When reading discussions of Gödel's theorems one is always heeded that just because formal system $F$ proves a theorem $T$ that doesn't necessarily mean that when one applies the intended interpretation the theorem is actually true. After all, even if one assumes that $F$ is consistent, it might still be unsound.
But this puzzles me. When setting up $F$ one always makes sure that all the axioms are true under the intended interpretation. Presumably, one would also make sure that all the rules of inference are valid. That is, if all the theorems proven so far are true, then the rules of inference only produce further true theorems. Is that too much to ask?
However, this straightforward construction immediately implies that I can't prove falsehoods under the intended interpretation. Further, I'd be hesitant to even call something for which the rules of inference aren't truth preserving an "interpretation".
But then it seems that unsound formal systems couldn't even exist. So where's the problem with this view?