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.

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),

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.
> 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 ∞.
First off, 'number' is merely a concept also. In standard analysis, you
don't get closer to infinity, you merely have larger and larger numbers.
> But valid what? Most people agree ∀n∈ℕ, n<∞.
Agreed. That's the nice thing about having a standard definition: not
only is what you said true; it's true by /definition/!
> Is x+1 not closer than x to infinity?
No it isn't.
> So, infinity ∞ must have arithmetic meaning. Here is one:
Infinity does have arithmetic meaning but not necessarily as a number.
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 -∞
Both claims are incorrect for standard analysis. These statements if
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.
Look up "limit" in a standard (calculus or analytic geometry) text book
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

wij <wyniijj2@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?

> 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.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 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 -∞

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 */

I might just talk to myself.
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.

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).

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.

On Monday, 15 August 2022 at 08:48:38 UTC+8, Jeff Barnett wrote:
> 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.
> > 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 ∞.
> First off, 'number' is merely a concept also. In standard analysis, you
> don't get closer to infinity, you merely have larger and larger numbers.
> > But valid what? Most people agree ∀n∈ℕ, n<∞.
> Agreed. That's the nice thing about having a standard definition: not
> only is what you said true; it's true by /definition/!
> > Is x+1 not closer than x to infinity?
> No it isn't.
> > So, infinity ∞ must have arithmetic meaning. Here is one:
> Infinity does have arithmetic meaning but not necessarily as a number.
> 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 -∞
> Both claims are incorrect for standard analysis. These statements if
> 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.
> Look up "limit" in a standard (calculus or analytic geometry) text book
> 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

You are addressing mostly about unrelated stuff.
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.

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

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.

wij <wyniijj2@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
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

Your algebra is wrong here. What rule of algebra do you think let's you
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

(0.999...)^n = 1 for all n >= 0.

> (1.000...1)^n approaches e (or defined as e)

1.000...1 is a non-standard definition. I know of two meanings one might
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.

Inductive definitions like this define A[k] for all k in N. They don't
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).

Yes. And open intervals in R have a least upper bound in R. For
well-behaved sequences like the partial sums 0.999... the limit is the
same as the least upper bound.

--
Ben.

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)
<=> 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?

wij <wyniijj2@gmail.com> writes:

> 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.

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.

--
Ben.

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?

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

On Monday, 15 August 2022 at 11:47:33 UTC+2, wyni...@gmail.com wrote:
> On Monday, 15 August 2022 at 08:48:38 UTC+8, Jeff Barnett wrote:
> > 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..
> > > 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 ∞.
> > First off, 'number' is merely a concept also. In standard analysis, you
> > don't get closer to infinity, you merely have larger and larger numbers..
> > > But valid what? Most people agree ∀n∈ℕ, n<∞.
> > Agreed. That's the nice thing about having a standard definition: not
> > only is what you said true; it's true by /definition/!
> > > Is x+1 not closer than x to infinity?
> > No it isn't.
> > > So, infinity ∞ must have arithmetic meaning. Here is one:
> > Infinity does have arithmetic meaning but not necessarily as a number.
> > 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 -∞
> > Both claims are incorrect for standard analysis. These statements if
> > 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.
> > Look up "limit" in a standard (calculus or analytic geometry) text book
> > 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
> You are addressing mostly about unrelated stuff.
> 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.
You really need to stop listening to all the fossils in the group. They can't help you - they don't know any better.

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.

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"

On Monday, 15 August 2022 at 16:52:32 UTC+2, wyni...@gmail.com 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)
> <=> 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.

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!

On Monday, 15 August 2022 at 23:45:59 UTC+8, Ben Bacarisse wrote:
> wij <wyni...@gmail.com> writes:
>
> > 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.
> 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.
>
> --
> Ben.

Skep Dick <skepdick22@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.

--
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.
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)

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.
>
> --
> Ben.
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.

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

let a = ∞
lim(x -> a)

wij <wyniijj2@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.

Skep Dick <skepdick22@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. 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.

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:
> > 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)
> > <=> 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.
>
> 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!

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.

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.

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