The meaning of infinity

[DBT]

Rationals as a conceptual representation of a space that would be infinitely divisible being a mathematical representation, a mathematical construct, a concept of infinite divisibility that does not necessarily relate to the physical world, physical infinity, an actual Physically Infinite Universe.

So, essentially, you're saying that, as a matter of principle, all mathematical models of the physical world that scientists could possibly produce, like QM and General Relativity, could not possibly compare to the real physical world?!

What I am saying, if it wasn't clear enough for you, is that Rationals can be applied infinitesimally, I'm not disputing that. I don't doubt it.

What I'm saying is that this is an abstract exercise. You can divide an inch, for example, zero to one, to the nth degree, infinitesimally, yet the inch was and alway remains a finite distance between zero and one.

Whoa! Maybe you should start to think seriously about maybe telling them.

No need, I'm sure they understand that an inch is a finite measurement regardless of the number of divisions using rationals. Maybe you should try to understand that simple distinction before trying to appear clever, as is your unfortunate habit.

 

[steve_bank]

Stumbled on this book.

https://en.wikipedia.org/wiki/One_Two_Three..._Infinity


Author George Gamow
Illustrator George Gamow
Country United States
Language English
Subject Science, mathematics
Published 1947 (Viking Press)


Overview
Beginning with an exploration of elementary numbers, the book opens with a description of the "Hottentots" (Khoikhoi), said to have words only for "one", "two", "three", and "many", and builds quickly to explore Georg Cantor's theory of three levels of infinity—hence the title of the book. It then describes a simple automatic printing press that can in principle (given enough paper, ink, and time) print all the English works that have ever been, or ever will be, printed (a more-systematic version of the infinite monkey theorem). The author notes that if all the atoms in the Universe, as known in Gamow's time, were such printing presses working in parallel "at the speed of atomic vibrations" since the beginning of known time, only an infinitesimal fraction of the job could have yet been completed.[1]

Gamow explores number theory, topology, spacetime, relativity, atomic chemistry, nuclear physics, entropy, genetics, and cosmology.

 

[steve_bank]

So let x = 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 90/10 + 90/100 + 90/1000 + ...

Therefore, 10x = 9/1 + 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 9/1 + x

No.
10x = 90/10 + 90/100 + 90/1000 + ...
10 x = 9 + .9 + .09 + .009
10x= 9.999
x = .9999

No matter ho many terms you use applying the distributive principle changes nothing.

I did not pick up your mistake and that led to mine.

What are you talking about? I didn't apply the distributive principle in that step. That step merely replaces a subexpression of the equation with an equal expression. I took out "9/10 + 9/100 + 9/1000 + ..." and put "x" in its place, which is a legitimate operation because those expressions are equal to each other according to an earlier step in the proof. Structurally speaking, Line 1 says "A = B", and Line 3 says "C = D + B", and Line 4 says "Therefore, C = D + A", and you're calling foul. What gives?

You started by multiplying both sides by 10. You made a math error as highlighted.

Therefore, 10x = 9/1 + 9/10 + 9/100 + 9/1000 + ....
Therefore, 10x = 9/1 + x Don't see how 9/10 + 9/100 + 9/1000 + ... reduces to x.

In Any case you can not algebraically manipulate an infinite series. Your proof fails. You either made an error or intentionally fudged it.

 

[Bomb#20]

So let x = 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 90/10 + 90/100 + 90/1000 + ...

Therefore, 10x = 9/1 + 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 9/1 + x

No.
10x = 90/10 + 90/100 + 90/1000 + ...
10 x = 9 + .9 + .09 + .009
10x= 9.999
x = .9999

No matter ho many terms you use applying the distributive principle changes nothing.

I did not pick up your mistake and that led to mine.

What are you talking about? I didn't apply the distributive principle in that step. That step merely replaces a subexpression of the equation with an equal expression. I took out "9/10 + 9/100 + 9/1000 + ..." and put "x" in its place, which is a legitimate operation because those expressions are equal to each other according to an earlier step in the proof. Structurally speaking, Line 1 says "A = B", and Line 3 says "C = D + B", and Line 4 says "Therefore, C = D + A", and you're calling foul. What gives?

You started by multiplying both sides by 10. You made a math error as highlighted.

Therefore, 10x = 9/1 + 9/10 + 9/100 + 9/1000 + ....
Therefore, 10x = 9/1 + x Don't see how 9/10 + 9/100 + 9/1000 + ... reduces to x.

It's the first line in the proof. "So let x = 9/10 + 9/100 + 9/1000 + ...". 9/10 + 9/100 + 9/1000 + ... reduces to x because that is the exact definition of x!

In Any case you can not algebraically manipulate an infinite series. Your proof fails.

Why can't I algebraically manipulate an infinite series? Does it lead to self-contradiction? Is there a rule against it in some algebra textbook? Or does it just offend your intuition?

 

[Phil Scott]

In Any case you can not algebraically manipulate an infinite series. Your proof fails. You either made an error or intentionally fudged it.

All of Bomb's manipulations are legal. My only complaint is that their legality has not been rigorously established. To do that, we need the sort of rigorous foundation I provide in the post you have ignored for the third time now. Until you read that, you're just blowing ill-informed and ill-educated hot air.

For anyone interested (and I fear it is just me by this point): bomb's steps are legal as follows:

Given a convergent series with limit l whose sequence of initial sums is x_n and, additionally, given a constant k, we can consider the series whose sequence of initial sums is x_n + k. This series converges with limit l + k. Indeed, given an error e > 0, we know there is an N such that |x_n - l| < e for n > N, and thus |(x_n + k) - (l + k)| < e.

That takes care of bomb saying that x + 9 = 9 + .9 + .09 + …

Now if we consider the series with limit l given by initial sums x_n and we take k, we can consider the series with sums kx_n. This has limit kl.

Indeed, given error e/|k| > 0, there is an N > 0 such that |x_n - l| < e/|k| for n > N and thus |k||x_n - l| = |kx_n - kl|< e.

That legitimises bomb's multiplication of the series by 10

All that remains is to show that 0.999… is convergent, a fact that I established in my other post where I also show that the limit is 1, rendering the digit manipulation proofs redundant.

See? Maths can be fun when you do it correctly. Or is it just me? My experience of threads like this indicates that people can have plenty of fun talking the nonsense they made up for themselves playing with calculators.

 

[Phil Scott]

Stumbled on this book.

https://en.wikipedia.org/wiki/One_Two_Three..._Infinity


Author George Gamow
Illustrator George Gamow
Country United States
Language English
Subject Science, mathematics
Published 1947 (Viking Press)

You stumbled across my post yet? It is short, not badly written if I say so myself, and aimed at you to replace the egregiously flawed understanding of real numbers you have displayed throughout the thread. I recommend it!

 

[Speakpigeon]

What I am saying, if it wasn't clear enough for you, is that Rationals can be applied infinitesimally, I'm not disputing that. I don't doubt it.

What I'm saying is that this is an abstract exercise. You can divide an inch, for example, zero to one, to the nth degree, infinitesimally, yet the inch was and alway remains a finite distance between zero and one.

We all know that and it's irrelevant.

We started this particular exchange on my statement that things can be finite in a way while infinite in another way. Everything you say here is entirely irrelevant to that. So, what are you talking about?! You appear to be replying but your response is irrelevant. How come?

Maybe you should try to understand that simple distinction before trying to appear clever, as is your unfortunate habit.

That distinction is irrelevant.

Phil Scott gave you one fine example of the idea that things can be finite in a way while infinite in another way. Yet, as of now, we're still not clear whether you understood the idea. All we've seen from you are irrelevancies and obfuscation. Nothing as to the substance of what you were pretending to reply to.
EB

 

[Artemus]

The problem people have with the fact that 0.9999... = 1 is a misunderstanding of what a convergent series actually represents. A convergent series does not "get closer and closer" to a value, it is in fact equal to that value. Zeno's paradox gives a physical example of why this must be true. Sequentially halving the remaining distance yields an infinite series. But when you cross an interval you don't get closer and closer to the other side, you reach the other side. The series must therefore sum to exactly the total distance. Same for crossing 90% of the remaining interval.

If this is not true, what is the number between 0.9999... and 1?

 

[Speakpigeon]

The problem people have with the fact that 0.9999... = 1 is a misunderstanding of what a convergent series actually represents. A convergent series does not "get closer and closer" to a value, it is in fact equal to that value.

What's the mathematical proof that convergent series are equal to their limits?

Zeno's paradox gives a physical example of why this must be true. Sequentially halving the remaining distance yields an infinite series. But when you cross an interval you don't get closer and closer to the other side, you reach the other side. The series must therefore sum to exactly the total distance. Same for crossing 90% of the remaining interval.

Must be true? So, you're not really sure... All you have is a physical allegory?!

If this is not true, what is the number between 0.9999... and 1?

I'd be happy with an infinitesimal.
EB

 

[Artemus]

What's the mathematical proof that convergent series are equal to their limits?



Must be true? So, you're not really sure... All you have is a physical allegory?!

If this is not true, what is the number between 0.9999... and 1?

I'd be happy with an infinitesimal.
EB

No, that is not all I have. People are either unable or unwilling to understand the math so I gave the physical analogy that should convince anyone who is willing to actually think about it. I gave a proof that 0.999... = 1. Others have given other proofs. Proofs that the sum of a convergent series is equal to the limit are given at the beginning of any introductory real analysis text book. What you are not happy with (ie do not understand) is irrelevant. Any one who questions it should just check out a book and read about it.

I'll give one more hint: If there was a remaining infinitesimal but non-zero value after the series was summed you could never cross an interval. And yet you can....

 

[Angra Mainyu]

What's the mathematical proof that convergent series are equal to their limits?

There is no need for a proof, because it is what it means by definition (in this context, anyway, so we're talking calculus). In other words, the expression
\( \sum\limits_{i=1}^{\infty}x_i \)
is defined as
\( \lim_{n\to +\infty}\sum\limits_{i=1}^{n}x_i, \)

provided that the limit exists. Or "exists and is finite", if you like: when it's not, it's also a matter of notation we can choose to leave it defined, or not. Also, sometimes we can speak say (for example) "If the series \( \sum\limits_{i=1}^{\infty}x_i \) converges..."), if we first define our notation so that that expression means that the limit in question exists - i.e., that there is x such that
\( \lim_{n\to +\infty}\sum\limits_{i=1}^{n}x_i=x, \)

Now if you are talking about formal series (for example) and you don't define an order, then it makes no sense to ask whether it's less than 1.

So, if you're asking for a proof that
\( \lim_{n\to +\infty}\sum\limits_{i=1}^{n}\frac{9}{10^{n}}=1, \)
some have already been given. Otherwise, what are you asking?

 

[steve_bank]

The problem people have with the fact that 0.9999... = 1 is a misunderstanding of what a convergent series actually represents. A convergent series does not "get closer and closer" to a value, it is in fact equal to that value. Zeno's paradox gives a physical example of why this must be true. Sequentially halving the remaining distance yields an infinite series. But when you cross an interval you don't get closer and closer to the other side, you reach the other side. The series must therefore sum to exactly the total distance. Same for crossing 90% of the remaining interval.

If this is not true, what is the number between 0.9999... and 1?

Thank you.

 

[steve_bank]

bomb

'It's the first line in the proof. "So let x = 9/10 + 9/100 + 9/1000 + ...". 9/10 + 9/100 + 9/1000 + ... reduces to x because that is the exact definition of x!'

I know that is what you did.

So let x = 9/10 + 9/100 + 9/1000 + ...
10x = 9/1 + 9/10 + 9/100 + 9/1000 + ....
10x = 9/1 + x

10x = 9.999
x = .9999

Carry it out to as many digits as you like and you will get 0.9999...



let x = 5/10 + 5/100 + 5/1000 + ...
10x = 5/1 + 5/10 + 5/100 + 5/1000 + ....
10x = 5/1 + x
x = .5555
or
10x = 5 + x
x = 5/9

If your technique is valid then 0.555... = 0.6


You are trying to apply finite algebra to an infinite series. Infinity is not a number. I believe a calculus for infinite sets is under cardinal numbers.

 

[Juma]

bomb

'It's the first line in the proof. "So let x = 9/10 + 9/100 + 9/1000 + ...". 9/10 + 9/100 + 9/1000 + ... reduces to x because that is the exact definition of x!'

I know that is what you did.

So let x = 9/10 + 9/100 + 9/1000 + ...
10x = 9/1 + 9/10 + 9/100 + 9/1000 + ....
10x = 9/1 + x

10x = 9.999
x = .9999

Carry it out to as many digits as you like and you will get 0.9999...



let x = 5/10 + 5/100 + 5/1000 + ...
10x = 5/1 + 5/10 + 5/100 + 5/1000 + ....
10x = 5/1 + x
x = .5555
or
10x = 5 + x
x = 5/9

If your technique is valid then 0.555... = 0.6


You are trying to apply finite algebra to an infinite series. Infinity is not a number. I believe a calculus for infinite sets is under cardinal numbers.

WTF? 5/9 is not 0.6 what ass did you pull that out of?

 

[fromderinside]

...a tub of roundup.

 

[bilby]

bomb

'It's the first line in the proof. "So let x = 9/10 + 9/100 + 9/1000 + ...". 9/10 + 9/100 + 9/1000 + ... reduces to x because that is the exact definition of x!'

I know that is what you did.

So let x = 9/10 + 9/100 + 9/1000 + ...
10x = 9/1 + 9/10 + 9/100 + 9/1000 + ....
10x = 9/1 + x

So what is wrong with saying, at this point:

9x = 9/1

Is there an error there? If not, who cares what else you could say; Let's go with that perfectly true and correct statement, and conclude:

9x = 9
x = 1

This MUST be true, unless there is an error in the above.

10x = 9.999

NO! You missed the ...

10x = 9.999...

Which is true, but tells us nothing interesting, so why say it?

x = .9999

Again, NO.

x = .999...

Which is back where we started. That's true, but not useful; We know that x = .999... and what we want to know is what ELSE is x. The proof above shows us that
x = .999...
and also that
x = 1
so we must conclude that
.999... = 1

Simples.

Carry it out to as many digits as you like and you will get 0.9999...

Indeed. Or 1, which is the same thing, as we just proved.

let x = 5/10 + 5/100 + 5/1000 + ...
10x = 5/1 + 5/10 + 5/100 + 5/1000 + ....
10x = 5/1 + x
x = .5555
or
10x = 5 + x

Or more simply:
9x = 5

x = 5/9

Correct. So far so good...

If your technique is valid then 0.555... = 0.6

NO!!!

5/9 != 0.6

You need to concentrate, and think about what you are saying, or you will continue to make foolish errors.

 

[Speakpigeon]

What's the mathematical proof that convergent series are equal to their limits?



Must be true? So, you're not really sure... All you have is a physical allegory?!

If this is not true, what is the number between 0.9999... and 1?

I'd be happy with an infinitesimal.
EB

No, that is not all I have. People are either unable or unwilling to understand the math so I gave the physical analogy that should convince anyone who is willing to actually think about it. I gave a proof that 0.999... = 1.

I seem to have missed that.

Others have given other proofs.

I must be happy with those.

Proofs that the sum of a convergent series is equal to the limit are given at the beginning of any introductory real analysis text book.

I doubt that very much just by looking at my own real analysis text book.

What you are not happy with (ie do not understand) is irrelevant. Any one who questions it should just check out a book and read about it.

What are you doing on a forum if you're not prepared to support your claims?

I'll give one more hint: If there was a remaining infinitesimal but non-zero value after the series was summed you could never cross an interval. And yet you can....

That's irrelevant.

Oh, well, never mind, I will have tried.
EB

 

[DBT]

What I am saying, if it wasn't clear enough for you, is that Rationals can be applied infinitesimally, I'm not disputing that. I don't doubt it.

What I'm saying is that this is an abstract exercise. You can divide an inch, for example, zero to one, to the nth degree, infinitesimally, yet the inch was and alway remains a finite distance between zero and one.

We all know that and it's irrelevant.

We started this particular exchange on my statement that things can be finite in a way while infinite in another way. Everything you say here is entirely irrelevant to that. So, what are you talking about?! You appear to be replying but your response is irrelevant. How come?

Maybe you should try to understand that simple distinction before trying to appear clever, as is your unfortunate habit.

That distinction is irrelevant.

Phil Scott gave you one fine example of the idea that things can be finite in a way while infinite in another way. Yet, as of now, we're still not clear whether you understood the idea. All we've seen from you are irrelevancies and obfuscation. Nothing as to the substance of what you were pretending to reply to.
EB

I'm saying that playing with rationals does not make a finite unit like an inch infinite in any way, shape or form. I am not questioning the use of rationals as a form of abstract modeling.

Abstract modeling, dividing a finite unit of measurement by scribbling ever smaller fractions, Rationals, does not mean the finite unit is infinite in any actual way, An inch is still an inch and has no relationship to actual infinity.

Mathematical models don't always relate to the attributes of physical world.

Cantors infinities are mathematical models.

I am not questioning the Math. I am questioning its relationship to the physical world.

 

[Speakpigeon]

What's the mathematical proof that convergent series are equal to their limits?

There is no need for a proof, because it is what it means by definition (in this context, anyway, so we're talking calculus). In other words, the expression
\( \sum\limits_{i=1}^{\infty}x_i \)
is defined as
\( \lim_{n\to +\infty}\sum\limits_{i=1}^{n}x_i, \)

provided that the limit exists. Or "exists and is finite", if you like: when it's not, it's also a matter of notation we can choose to leave it defined, or not. Also, sometimes we can speak say (for example) "If the series \( \sum\limits_{i=1}^{\infty}x_i \) converges..."), if we first define our notation so that that expression means that the limit in question exists - i.e., that there is x such that
\( \lim_{n\to +\infty}\sum\limits_{i=1}^{n}x_i=x, \)

Now if you are talking about formal series (for example) and you don't define an order, then it makes no sense to ask whether it's less than 1.

So, if you're asking for a proof that
\( \lim_{n\to +\infty}\sum\limits_{i=1}^{n}\frac{9}{10^{n}}=1, \)
some have already been given. Otherwise, what are you asking?

I was asking Artemus to clarify his post. No physical observation we're able to make could prove 0.999... = 1.

I'm happy with the proof given by others.

All but one of the formula in your post come out all garbled but thanks anyway.
EB

 

[Speakpigeon]

We all know that and it's irrelevant.

We started this particular exchange on my statement that things can be finite in a way while infinite in another way. Everything you say here is entirely irrelevant to that. So, what are you talking about?! You appear to be replying but your response is irrelevant. How come?



That distinction is irrelevant.

Phil Scott gave you one fine example of the idea that things can be finite in a way while infinite in another way. Yet, as of now, we're still not clear whether you understood the idea. All we've seen from you are irrelevancies and obfuscation. Nothing as to the substance of what you were pretending to reply to.
EB

I'm saying that playing with rationals does not make a finite unit like an inch infinite in any way, shape or form. I am not questioning the use of rationals as a form of abstract modeling.

Abstract modeling, dividing a finite unit of measurement by scribbling ever smaller fractions, Rationals, does not mean the finite unit is infinite in any actual way, An inch is still an inch and has no relationship to actual infinity.

Mathematical models don't always relate to the attributes of physical world.

Cantors infinities are mathematical models.

I am not questioning the Math. I am questioning its relationship to the physical world.

Sure, we all know that and that's irrelevant.
EB