 Revision processes help provide a semantics for circular  definitions.[14]  They can be used to define semantic notions such as “categorical truth” and logical notions such as “validity.” The characteristics of the logical notions we obtain depend crucially on one aspect of revision: the number of stages before objects settle down to their regular behavior in the revision process. A definition is said to be finite iff, roughly, its revision process necessarily requires only finitely many such  stages.[15]  For finite definitions, there is a simple logical calculus, \(\mathbf{C}_{0}\), that is sound and complete for the revision  semantics.[16]  With non-finite definitions, the revision process extends into the  transfinite.[17]  And these definitions can add considerable expressive power to the language. (When added to first-order arithmetic, these definitions render all \(\Pi^{1}_{2}\) sets of natural numbers definable.) Because of the expressive power, the general notion of validity for non-finite circular definitions is not axiomatizable (Kremer 1993). We can give at best a sound logical calculus, but not a complete one. The situation is analogous to that with second-order logic.