>The Incompleteness Theorem says that, given any consistent, computable set of axioms, there's a true statement about the integers that can never be proved from those axioms.
Upon reading something like this I immediately have questions like: if this is so, then how do we know that this statement about the integers is true at all? What does it mean for something to be true within a set of axioms when you can't prove it? Why don't we say that the truth of this statement, within those axioms, is undetermined? If we, on the outside, know that it's true, why can't we forcefully plug that truth back into the theory?
OK, I know that last one, you can do it but then you can also do an incompleteness proof for the new theory. But still, if the "problem" is only with self-referential statements, why can't we somehow isolate all self-referential statements and have a theory that's complete and consistent except for some caveats, which seems vastly better than just inconsistent, period?
Sorry if that makes no sense, I know this topic is famous for attracting cranky discourse.. It just feels like all the popular explanations stop just short of really grappling with the real weirdness of the theorem.