A *contradiction* occurs when the axioms and theorems of a logical system are used to prove both a statement and the opposite of that statement. Being able to prove a statement and its opposite means that anything follows. Symbolically, a contradiction can be represented by the statement (`X`)(~`X`) → `Y`, where `Y` can be any statement whatsoever.
This is a big problem, since being able to prove absolutely anything causes the entire system to collapse and renders it useless.

One important contradiction in the history of mathematics was Russell's paradox, which revealed a contradiction in set theory and thus forced a major reworking of set theory.