You’ve seen the sign for infinity, it looks like an 8 on it’s side. Infinity is not a number or a thing, but the idea behind many notions:
* no matter how high you count, you can always count higher
* no matter how long you draw a pair of parallel lines, they never meet
* if you start with a line of any length, you can divide it in half, then divide one of the pieces in half, and no matter how many times you repeat the process, you will always have another piece that you can divide in half again.
Our intuition tells us that these infinite things are theoretically possible, even though our experience, which is finite, tells us that it is impossible to demonstrate them in the physical world. Still, it is easier to imagine them being true than it is to imagine them not to be true. How could there be a “last number” when practically everyone knows how to add 1 to it?
Where does this leave us, though, when we accept these truths that involve infinity? Does it makes sense to have a number that is greater than the estimated number of particles in the universe, or to begin dividing a line into lengths shorter than the diameter of any particle known to atomic physicists? What if, in the vast unreachable universe that is larger than our senses can comprehend, parallel lines do eventually meet?
We might want to ask, then if it is “legal”, for mathematicians, whose field of study is founded on logic and proof to say, in effect, “Well, imagine that this line or this process does go on forever, and then at the end, the result would be…” Isn’t everything in mathematics supposed to be rigorous and concrete?
In attempting to grapple mathematically with these ideas, Georg Cantor surprised and puzzled himself by demonstrating, not only that infinities come in different sizes, but also that there are infinitely many of them!
The idea of infinity is a deep and confounding one, and it does not seem that mathematics will ever nail it down.