Abstract
To study the relative difficulty of classification problems, a recent trend has been to encode them as countable Borel equivalence relations, and to then preorder the equivalence relations under a notion called Borel reducibility. This preorder has a minimal element (equality on the reals), and this minimal element has a unique successor called E_0 (equality mod rationals on the reals). Although it is known that this preorder is not linear, it is still a major open question to determine if E_0 has a unique successor or not. We will give an overview of the first major result in this direction due to Conley and Miller, which states that under the weaker notion of measure reducibility, not only is there no unique successor of E_0, but in fact any basis above E_0 must be uncountable.