Definition
A formal system is consistent if it never proves both a statement A and its negation ~A. Equivalently — in classical logic, where everything follows from a contradiction — a system is consistent if there is at least one statement it does not prove.
Consistency is the most basic correctness property of a logical system. An inconsistent system proves everything (because A ∧ ~A ⊃ B is a tautology for any B), so its theorems carry no information. Hilbert's program of the 1920s aimed to prove the consistency of formal mathematics from within a safer fragment of mathematics; Gödel's second incompleteness theorem showed this could not be done.
Why it matters
How it works
To prove consistency, you typically construct a model — a structure in which every axiom and every consequence is true. Since true statements never contradict each other, a system with a model is consistent. The catch: to construct the model you need another formal system, and to know that system is consistent you need yet another. Gödel's second theorem makes this regress unavoidable.
In practice, working mathematicians trust the consistency of ZFC set theory because no contradiction has been found in a century of vigorous use, and because relative consistency results show ZFC is at least as consistent as several simpler systems.