6. On Computable Numbers
This one I never actually read entirely in college, but it is quite a beautiful proof. First, it defines what a computer is mathematically. Then, it elegantly shows that the definition of this computer can be subbed in to Cantor's diagonalization argument to show that one could not determine if every computer "halts" or not.
There are a bunch of other interesting tidbits here, like the use of semicolons to differentiate instructions. Also, the definition of a computer is just a person who performs computations, since, after all, that is what the computer set out to do and still does, despite how magical it seems nowadays.
Author: Alan Mathison Turing
Original date: 1936