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There are lots of problems with trying to talk about infinity. Most of them spring from the fact that most of the things that sound reasonable are actually wrong.

This is actually a bit of a rant. I hope you will bear with me.

People have an "anything can happen" attitude towards infinity. But it is not necessarily true, for example, that an infinite number of monkeys at an infinite number of typewriters will eventually produce Hamlet, mostly because that would only be true if they didn't keep typing roughly the same thing. I suspect they would, and that asdfg would figure prominently in all of their writings.

Saying that there are infinitely many of something does not make all statements about it true. If I give you an infinite number of even numbers, you will not be able to find an odd number.

For example, infinity minus one is still infinity (for any size of infinity). But this does not mean that there is only one size.

As brock mentions on the infinity node, there are several "levels" of infinity. In fact, there are at least a countably infinite number of them, because the power set of an infinite set always has a larger cardinality than the original set. (This is because it has been proven that there never exists a set such that there is a one-to-one mapping from its elements to the elements of its power set. Not even for the empty set!)

So, take the integers. They have a countably infinite cardinality. Take their power set. That set has a larger cardinality. Take its power set. Lather, rinse, repeat. I think it is fairly obvious that you can find a countably infinite number of such cardinalities, because presented with one I can always give you another.

There are a lot of really cool, but counter-intuitive results about infinite sets. The fact that so many of the coolest results are counter-intuitive, however, just highlights the fact that we should be very cautious about placing confidence in intuitive results.

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