What is "equiconsistency"

In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other".

In general, it is not possible to prove the absolute consistency of a theory T. Instead we usually take a theory S, believed to be consistent, and try to prove the weaker statement that if S is consistent then T must also be consistent—if we can do this we say that T is consistent relative to S. If S is also consistent relative to T then we say that S and T are equiconsistent.