Abstract by Michael Andersen
Many mathematical objects can be constructed using only countably many choices, but inscrutable objects require more than countably many choices in their construction (e.g. a non-Lebesgue measurable subset of the real numbers). Inscrutable objects are the bane of young mathematicians, who are prone to using concrete examples to build their intuition; inscrutable classes elude attempts to capture a concrete example. We study a few classes of inscrutable classes in this talk.
The usefulness of the concept of inscutability was first demonstrated by the proof that the class of group homomorphisms which are not in the image of π1 is an inscrutable class (i.e. the class Hom(GRP)\\Im(π1) is inscrutable). This allows one to tell during the construction of a group homomorphism whether or not it is induced by a continuous function under π1. Similar inferences can be drawn about other inscrutable classes.
An inscrutable class is a class P such that the axiomatic system ZF + DC + "P is empty" is equiconsistent with ZFC (i.e. the consistency of each of these axiomatic systems implies the consistency of the other) and ZFC implies that P is not empty.