in R.
> Most of the theorems he wants hold in *R

Not so.

> Why are you dragging him down instead of pulling him up?!?
>
>>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.
>
> Bullshit. 0.999... is meaningless in *R. It's a syntax error.

No. 0.990... means the same in R and in *R. It's an infinite sum. The
sum is 1 in both R and in *R. Just saying stuff is not how to do
mathematics.

--
Ben.

Skep Dick <skepdick22@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.

--
Ben.

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.

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