  But large cardinal axioms have additional features that singles them out from the class of natural theories in the wellordered sequence of degrees.  To begin with they provide the most natural way to climb the hierarchy of interpretability—they are the simplest and most natural manifestation of pure mathematical strength.  But more important is the second component mentioned above, namely, the large cardinal axioms act as intermediaries in comparing theories from conceptually distinct domains.  For recall how this works: Given ZFC+φ and ZFC+ψ one finds large cardinal axioms Φ and Ψ such that (using the methods of inner and outer models) ZFC+φ and ZFC+Φ are mutually interpretable and ZFC+ψ and ZFC+Ψ are mutually interpretable.  One then compares ZFC+φ and ZFC+ψ (in terms of interpretability) by mediating through the natural interpretability relationship between ZFC+Φ and ZFC+Ψ.