What are some examples of a mathematical result being counterintuitive? As I procrastinate studying for my Maths Exams, I want to know what are some cool examples of where math counters intuition.
My first and favorite experience of this is Gabriel's Horn that you see in intro Calc course, where the figure has finite volume but infinite surface area (I later learned of Koch's snowflake which is a 1d analog). I just remember doing out the integrals for it and thinking that it was unreal. I later heard the remark that you can fill it with paint, but you can't paint it, which blew my mind.
Also, philosophically/psychologically speaking, why does this happen? It seems that our intuition often guides us and is often correct for "finite" things, but when things become "infinite" our intuition flat-out fails.

The fact that one can easily prove the existence of uncountably infinite (as opposed to countably infinite) sets is counterintutive to me.  Not that fact that uncountably infinite sets exist, but the fact that the proof is so simple.  I was astonished when I first learned of it.  I was in ninth grade.  I think it was in a book by Vilenkin that I read the proof.
Similarly the fact that one can easily prove that the square root of $2$ is irrational.  I hadn't expected that to be so simple.  And the mere existence of irrational numbers seems counterintuitive: why shouldn't fractions be enough to fill up all the space between integers?
