Characterizations of interpretability in bounded arithmetic

This paper deals with three tools to compare proof-theoretic strength of formal arithmetical theories: interpretability, ${\mathrm{\Pi }}_1^0$-conservativity and proving restricted consistency. It is well known that under certain conditions these three notions are equivalent and this equivalence is often referred to as the Orey-H'ajek characterization of interpretability. In this paper we look with detail at the Orey-H'ajek characterization and study what conditions are needed and in what meta-theory the characterizations can be formalized.