Consistency

A math system is consistent if no contradictory statements can be derived in it. If you can derive p ∧ ¬p, which is a contradiction, then a logic system is not consistent. For, if one stumbles onto a contradiction, then anything would follow, meaning anything can be proven from a contradiction (ex falsum quodlibet principle).

Hilbert's first concern was the completeness of mathematics: is there a way to prove all true propositions within a system of math? Hilbert's second question was about the consistency of mathematics: is there a way to prove that no contradictory statements arise in a system of math? Hilbert's third question was about decidability: is there an algorithm that can always determine whether a statement follows from the axioms?