The meaning of infinity

[DBT]

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

It's relevant to me. Which is why I made the comment. That done.....carry on.

 

[Phil Scott]

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

Angra Mainyu has answered this well enough, but if you are still confused, I would interrogate him on the matter. I'm sure he'd be happy to help, and he's one of the few in the thread who knows what they're on about.

True by definition might sound unsatisfying, but I am afraid this is how the recurring debate is solved, by fiat not proof. If you need more, I recommend beero's OP on how the various number systems arise by natural extensions of others.

In particular, the reals arise when we ask for an extension of the rationals that contains the least upper bounds of all subsets that have some upper bound. It was realised that this crucial property could yield a comprehensive theory of the otherwise slippery 18th notion of limits. This stuff is worth hearing out.

Now the existence of extensions of the rationals containing least upper bounds is not a matter of divine fiat, but a matter of rigorous proof. It is also a matter of proof that such extensions are unique up to isomorphism. It is also a matter of proof that all of traditional classical calculus and algebraic geometry can be rigorously formalised over this extension, including the first rigorous proof of the Fundamental Theorem of Calculus.

We name the extension "the real numbers" and, based on its fit for purpose, we advance the thesis that it is a model of continuous 1 dimensional space: the continuum.

That is the only springboard for further discussion about real numbers.

Now for infinite series: their usefulness is justified as a tool in calculus. But here's the important observation for this discussion: every decimal can be identified with the limit of an infinite series. And moreover: every real can be identified with such a decimal. Again, this is a matter of proof. Moreover, arithmetic operations and comparison can be extended to decimals, and so with the caveat that decimals ending in a string of 9s like 0.999… are identified with decimals ending in a string of 0s like 1.0…, the decimals form not just a workable infinite notation for reals, but an example of the very extension of rationals we sought earlier.

It is this sort of reasoning which compels us to adopt the identity 0.9…=1.

Final note, since you mention infinitesimals: series work differently in the hyperreals, but that just reminds us to sort out the definitions before we embark on the proofs.

 

[Speakpigeon]

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.

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.

I would have thought it's long been obvious that this particular case is beyond reprieve.

Anyone so far gone into denying the obvious is beyond reprieve.

Maybe it's testament to the human psychology that we are desperate to believe we can operate miracles and cure the dead of their illness.

Or is it that some people here come to incarnate what's really wrong with "other people". To give up on them may seem like to give up on a better future altogether. If we could just get them to change, there might still be hope for our children.

Or it is that what the guy says is so obviously wrong that anyone can feel he should be able to put it right?

Me, all I see is that the failures of the initial attempts have shown this would be delusional. It can only gets worse and worse. Capital letters and triple exclamation points are a good indication of that.

On the other end, if ignoring the dude didn't stop the garbage coming in, I understand the frustration.

Still, it would be more interesting letting Steve and UM exchange profound truths without human beings constantly interfering.
EB

 

[Speakpigeon]

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

It's relevant to me. Which is why I made the comment. That done.....carry on.

So, you're basically responding to yourself?!

I would worry talking to myself.
EB

 

[Speakpigeon]

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

Angra Mainyu has answered this well enough, but if you are still confused, I would interrogate him on the matter. I'm sure he'd be happy to help, and he's one of the few in the thread who knows what they're on about.

True by definition might sound unsatisfying, but I am afraid this is how the recurring debate is solved, by fiat not proof. If you need more, I recommend beero's OP on how the various number systems arise by natural extensions of others.

In particular, the reals arise when we ask for an extension of the rationals that contains the least upper bounds of all subsets that have some upper bound. It was realised that this crucial property could yield a comprehensive theory of the otherwise slippery 18th notion of limits. This stuff is worth hearing out.

Now the existence of extensions of the rationals containing least upper bounds is not a matter of divine fiat, but a matter of rigorous proof. It is also a matter of proof that such extensions are unique up to isomorphism. It is also a matter of proof that all of traditional classical calculus and algebraic geometry can be rigorously formalised over this extension, including the first rigorous proof of the Fundamental Theorem of Calculus.

We name the extension "the real numbers" and, based on its fit for purpose, we advance the thesis that it is a model of continuous 1 dimensional space: the continuum.

That is the only springboard for further discussion about real numbers.

Now for infinite series: their usefulness is justified as a tool in calculus. But here's the important observation for this discussion: every decimal can be identified with the limit of an infinite series. And moreover: every real can be identified with such a decimal. Again, this is a matter of proof. Moreover, arithmetic operations and comparison can be extended to decimals, and so with the caveat that decimals ending in a string of 9s like 0.999… are identified with decimals ending in a string of 0s like 1.0…, the decimals form not just a workable infinite notation for reals, but an example of the very extension of rationals we sought earlier.

It is this sort of reasoning which compels us to adopt the identity 0.9…=1.

Final note, since you mention infinitesimals: series work differently in the hyperreals, but that just reminds us to sort out the definitions before we embark on the proofs.

Thanks. This does answer my question and some. I will mull on that.
EB

 

[DBT]

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

It's relevant to me. Which is why I made the comment. That done.....carry on.

So, you're basically responding to yourself?!

I would worry talking to myself.
EB

You do it all the time. You try to puff yourself up whenever the opportunity arises. You are doing it now.

 

[Artemus]

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.

One last time: If 0.9999.... is not equal to 1 there must be another real number that is greater than 0.9999... and less than 1. There is none. State a number that is between the two (without misusing the term "infinitesimal" this time).

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.

OK, I'll concede that it has been a while. Perhaps it is accepted as an axiom or definition. Angra Mainyu or one of the others could help here.

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?[/UQOTE]

There goes another irony meter.

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.

No, this is extremely relevant. It shows that it the sum of the series absolutely, positively must be equal to the limit to describe the physical world. Zeno's paradox was resolved.

 

[untermensche]

No, this is extremely relevant. It shows that it the sum of the series absolutely, positively must be equal to the limit to describe the physical world. Zeno's paradox was resolved.

By pretending you have a final product based on introducing the concept of limits?

The paradox shows it is impossible for an infinity to exist in the real world.

If you can move an infinitely smaller distance you can never reach your destination.

That is not really a paradox. It is a logical conclusion.

The only way you can reach your destination is if there is a finite smallest possible movement to make.

 

[untermensche]

Looking at something and saying it is infinitely close to something else is not saying it is equal to that something else.

Saying it is equal is just something that can arbitrarily be done in the mind because the difference is so insignificant. Infinitely insignificant.

While in symbol form the difference always exists.

0.999... will never reach 1.000... on a line.

 

[Artemus]

No, this is extremely relevant. It shows that it the sum of the series absolutely, positively must be equal to the limit to describe the physical world. Zeno's paradox was resolved.

By pretending you have a final product based on introducing the concept of limits?

It's no more pretend than -2 x -2 = 4 or even 2 + 2 = 4. If it correctly describes what happens in the physical world, it is correct.

Can someone tell me what 1 - 0.9999.... equals?

 

[untermensche]

No, this is extremely relevant. It shows that it the sum of the series absolutely, positively must be equal to the limit to describe the physical world. Zeno's paradox was resolved.

By pretending you have a final product based on introducing the concept of limits?

It's no more pretend than -2 x -2 = 4 or even 2 + 2 = 4. If it correctly describes what happens in the physical world, it is correct.

Can someone tell me what 1 - 0.9999.... equals?

There are no negative twos in the real world.

1 - 0.9999... is an infinitely shrinking value.

It is not really a reasonable request since 1 is defined and 0.999... is not.

It is like saying 2 - giraffe = what?

 

[Phil Scott]

1 - 0.9999... is an infinitely shrinking value.

Nope. It's zero, because 1 = 0.9999...., and anything minus itself is zero.

"1" and "0.9999..." refer to the same number. They are a redundancy in decimal notation, like how "1/2" and "2/4" are a redundancy in fractional notation.

If you don't like it, eschew recurring decimals entirely, and stick to fractions. They're far more elegant, anyway. To convert an ugly decimal to an elegant fraction, take the non-recurring prefix and divide it by 10 to the power of the number of non-recurring digits. Then take the recurring part and divide it by that same power, and divide it further by 10 to the power the number of recurring digits - 1. So

\(0.50\overline{142857}\)

becomes \(\frac{50}{100} + \frac{142857}{99999900}\)

which simplifies to \(\frac{351}{700}\).

If you follow this algorithm in the trivial case of 0.999..., you get 9/9=1.

And if you start with rationals, you can calculate freely using +, -, * and divide without ever having to muck around with an infinite decimal. If you have a programming language with a decent library for arbitrary precision arithmetic, you can do the same, without ever losing precision.

 

[Juma]

It's no more pretend than -2 x -2 = 4 or even 2 + 2 = 4. If it correctly describes what happens in the physical world, it is correct.

Can someone tell me what 1 - 0.9999.... equals?

There are no negative twos in the real world.

1 - 0.9999... is an infinitely shrinking value.

It is not really a reasonable request since 1 is defined and 0.999... is not.

It is like saying 2 - giraffe = what?

There are no 1:s and 2:s in the real world either.

 

[untermensche]

1 - 0.9999... is an infinitely shrinking value.

Nope. It's zero, because 1 = 0.9999...., and anything minus itself is zero.

"1" and "0.9999..." refer to the same number. They are a redundancy in decimal notation, like how "1/2" and "2/4" are a redundancy in fractional notation.

If you don't like it, eschew recurring decimals entirely, and stick to fractions. They're far more elegant, anyway. To convert an ugly decimal to an elegant fraction, take the non-recurring prefix and divide it by 10 to the power of the number of non-recurring digits. Then take the recurring part and divide it by that same power, and divide it further by 10 to the power the number of recurring digits - 1. So

\(0.50\overline{142857}\)

becomes \(\frac{50}{100} + \frac{142857}{99999900}\)

which simplifies to \(\frac{351}{700}\).

If you follow this algorithm in the trivial case of 0.999..., you get 9/9=1.

And if you start with rationals, you can calculate freely using +, -, * and divide without ever having to muck around with an infinite decimal. If you have a programming language with a decent library for arbitrary precision arithmetic, you can do the same, without ever losing precision.

You have a little trick where the tiniest bit of rounding is hidden.

You can have 1 apple.

You cannot have 0.999... of anything.

They are not the same thing.

- - - Updated - - -

It's no more pretend than -2 x -2 = 4 or even 2 + 2 = 4. If it correctly describes what happens in the physical world, it is correct.

Can someone tell me what 1 - 0.9999.... equals?

There are no negative twos in the real world.

1 - 0.9999... is an infinitely shrinking value.

It is not really a reasonable request since 1 is defined and 0.999... is not.

It is like saying 2 - giraffe = what?

There are no 1:s and 2:s in the real world either.

You can have one apple or two apples.

But you cannot have negative one apples.

And you cannot have infinity.

It is not an amount.

To have something is to have an amount of it.

 

[Juma]

You have a little trick where the tiniest bit of rounding is hidden.

You can have 1 apple.

You cannot have 0.999... of anything.

They are not the same thing.

- - - Updated - - -

It's no more pretend than -2 x -2 = 4 or even 2 + 2 = 4. If it correctly describes what happens in the physical world, it is correct.

Can someone tell me what 1 - 0.9999.... equals?

There are no negative twos in the real world.

1 - 0.9999... is an infinitely shrinking value.

It is not really a reasonable request since 1 is defined and 0.999... is not.

It is like saying 2 - giraffe = what?

There are no 1:s and 2:s in the real world either.

You can have one apple or two apples.

But you cannot have negative one apples.

And you cannot have infinity.

It is not an amount.

To have something is to have an amount of it.

And if I need 2 apples I have -2 apples.
Amounts are something we have invented.

 

[untermensche]

And if I need 2 apples I have -2 apples.
Amounts are something we have invented.

In the real world to have something means something different.

You cannot have negative two apples.

Having something means something.

It means you can show it to me. Can you show me the negative two apples you claim to have?

 

[Juma]

And if I need 2 apples I have -2 apples.
Amounts are something we have invented.

In the real world to have something means something different.

You cannot have negative two apples.

Having something means something.

It means you can show it to me. Can you show me the negative two apples you claim to have?

Easy: I show you the space where they should have been.
Showing the amount is only a practicality: if you had 35536464674 apples, could you show them?

 

[Artemus]

It is not really a reasonable request since 1 is defined and 0.999... is not.

Having an infinite number of digits when written in decimal notation does not mean it is undefined. Pi has an exact, rigorously defined physical value but it cannot be represented with a finite number of digits either. That doesn't mean that pi approaches C/2R; it is exactly equal to it. Similarly, 0.999... has an exact value, which happens to be 1.

 

[untermensche]

And if I need 2 apples I have -2 apples.
Amounts are something we have invented.

In the real world to have something means something different.

You cannot have negative two apples.

Having something means something.

It means you can show it to me. Can you show me the negative two apples you claim to have?

Easy: I show you the space where they should have been.
Showing the amount is only a practicality: if you had 35536464674 apples, could you show them?

You then will show me space not negative two apples.

Your position is not describing the real world.

 

[untermensche]

It is not really a reasonable request since 1 is defined and 0.999... is not.

Having an infinite number of digits when written in decimal notation does not mean it is undefined. Pi has an exact, rigorously defined physical value but it cannot be represented with a finite number of digits either. That doesn't mean that pi approaches C/2R; it is exactly equal to it. Similarly, 0.999... has an exact value, which happens to be 1.

The number is undefined.

All that is defined in an infinite operation. The number can never exist.

Every representation will always be a partial representation.

Does 0.9 = 1.0?

Does 0.99 = 1.0?

Does 0.999 = 1.0?

All defined values will not equal 1.0.

This undefined value 0.999... is only thought to equal 1.0.