Consistency

A consistent theory is one in which there are well-formed statements that you cannot prove from its axioms; or equivalently, that there is no \(X\) such that \(T\vdash X\) and \(T\vdash \neg X\).

From the point of view of model theory, a consistent theory is one whose axioms are satisfiable. Thus, to prove that a set of axioms is consistent you can resort to constructing a model using a formal system whose consistency you trust (normally using set theory) in which all the axioms come true.

Arithmetic is expressive enough to talk about consistency within itself. If \(\square_{PA}\) represents the standard provability predicate in Peano Arithmetic then a sentence of the form \(\neg\square_{PA}(\ulcorner 0=1\urcorner)\) represents the consistency of \(PA\), since it comes to say that there exists a disprovable sentence for which there is no proof. Gödel’s second incompleteness theorem comes to say that such a sentence is not provable from the axioms of \(PA\).