Logic has this limitation: no logic can be both complete and consistent. Flipped on its head: there will always be statements that are independent of any logical system. For the rest of this piece I’m going to call a “logical system” a logic.
The easiest and most intuitive example of a statement that is always logically independent comes from Gödel: “This logic is consistent.” In fact, Gödel showed that any logic that can make a statement about its own consistency is automatically inconsistent.
It’s interesting to consider that term “inconsistent” without automatically equating it to “false.” While it’s possible that the logical system used for many mathematical proofs (Zermelo-Fraenkel) could be found to have an inconsistency, it’s difficult to know what this would portend. Which of the many theorems proven with this logic would need to be considered unproven?
In practice, since the consistency of such a system is impossible to prove, it may not be worth spending a lot of thought on it, but I will offer this insight of my own: I don’t think it matters much. It’s okay to use a logic, even if it has limitations. It’s a good thing, too, because every logic has this characteristic.
In the universe of all possible statements, the fraction of them that are provable in any given logic may be infinitesimally small. That’s also okay. It’s what makes that logic so special.