equiconsistent

Equiconsistent is a term in mathematical logic used to compare the strength of formal theories. Let T1 and T2 be theories in the same language. They are equiconsistent if either both are inconsistent or both are consistent; equivalently, they have the same consistency strength. In practice, equiconsistency is assessed relative to a background meta-theory (often a fragment of arithmetic like PA).

To establish equiconsistency, logicians use several standard methods. One common approach is to show that one theory is a conservative extension of the other, meaning that every theorem of the original language provable in the extended theory was already provable in the base theory. If T2 is a conservative extension of T1, then T1 and T2 are equiconsistent. Another method is to demonstrate mutual interpretability or to construct models of one theory from models of the other, thereby aligning their consistency status.

Equiconsistency is central to relative consistency results and to understanding how adding axioms affects strength. It allows researchers to argue that extending a theory with certain axioms does not increase its consistency strength, even if it increases expressive power. Conversely, if adding an axiom is known to imply the consistency of the base theory, equiconsistency typically fails.

Notes: The notion is meta-theoretical and depends on the chosen language and background theory. Statements about equiconsistency can be subtle when theories use different signatures or when formalizing internal notions of proof and consistency.