Infinity

KALSTER · 09-03-2008

This seems to be a widely misunderstood concept (including by me), yet one commonly used wrongly. One of the common mistakes, for instance, is describing something as simply infinite without specifying exactly what quality of the thing is supposed to be infinite. So let me make some statements and ask some questions regarding infinity and I would like anyone properly knowledgeable to verify their validity, if not make some corrections.

It seems to me that structure can exist within infinity. One example of this would be that in a hypothetical universe that is infinite in time, space and matter, that matter could exist as one particle per light year or as an infinite gas cloud and that both scenarios would have infinite matter.

As I understand it, it is considered in mathematics that $0.99999^. = 1$ . But does this necessarily have to be the case in reality? If you have an infinite data set where $0.99999^.$ is supposed to describe the probability of a certain property to be exhibited by each element of the data set, does that mean that among any number of elements in the infinite data set you would care to consider that all of them would exhibit this particular property? (hope that was coherent :? )

Am I wrong though in saying that limits are only a way for us to deal with infinity? It involves choosing an infinitesimal that has to be added after a certain number of iterations of the series, no? The smaller the infinitesimal the more accurate the result that is attained. In reality you could reduce the size of the infinitesimal infinitely, ending up right back where you started. Or do I misunderstand something?

Tormod · 09-03-2008

Quote:

Originally Posted by KALSTER

As I understand it, it is considered in mathematics that $0.99999^. = 1$ .

It's a strange claim. Do you have a source for it?

I think we can say that in general, 0.9999.... is rounded up to 1 when it's practical to do so, but it is not considered equal to 1.

KALSTER · 09-03-2008

I agree, it does not intuatively sit right with me either. It seems almost as if they assume that the infinite set eventually reaches infinity. Infinity is not a number though, as you well know. So I am not sure how they reach their conclusion. But, take a look at THIS.

Buffy · 09-03-2008

Quote:

Originally Posted by KALSTER

It involves choosing an infinitesimal that has to be added after a certain number of iterations of the series, no? The smaller the infinitesimal the more accurate the result that is attained. In reality you could reduce the size of the infinitesimal infinitely, ending up right back where you started. Or do I misunderstand something?

You should look up the mathematical concept "epsilon": uses the notation " $\epsilon$ ".

Restating your equation above, in math you can get across this concept by saying (still incorrectly by the way):

$0.9999... + \epsilon = 1.0$

where traditionally the "..." is translated as "with as much precision as you wish", and epsilon is translated as "a positive quantity as small as you wish"

The more rigorous notation for your concept however is

$0.\overline{9999}$

which more correctly translates to "an infinite expansion of the digit 9" which is still not equal to 1. What you're looking for is this:

$\lim_{n\to\infty}\sum_{x=1}^n 9\times10^{-x} = 1.0$

where the notion of limit gets you the result you're looking for, but substituting for infinity:

$\sum_{x=1}^\infty 9\times10^{-x} \neq 1.0$

So your question:

Quote:

Originally Posted by KALSTER

Am I wrong though in saying that limits are only a way for us to deal with infinity?

is actually the right one, but not in the way I think you really meant it!

Infinity is a hard to grasp topic, especially because there are so many kinds!

So the question now becomes, what aspect of infinity are you trying to ponder?

BTW: In reference to the wiki page you linked, if you read it carefully, it doesn't actually say that any of the proposed "proofs" are true!

Civilization is the process of reducing the infinite to the finite,

Buffy

Erasmus00 · 09-03-2008

Quote:

Originally Posted by Buffy

The more rigorous notation for your concept however is

$0.\overline{9999}$

which more correctly translates to "an infinite expansion of the digit 9" which is still not equal to 1.

Actually its pretty well accepted that $0.\overline{9999}$ is an equivalent decimal expansion to 1. These are the same. While the proofs in wiki page are of varying degrees of rigor, there aren't any nonzero infinitesimal numbers in the reals, so these must be the same.
-Will

Buffy · 09-03-2008

Quote:

Originally Posted by Erasmus00

Actually its pretty well accepted that $0.\overline{9999}$ is an equivalent decimal expansion to 1. These are the same. While the proofs in wiki page are of varying degrees of rigor, there aren't any nonzero infinitesimal numbers in the reals, so these must be the same.

Sure, but it the proofs on both sides are so unsatisfying, and at the same time fun to manipulate!

I actually argue that the proofs promoting the equality verge on the axiomatic (cf. $y=\frac 1 x$ at $x=0$ ), and thus I'm happy to take the contrarian side because its the "popular" one, especially among us "finite-representationally-handicapped" computer scientists!

Anyone game to post a "pro-" proof?

Of course some people do go both ways,

Buffy

KALSTER · 09-04-2008

Well, I want true infinity. One that is not a compromised version. But can we deal with that without all the cardinality business? An infinity with infinite cardinality (did I use that correctly?).

Let's for argument's sake say that the universe is indeed infinite in space and time and that the big bang is a local event. Forgetting about other concerns (like infinite light would hit us or whatever) and focusing on the infinity aspect, I can't really see a problem with it, as I tried to illustrate in the OP.

This scenario would then be one where the mathematical and the physics version of infinity could coexist IMO. The thing that I am wondering about, is the emergence of structure in such a setup.

To quote myself (with fixes):
" If you have an infinite data set where $0.\overline{9999}$ is supposed to describe the probability of a certain property to be exhibited by each element of the data set, does that mean that among any number of elements in the infinite data set you would care to consider that all of them would exhibit this particular property?"

(concerning the above) If you were to limit the data set to say $10^{100}$ elements, you would work out that all the elements would exhibit the particular property every cycle of unit time (as I've gathered). But if the probability for this property to be exhibited per cycle were to be < 1, with any number of 9's after the comma, and you could leave the data set for as long as you wished and the number of elements that did not exhibit the property would start going up. The nature of probability (as I understand it) is that the possibility exists that none of the elements in the $10^{100}$ element data set could at any time exhibit the property, since the probability of it happening is extremely small, but still there.

Now, if all I said above made sense and is correct, then you could have an infinite number of localized data sets behaving in this way on their own, but then how would one then describe the behavior of an infinite number of them collectively?