Infinity is really a facanating concept.
Infinity is not a number. I won't give you the formal definition because it would take a lot of explanation if you have not had calculus (and I assume you don't from your question), but essentially infinity means "unbounded."
Quick example. Take the sequence s_n={1,2,3,4,..,n}. If n=3, what does the sequence approach? 3. Now let us take the sequence s_n={1\n}, n>=0. What happens as we apprach 0? Well, if we apprach from the positive numbers we get somthing like {1/1, 1/.5, 1/.05, .., 1/n} = {1,2,20,..}. As we get closer to zero, our numbers are growing larger*.
Now for the interesting stuff. Not all infinites are the same.
Consider this. Let's take a whole bunch of objects. Can we count them? To answer this, we need to define count a little better. Let us define counting in the following way :
If there exists a one-to-one and onto mapping between the Natural Numbers (1,2,3,..) and the objects in our set, then the set is countable.
So, are the natural numbers countable? Yes. 1<->1, 2<->2, 3<->3, .., n<->n. Is N^2 countable? By N^2 I mean something like {(1,1),(1,2),...(1,n),(2,1),(2,2),..,(2.n),..,(n,n)}. Yes! A formal proof is pretty complicated (most formal proofs of coutability are), but think of N^N as a giant grid, and figure out how you can trace every point of that grid using a single line to give you an idea of how this proof works.
Now, is R countable? Nope. The proof of this is very difficult, and quite frankly the result is much more interesting then the proof is.
Because we have N as countable, and R as uncountable, it seems that all infinities are not equal. In fact, it is pretty easy to show that there are "infinity" infinities. Just take your "infinity" set, and create a new set that contains all the possible subsets of this set.
Countability also leads us to one of the most important results in modern math. The "cardnality" of a set is essentially how many elements a set has. Quick example, the cardinality of {1,2,3,4,178} is 5, since there are 5 elements. Now, let us call the cardinality of N alph_0, and let us call the cardinality of R alph_1.
Now the question. Does a set exist with cardinality between alph_0 and alph_1? The answer is truly surprising.
The answer is that there is no answer. It is impossible to prove this true or false. This question was the inspiration that led to Godel's idea of incompleteness, which says the following :
Every axomatised system is either incomplete or inconsistance.
Incomplete means that no false statement is provable true, and no true statement is provable false, but some statements are unprovable true or false.
Inconsistant means every true statement is provable true, but some true statements are also provable false and some false statements are provable true.
-Chu
P.S., btw, this is INCREDIBLY informal. If any of my math professors caught my extreemly loose definition of limits or sets or Godel's Incompleteness Theorem on a test, I would hate to see my grade . . .
* : This is the only place I will even be a little formal. Given a sequence, the limit is defined as follows. x = lim(s_n), n->t <=> For all eps>0, there exists a natural number n such that N>n implies |x - s_N| < eps.
