Inner model theory aims to produce “L-like” models that contain large cardinal axioms. For each large cardinal axiom Φ that has been reached by inner model theory, one has an axiom of the form V = LΦ. This axiom has the virtue that (just as in the simplest case of V = L) it provides an “effectively complete” solution regarding questions about LΦ (which, by assumption, is V). Unfortunately, it turns out that the axiom V = LΦ is incompatible with stronger large cardinal axioms Φ'. For this reason, axioms of this form have never been considered as plausible candidates for new axioms. 