[TheRealPomax]

Unprovable statements. Not paradoxical statements. Paradoxes confound the reader, but can be explained as either being true or false due to rules that were implicitly used to construct the statement. Once all the steps are explicitly shown, with each rule that applies at each step, there is no more paradox.

On the other hand, unprovable statements never make it to the "but can be explained" stage: either the formal system is consistent, and these expressions by definition cannot have an answer, or they can be answered but the formal system is demonstrably inconsistent.

The formal system also needs to be a little more precise than "significantly expressive": it needs to allow for self-referential expressions. In effect, it must be possible to create an expression that can be mapped to the natural language sentence "This statement has no proof in this formal system".

While in natural language puzzles that might be a fun paradox to try to figure out, where the challenge is to determine whether, given the rules of the context, the answer is actually "true" or "false", when it comes to Godel sentences for formal systems there is never a paradox. Either they are undecidable, or the formal system is inconsistent. And we don't get an "out" by saying "but in reality, the answer is ..." because the whole point of formal systems is to be self-contained. There is nothing "outside" of them that we can use to make claims about expressions within that formal system. We only get the axioms of the formal system itself.