When I write a program, I don't want it to be just correct (true); I want it to be _provably_ correct; I want to be able to be convinced that it is correct, at least in principle, given enough time and whole specification of the system. Programs which are correct, but not provably so, should not pass code review and might as well be lumped together with those which are wrong. It doesn't matter if you are using a full-blown theorem prover or thinking about the code in your head; Gödel's theorems are not really relevant to programming, even when the code uses deep mathematics.

I'm not sure I agree with "theorems that must be verified at compile time can never account for data that are provided only after compilation". At compile time, you prove the assertion "for every x, the program outputs correct answer for x". Now, you don't know that the user will enter say x=5 at runtime, but since you proved a theorem about _every_ x, this includes x=5. You cannot predict the future (the path that will be taken by the program), but once you prepare for all possible futures, you're safe.