  We have already noted that ZFC+¬PM is mutually interpretable with ZFC+LC where LC is the large cardinal axiom “There is a strongly inaccessible cardinal” and that this is shown using the dual techniques of inner and outer model theory.  It is a remarkable empirical fact that for any “natural” statement in the language of set theory φ one can generally find a large cardinal axiom Φ such that ZFC+φ and ZFC+Φ are mutually interpretable.  Again, this is established using the dual techniques of inner and outer model theory only now large cardinals enter the mix.  To establish that ZFC+Φ interprets ZFC+φ one generally starts with a model of ZFC+Φ and uses forcing to construct a model of ZFC+φ.  In many cases the forcing construction involves “collapsing” the large cardinal associated with Φ and arranging the collapse in such a way that φ holds in the “rubble”.  In the other direction, one generally starts with a model of ZFC+φ and then constructs an inner model (a model resembling L but able to accommodate large cardinal axioms) that contains the large cardinal asserted to exist by Φ. The branch of set theory known as inner model theory is devoted to the construction of such “L-like” models for stronger and stronger large cardinal axioms.