Incompleteness & Inconsistency
Among the most important and philosophically intriguing results in 20th-century logic are the limitative theorems such as Gödel’s incompleteness theorem and Tarski’s demonstration of the indefinability of truth in certain languages. A wide variety of approaches to resolving fundamental mathematical and semantical paradoxes have emerged in the wake of these results, as well as a variety of alternative logics including paraconsistent logics in which contradictions are tolerated.