  However, just as in the previous subsection, one would like some assurance that there are not incompatible axioms that share this fruitful consequence. Remarkably, this turns out to be the case—axioms of definable determinacy are actually equivalent to axioms asserting the existence of inner models of large cardinals. We discuss what is known about this connection, starting with a low level of boldface definable determinacy and proceeding upward. It should be emphasized that our concern here is not merely with consistency strength but rather with outright equivalence (over   ZFC).[12] 