The Gödel sentence (which is not otherwise of any interest) is true but unprovable within that axiomatic system. The Continuum hypothesis (which is of great interest) is true or false only by stipulation.

The fact is that neither Gödel's theorems nor the Halting Problem have any actual real world consequences (outside of people talking about them). People say "oh, we can't write a verifier because of the Halting Problem", but that's simply not true since all of our programs are actually FSMs with physically limited data, and the HP is solvable for that subset of TMs. The real limitations are time and space, so this is an engineering problem -- and people who aren't suckered by Halting Problem Hysteria find engineering solutions that work on real programs, bailing if memory or time thresholds are exceeded.