[04/09, 12:24] Pankaj: Imagine the whole of mathematics as a huge sack, and inside are all the possible things math can do. It’s a mighty big sack, indeed. What Gödel proved is that, first, there exists in this sack a set of things which cannot be proven or disproven, such as axioms. Second, there is no possible way to prove these axioms from within that sack. It’s impossible for math, on its own, to prove its own axioms. Essentially, it’s a problem of self-reference. It’s an issue seen, too, in Russell’s paradox about sets. More famously, the liar paradox imagines a sentence like, “This sentence is false.” When you examine it closely, it creates a logical circularity. If the sentence is true, then it’s false; but then if it’s false, it’s true. It’s enough to make a robot’s brain explode [04/09, 12:24] Pankaj: Gödel applied a similar logic to the whole system of mathematics. He took the sentence, “This statement is unproven,” and converted it into a numb...