Gödel's incompleteness theorems are more of a theoretical limitation than a practical one. Roughly speaking, Gödel's incompleteness theorems say that all proof

Gödel's incompleteness theorems are more of a theoretical limitation than a practical one. Roughly speaking, Gödel's incompleteness theorems say that all proof systems for number theory are limited in some way. For example, first-order Peano arithmetic is such a system, and one of its limitations is that it doesn't support transfinite induction. (It doesn't matter if you don't know what it is.) In other words, if you want to translate a mathematical proof to Peano arithmetic, you have to come up with a way to do it without transfinite induction. Sometimes, such in the case of Goodstein's theorem, this is impossible. To prove Goodstein's theorem, you have to choose a stronger proof system to begin with. So, Gödel's theorems guarantee that no matter how strong you proof system is, there are always number-theoretic statements that are beyond its reach. But for reasons that are not currently completely understood, this doesn't really happen in practice. "Naturally occurring" examples of number-theoretic statements almost always turn out to be provable in surprisingly weak systems. Instead, you run into practical problems: the theorem is provable in the system, but actually writing out the proof is utterly inconvenient. As an analogy, there are Turing-complete programming languages that don't have the concept of functions. In theory, they are capable of all kinds of computations, but in practice you don't want to use them. And if, instead of mathematics, we concentrate on proofs of correctness, then this is even less of a practical problem. To quote Leslie Lamport, proofs of correctness "are seldom deep, but usually have considerable detail." The proofs may be long and complicated, but as long as they don't use any kind of ridiculously abstract techniques, they are just the kind of proofs where computers can have an advantage over humans.