I'm currently reading Mathematical Carnival, by Martin Gardner. The topics have a wide range of complexity, but all have a certain bizarre unintuitiveness to them that I find charming.
One such topic is that of Alephs, or levels of infinity. Essentially, the idea is that there is a number higher than infinity. This is Aleph-One. (Infinity itself is known as Aleph-Null.) Yes, this is provable, and theoretically useful acording to Set Theorists and other people who don't do real work. Luckily for the field of mathematics, I have found a practical application.
Have you ever had one of those arguments along the lines of "yuh-huh" "nuh-uh" "double yuh-huh" "nuh-uh infinity!" "yuh-huh infinity plus one?" We know this is folly; the value of infinity and infinity plus one are the same, thus making the yuh-huh no better than the nuh-uh.
Instead, you would be wise to reply "yuh-huh aleph-one," which is greater than your opponent's response by an entire magnitute of infinity! That will show them who's boss.
Of course, using principles of game theory, one can see the logical result of such a stragem. The opponent will indubidably respond with something to the effect of 'nuh-uh aleph-two,' or more effectively, 'nuh-uh aleph-infinity!' To which, of course, one can respond with something to the effect of 'yuh-huh aleph-aleph-infinity!'
The truly astute reader will note that such a non-terminating argument will end the same way as all other non-terminating arguments: in a fist fight. This is where the true viability of higher order infinites is shown: while your debate opponent is standing confused at your mention of alephs, you can take the opportunity to punch them in the face.