  Our aim is to investigate the space of mathematical theories (construed as recursively enumerable axiom systems).  The ordering on the space of such theories that we will consider is that of interpretability.  The informal notion of interpretability is ubiquitous in mathematics; for example, Poincaré provided an interpretation of two dimensional hyperbolic geometry in the Euclidean geometry of the unit circle; Dedekind provided an interpretation of analysis in set theory; and Gödel provided an interpretation of the theory of formal syntax in arithmetic.