I was reading about Gödels Theorems, and have recently gotten super interested in the shortcomings of classical logic and the formal systems based on it. The way I see it, everything is based on logic, so the theorems are pervasive in reality. Are there alternatives to classical logic? Do any of those systems circumvent incompleteness/inconsistency without the requiring logic outside the system?

Lets make it a logic general thread to talk about weird paradoxes and peculiarities, and all things logic related.