This doesn't necessarily imply that "all problems aren't solvable." Only that "all problems aren't solvable within the framework of any particular effective formal system."

For instance, there doesn't exist any single effective formal system which could determine if any given Diophantine equation has a solution. However, it could be the case that for every Diophantine equation D there exists an effective formal system F which proves that D does/doesn't have a solution.

The reason I say "effective" formal system is because, technically speaking, you could always just treat it as an axiom that D has a solution, but this wouldn't be considered an adequate "proof" because it assumes the truth of a statement which has a controversial truth value. So, by "effective" I mean that it is computable and believed to be consistent with the canonical model of arithmetic or founded in axioms which are strongly believed to be true in true arithmetic.

I feel like I've rushed this explanation, but hopefully you see that this issue is much more complicated than you're letting on.