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Aug 14, 2022, 7:35:27 PMAug 14

to

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.

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.

Aug 14, 2022, 8:34:39 PMAug 14

to

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),

Aug 14, 2022, 8:48:38 PMAug 14

to

On 8/14/2022 5:35 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?

It has a well-defined meaning in standard analysis. There is no mystery.
> 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 ∞.

don't get closer to infinity, you merely have larger and larger numbers.

> But valid what? Most people agree ∀n∈ℕ, n<∞.

only is what you said true; it's true by /definition/!

> Is x+1 not closer than x to infinity?

> So, infinity ∞ must have arithmetic meaning. Here is one:

There are ways to adjoin it to the numbers but then you must change the

axioms defining real fields.

> The multiplicative inverse of ∞ is 1/∞, the additive inverse is -∞

allowed, would lead to inconsistencies. And that means you would have a

broken system.

> 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.

and the mysteries that so plague you will disappear.

There are interesting systems that do some of what you want or believe

that have been developed. But, note well, they were developed by folks

who know well what the pitfalls were and they went into the work with

enough background to not make an utter mess of it.

What I'm suggesting is that you (1) learn standard analysis, (2) study

alternative formulations, and (3) then and only then do you try to

invent a system to your own taste. You might even enjoy the learning

experience.

--

Jeff Barnett

Aug 14, 2022, 10:37:13 PMAug 14

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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
> 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 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?

from ∞-x?

> 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.

multiplicative inverse is 1/∞ and its additive inverse is -∞?

OK, but if I were to define ∞, I'd probably try to come up with a

definition that doesn't apply equally well to 8. (Just in case my point

wasn't clear, the multiplicative inverse of 8 is 1/8 and its additive

inverse is -8.)

--

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 */

Aug 15, 2022, 5:17:38 AMAug 15

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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.
> 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?

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 -∞

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).

> > 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.

> Is that supposed to be a *definition* of ∞? Just that it's

> multiplicative inverse is 1/∞ and its additive inverse is -∞?

>

> OK, but if I were to define ∞, I'd probably try to come up with a

> definition that doesn't apply equally well to 8. (Just in case my point

> wasn't clear, the multiplicative inverse of 8 is 1/8 and its additive

> inverse is -8.)

>

> --

> 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 */

Definition is an arbitrary thing. Like 'programming theory', We ask what kind of

problem it solves, efficiency, benefit and deficiency. With number theory, logic

is a must. I think the notion conveyed by limit is accepted is because of these

qualities except the last one mentioned.

Aug 15, 2022, 5:38:43 AMAug 15

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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).

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

<=> 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.

Aug 15, 2022, 5:47:33 AMAug 15

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If your are talking the inconsistency and incorrectness hidden in your standard

analysis, I agree. In my theory and view, many kind of standard-analysis are

actually trying to solve problems caused by the limit theory. No need and

unnecessary to build lies on lies.

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.

Aug 15, 2022, 7:22:25 AMAug 15

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On 15/08/2022 01:48, Jeff Barnett wrote:

> On 8/14/2022 5:35 PM, wij wrote:

[...]
> On 8/14/2022 5:35 PM, wij wrote:

> What I'm suggesting is that you (1) learn standard analysis, (2)

> study alternative formulations, and (3) then and only then do you try

> to invent a system to your own taste. You might even enjoy the

> learning experience.

Seconded. But, sadly, experience is that Wij and the others
> study alternative formulations, and (3) then and only then do you try

> to invent a system to your own taste. You might even enjoy the

> learning experience.

here who seem to think that they have new insights into mathematics

will take no notice, and instead convince themselves that everyone

else is simply hidebound, a "learned-by-rote", and incapable of new

thoughts.

Meanwhile, I commend [yet again] the study of the surreal

numbers; much easier than the hyperreals, and more useful. The

links with games can be explained to anyone who is not totally

innumerate, and infinity/infinitesimals arise in a very natural

and practical way. For some hints and further references, see

https://www-cs-faculty.stanford.edu/~knuth/sn.html

or, of course, google. Also, any book by [or co-authored by] the

late, great John Conway is worth reading.

--

Andy Walker, Nottingham.

Andy's music pages: www.cuboid.me.uk/andy/Music

Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Dvorak

Aug 15, 2022, 8:02:58 AMAug 15

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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

divided by zero depending on the steps you did to make that transition.

This is the problem of assuming that "infinity" is a number.

Aug 15, 2022, 8:32:00 AMAug 15

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

> 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

(Technically, it's the two lines above that are the assumptions. Once
> 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

you have written Let A = <stuff> = 0.999... and B = <other stuff> =

0.999... the line A=B is not an assumption anymore since it follows from

the earlier two.)

> <=> lim(n->∞) 1-1/2^n = lim(n->∞) 1-1/10^n

> <=> lim(n->∞) 1/2^n = lim(n->∞) 1/10^n

> <=> lim(n->∞) 1 = lim(n->∞) 1/5^n

take this step? It's not multiplying by 2, and you can't multiply by

2^n as n is a bound variable.

> <=> 1=0

This was pointed out before (by at least two posters). Why do you keep

posting "proofs" with basic mistakes in them?

> -----------------------

> 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)

give it, and in neither case does (1.000...1)^n approaches e. Anyway,

you should not use ambiguous notations without a some explanation of

what the notation means.

> -----------------------

> 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.

define A[z] for any z not in N. You can extend the definition to

include something called oo (so that, for example, A[oo] = 42 if you

like), but a more natural extension would be to define A[oo] = 1.

Either way, it's up to anyone making the extension to defend it.

> 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).

well-behaved sequences like the partial sums 0.999... the limit is the

same as the least upper bound.

--

Ben.

Aug 15, 2022, 10:52:32 AMAug 15

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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/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?
<=> 1=0

Aug 15, 2022, 11:45:59 AMAug 15

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> 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)

real number. lim(n->∞) 2^n is not a real number.

> <=> 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?

doubt anyone who points them out.

--

Ben.

Aug 15, 2022, 11:54:20 AMAug 15

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On Monday, 15 August 2022 at 02:48:38 UTC+2, Jeff Barnett wrote:

> What I'm suggesting is that you (1) learn standard analysis

Why are you directing him to standard analysis when the system DOESN'T do anything of what he's asking/saying ?!? Are you just trying to waste his time?
> What I'm suggesting is that you (1) learn standard analysis

Why aren't you suggesting that he learns NONstandard analysis instead, which does PRECISELY what he's trying to do?

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

Aug 15, 2022, 12:00:40 PMAug 15

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You are right. Don't let them guilt you into believing otherwise - they don't understand how to compute/program with Real numbers!

They don't understand parametricity. https://en.wikipedia.org/wiki/Parametricity and why it's directly relevant to the syntax of "lim(x->∞)" if you are passing "∞" as a parameter to a function then "∞" is BOUND TO A VARIABLE.

If you are binding objects to free variables - you are treating those objects as numbers/values. If "∞" is not a number then "lim(x->∞)" is a syntax error.

Aug 15, 2022, 12:02:31 PMAug 15

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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
> Obviously if you don't understand the basics of real analysis, you will

> doubt anyone who points them out.

you will doubt anyone who points out that "lim(x -> ∞)" is a syntax error IF "∞ is not a number"

Aug 15, 2022, 12:14:19 PMAug 15

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I will repeat it until the cows come home. Despite having said it three times already.

You can't parametrize functions by outside the domain of thefunction!

So you can parametrize lim(x -> y) by y = ∞ then ∞ is in the domain of lim(x -> y).

If ∞ is in the domain of lim(x -> y) then ∞ is a number!

Aug 15, 2022, 12:20:56 PMAug 15

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Aug 15, 2022, 12:25:25 PMAug 15

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are well-established form used to mean two quite different kinds of

limits, despite the similarity in the syntax.

--

Ben.

Aug 15, 2022, 12:27:34 PMAug 15

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lim(x -> a) means EXACTLY

let a = ∞

lim(x -> a)

Aug 15, 2022, 12:29:56 PMAug 15

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On Monday, 15 August 2022 at 18:25:25 UTC+2, Ben Bacarisse wrote:

Yes, I do get to say it. And if you don't like me saying it - I will repeat it louder.

Aug 15, 2022, 12:34:41 PMAug 15

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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.

Aug 15, 2022, 12:39:33 PMAug 15

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(rules he or she does not properly understand) to support this bogus

proof. 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.

--

Ben.

Aug 15, 2022, 12:40:15 PMAug 15

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me confidence I am not crazy (and I used to play electronics. I think I understand

how real thing works).

I feel ℝ is not closed and incomplete. But I am a programmer, just learn what I

feel need to learn (for time/learning efficiency reason).

You have mentioned Hyperreal several times. After seeing what my idea is,

should I really learn it? What would I get?

Aug 15, 2022, 12:43:06 PMAug 15

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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!
> 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.

None of the theorems he wants hold in R.

Most of the theorems he wants hold in *R

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.

Aug 15, 2022, 12:48:17 PMAug 15

to

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.
> as it happens, 0.999... = 1 in *R as well as in R), they want

> conventional wisdom to be wrong.

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.

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

What "cranks" want is the theorem 0.999... = 1 - ε

Aug 15, 2022, 12:50:43 PMAug 15

to

Aug 15, 2022, 1:26:18 PMAug 15

to

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.
> Thank you. I kind of lost, wondering what the world is.

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).

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.

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).

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 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!

Aug 15, 2022, 1:39:35 PMAug 15

to

/Flibble

Aug 15, 2022, 1:52:13 PMAug 15

to

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!
> Infinitesimals don't exist: 1 / infinity = 0

0 is undefined.

The successor function is undefined.

Now fuck off.

Aug 15, 2022, 2:18:59 PMAug 15

to

/Flibble

Aug 15, 2022, 2:25:36 PMAug 15

to

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

> On Monday, 15 August 2022 at 10:37:13 UTC+8, Keith Thompson wrote:

>> ∞ 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?

>> 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.T...@gmail.com

Working, but not speaking, for Philips

void Void(void) { Void(); } /* The recursive call of the void */

> 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
>> > 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 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.T...@gmail.com

Working, but not speaking, for Philips

void Void(void) { Void(); } /* The recursive call of the void */

Aug 15, 2022, 2:38:14 PMAug 15

to

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?
> 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 do we constantly have to re-invent the wheel?

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

Aug 15, 2022, 2:53:59 PMAug 15

to

have to specify which one you're using.

Aug 15, 2022, 3:09:10 PMAug 15

to

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!
> If there are multiple possible libraries you could import, you still

> have to specify which one you're using.

Aug 15, 2022, 3:11:31 PMAug 15

to

no importing involved. For example: I studied Perturbations of Operators

on Banach Spaces. Nothing extraneous was needed.

Aug 15, 2022, 3:16:22 PMAug 15

to

> 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
> should I really learn it? What would I get?

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.

Aug 15, 2022, 3:17:57 PMAug 15

to

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?
> 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.

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?

Aug 15, 2022, 3:21:08 PMAug 15

to

> 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
> responding to you.

(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.

Aug 15, 2022, 4:14:14 PMAug 15

to

> since every natural number is less than ∞.

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.

> 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

>

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?

∞-(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.

>

usual numbers, implied and sufficient. Defining them will cause ambiguous problems.

Did these answer you?

Aug 15, 2022, 4:50:43 PMAug 15

to

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
> 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.

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.

> > 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, 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.

> 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!

We may have many idea to exchange. You can email me or my C++ library project

at https://sourceforge.net/projects/cscall/

Aug 15, 2022, 4:54:53 PMAug 15

to

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.

Aug 15, 2022, 5:36:34 PMAug 15

to

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)

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.

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.

Aug 15, 2022, 6:12:15 PMAug 15

to

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

> 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

Aug 15, 2022, 6:50:41 PMAug 15

to

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

> On Monday, 15 August 2022 at 17:45:59 UTC+2, Ben Bacarisse 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.

>

>> 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
> you will doubt anyone who points out that "lim(x -> ∞)" is a syntax error IF "∞ is not a number"

>

infinity.

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

don't exist.

Aug 15, 2022, 6:53:59 PMAug 15

to

On 8/15/22 12:27 PM, Skep Dick wrote:

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 Monday, 15 August 2022 at 18:25:25 UTC+2, Ben Bacarisse wrote:

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

>>

>> 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"

>>>> 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.

>> 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.
> let a = ∞

> lim(x -> a)

>

>

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.

Aug 15, 2022, 7:15:55 PMAug 15

to

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:

> 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<∞.

>>>>>

>>>>> 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 -∞

>>>>>

>>>>>

>>> 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

>>> 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.

>> 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

>>>

>>> <=> 1=0

>>>

> 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
> 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)

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.
> <=> 1=0

>

> I wonder how much does you guys really understand you are talking?

Aug 15, 2022, 8:36:01 PMAug 15

to

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.

Aug 15, 2022, 9:20:20 PMAug 15

to

Skep Dick <skepd...@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.
> 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

> 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.

sum is 1 in both R and in *R. Just saying stuff is not how to do

mathematics.

--

Ben.

Aug 15, 2022, 9:46:58 PMAug 15

to

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:

>>

Say it again and again, please.

> lim(x -> ∞) means EXACTLY THE SAME THING AS

reasons.

--

Ben.

> 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.
>> >> 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.

Why would you do that? Anyway, don't worry; I do like you saying it.
> Yes, I do get to say it. And if you don't like me saying it - I will

> repeat it louder.

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
> let a = ∞

> lim(x -> a)

reasons.

--

Ben.

Aug 16, 2022, 2:54:40 AMAug 16

to

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

Aug 16, 2022, 4:42:45 AMAug 16

to

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...
> Say it again and again, please.

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 ∞

Aug 16, 2022, 4:56:55 AMAug 16

to

Op 15.aug..2022 om 01:35 schreef wij:

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.

> 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
> 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.

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.

Aug 16, 2022, 6:31:39 AMAug 16

to

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.
> If you don't follow the rules of the Math you are using, you get

> unreliable results.

Aug 16, 2022, 7:22:14 AMAug 16

to

>

> 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

>

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.

Aug 16, 2022, 7:23:41 AMAug 16

to

universe radically changes when you do.

Aug 16, 2022, 7:29:22 AMAug 16

to

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.

Aug 16, 2022, 7:54:49 AMAug 16

to

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!?!
> Nope, because you CAN'T just ignore that discontinuity, because the

> universe radically changes when you do.

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

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

Aug 16, 2022, 8:08:20 AMAug 16

to

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

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.

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.

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.

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.

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

Aug 16, 2022, 8:28:52 AMAug 16

to

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?
> Lets see your complier enforce a speed limit.

I am talking about the abstract compiler in your head.

> The rules are well established, and well known.

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.

> Perhaps one issue is that some are semantic, so can't always be detected

> at compile time depending on your type system.

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.

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.

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.

Aug 16, 2022, 8:35:22 AMAug 16