Or you could use Gödel's incompleteness theorems and say the universe cannot be a computer (as we understand it) because computers (as we understand them) are part of the universe and therefore cannot describe the universe. Then the discussion could go on and you could reach a definition of "computer" vague enough to be able to say "the universe is a computer".
...that is not what the incompleteness theorem says.
Yes. I'm not a mathematician, but I believe Gödel's incompleteness theorems only address arithmetic. I heard of a book that focuses on common misuses of Gödel's theorems, but don't have a reference. Anyone know of such a book?