Denumerable Infinities Are The Same Size; No Mind Emerges Through Sorting
My favorites: Cantor's explanation of why all denumerable infinities are the same size—why, eg, the set of all integers is the same size as the set of all positive integers, or all even integers—and why some infinities are bigger than others. (The set of all rational numbers is the same size as the set of all integers, but the set of all real numbers—terminating plus non-terminating decimals—is larger.) The set of all positive integers is the same size as the set of all even, positive integers—to see that, just line them up, one by one. 1 is paired with 2 (the first even positive integer), 2 is paired with 4, 3 with 6, 4 with 8 and so on.
You would think there would be more positive integers than even ones; but this pairing-off shows that no positive integer will ever be left without a partner. (And so they all dance happily off & there are no wallflowers.) The other proofs are similar in their stunning simplicity, but much easier to demo on a blackboard than to describe in words.
Equally favorite: Searle's proof that no digital computer can have mental states (a mental state is, eg, your state of mind when I say "picture a red rose," and you do); that minds can't be built out of software. A digital computer can only do trivial arithmetic and logical instructions. You can do them too; you can execute any instruction that a computer can execute. You can also imagine yourself executing lots & lots of trivial instruction, & then ask yourself: can I picture a new mind emerging on the basis of my doing lots & lots & lots of trivial instructions? (No.) Or: imagine yourself sorting a deck of cards.
Sorting is the sort of thing digital computers do. Now imagine sorting a bigger & bigger & bigger deck; can you see consciousness emerging at some point, when you sort a large enough batch? Nope. (And the inevitable answer to the inevitable first objection: but neurons only do simple signal transmission; can you imagine consciousness emerging out of that? An irrelevant question. The fact that lots of neurons make a mind has no bearing on the question of whether lots of anything else make a mind. I can't imagine being a neuron, but I can imagine executing machine instructions. No mind emerges no matter how many of those instructions I carry out.)