Question: are there proportions in infinity?

[John Cxnnor]

Let me preface this by saying Math is not my strongest subject.

I am curious if there is any relationship between proportionality and the term infinity.

This question stems from an online discussion about whether or not the physical universe is infinite or not. (Obviously its not, but I have a hard time explaining this to the new agers).

Lets say that on average, for every 100 stars in the universe, there is a corresponding black hole giving us a 100:1 ratio.

Now let's say that the universe is infinite. Meaning there would be an infinite number of stars and black holes.

Could you then say there is an infinite number of stars and black holes at a proportion of 100:1?
I know how non-sensical that sounds, even to a math layman such as myself.

But is there anything in the mathematics of infinity that has to do with proportionality?

Could you describe the rate at which numbers appear, that are divisible by 2 versus the rate at which numbers appear that are divisible by 3 on an infinite number line?

Thanks in advance!

 

[Stephen Tashi]

Some forum members fearlessly answer questions like "What is the proability that a positive integer selected at random is even?" - as if it were possible to make such a selection and as if the infinity of positive integers had a fraction of even integers equal to 1/2.

However, I don't know of any mathematical theory that correctly deals with such questions except by using limits of finite quantities. For example if you say:

Let p(n) = the probability that an integer selected at random from the first n integers is even
or
Let p(n) = the fraction of the first n integers that are even

then you can sensibly talk about taking "The limit of p(n) as n approaches infinity". In that phrase, the word "infinity" has no independent meaning. The whole phrase has a mathematical definition, but the individual terms like "infinity" or "approaches" have no precise definition.

By the way, Benoit Madelbrot, in one of his books on fractals, remarks that the standard arguments that the universe is finite (such as "if it were infiinite then the sky would be infinitely bright") are incorrect if we consider certain fractal distributions of stars.

 

[John Cxnnor]

Thank you for the clarification and an easily understandable explanation!

I have read about the fractal placement of stars as a possible solution to olbers paradox.

Which confuses me even more. Have we even noticed anything remotely close to a fractal pattern in the placement of stars? Or maybe its difficult to discern from our scale/point of view.

But, wouldn't an infinite universe = infinite mass = infinite gravity?

 

[Stephen Tashi]

But, wouldn't an infinite universe = infinite mass = infinite gravity?

I don't think cosmology can settled by simplistic logic - but I'm not an astronomer. I don't know of any physical or mathematical laws that confirm your statement. (I take it that by "=", you don't mean "equal", but rather "imply".)

 

[skeptic2]

then you can sensibly talk about taking "The limit of p(n) as n approaches infinity". In that phrase, the word "infinity" has no independent meaning. The whole phrase has a mathematical definition, but the individual terms like "infinity" or "approaches" have no precise definition..

I also have questioned the meaning of "approaches" as in "The limit of p(n) as n approaches infinity". The implication is that to approach infinity n must increase. But is 100 or 1 million any closer to infinity than is 1? Wouldn't the phrase have more meaning if it were stated as "The limit of p(n) as 1/n approaches zero".

 

[micromass]

I also have questioned the meaning of "approaches" as in "The limit of p(n) as n approaches infinity". The implication is that to approach infinity n must increase. But is 100 or 1 million any closer to infinity than is 1? Wouldn't the phrase have more meaning if it were stated as "The limit of p(n) as 1/n approaches zero".

The limit as n approaches infinity has a well-defined technical meaning. The distance to infinity is not relevant to limits. We say that $p(n)\rightarrow x$ as n approaches infinity if for all $\varepsilon >0$, there exists an N such that for all $n\geq N$ holds that the distance between p(n) and x is smaller than $\varepsilon$.

With this definition, we don't need to talk about infinity or distance to infinity.

 

[John Cxnnor]

I don't think cosmology can settled by simplistic logic - but I'm not an astronomer. I don't know of any physical or mathematical laws that confirm your statement. (I take it that by "=", you don't mean "equal", but rather "imply".)

Yes, its probably not wise to use such simplistic logic for such an immense question.

And yes I should have used the word imply rather than =.

Thanks for the help. I was unsure which math forum to pose the initial question in.

Maybe I should continue this in one of the physics/cosmology forums.

Again, I appreciate that you took the time to answer my questions.

 

[smize]

Some people say infinity is infinity. There is nothing greater than infinity not even infinity + 1. But, others argue that there is a greater infinity of real numbers between 0 and 1 than there are integers between 0 and infinity. It is all how you interpret it.

 

[chiro]

Some people say infinity is infinity. There is nothing greater than infinity not even infinity + 1. But, others argue that there is a greater infinity of real numbers between 0 and 1 than there are integers between 0 and infinity. It is all how you interpret it.

There are different classes of infinities: the idea of limits helps establish things like this.

The thing you have to remember is that infinity in no way acts like a finite number. A finite number has a distinct definition in that it can be compared to any other number that has some kind of ordering or ranking.

Infinities don't work like that because they are not defined the same way.

You can use some tools to rank different infinities but it requires a different way than ranking finite numbers. The first person to look at this in serious detail (in modern mathematics) was George Cantor and you can read his work on the subject (and later work from others that followed).

There are things like limit laws that deal with very precise definitions involving infinity so it's completely misleading to say that it's all down to "interpretation".

 

[student34]

Some people say infinity is infinity. There is nothing greater than infinity not even infinity + 1. But, others argue that there is a greater infinity of real numbers between 0 and 1 than there are integers between 0 and infinity. It is all how you interpret it.

Check out Cantor's diagonal argument
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument . It is very convincing that there are larger infinities. Specifically, it shows how there is an infinite set larger than the countably infinite set aleph-naught that we are used to seeing such as the set of all natual numbers {1, 2, 3,...}.

 

[smize]

Check out Cantor's diagonal argument
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument . It is very convincing that there are larger infinities. Specifically, it shows how there is an infinite set larger than the countably infinite set aleph-naught that we are used to seeing such as the set of all natual numbers {1, 2, 3,...}.

The interpretation of Cantor's result will depend upon one's view of mathematics. To constructivists, the argument shows no more than that there is no bijection between the natural numbers and T. It does not rule out the possibility that the latter are subcountable. In the context of classical mathematics, this is impossible, and the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers.

Yes, but it is still debated between many people (I personally am on the side of different sizes of infinity).

 

[Studiot]

I am curious if there is any relationship between proportionality and the term infinity.

Of course there is, since there is more than one infinity.

That is why the ratio of two (different) infinities can be deduced as a real number eg

Wallis' formula.

 

[Bacle2]

Maybe you'll find this interesting; it is somewhat related (If I understood your

question correctly) :

http://en.wikipedia.org/wiki/Cardinal_arithmetic#Cardinal_arithmetic

Look up the section on multiplication and division--which you may relate

to proportion-- but remember that cardinal

arithmetic does not follow the standard rules that everyday numbers follow.