The axiom AD^+, a structural strengthening of the Axiom of Determinacy (AD), was introduced by Hugh Woodin in the 1980's. AD^+ resolves many basic structural questions unsettled by AD. However, there are still many basic questions not answered by AD^+. One such class of questions concerns comparing cardinalities of sets under AD^+: given any two sets X and Y, how can we compare |X| and |Y|? One concrete instance of this is the following conjecture. 

Conjecture (the ABCD conjecture): suppose \alpha,\beta,\gamma,\delta are infinite cardinals such that \beta \leq \alpha and \delta\leq \gamma. Then |\alpha^\beta| \leq |\gamma^\delta| if and only if \alpha\leq \gamma and \beta \leq \delta.

The ABCD Conjecture is false under ZFC. It is open whether AD^+ implies the conjecture holds, but many instances of the conjecture have been established (by work of Woodin, Chan-Jackson-Trang etc). We introduce a structural strengthening of the axiom AD^+, called AD^{++}. AD^{++} implies the ABCD Conjecture and appears to have other interesting consequences not known to follow from AD^+. We do not know if AD^+ implies AD^{++} but some special cases have been proved. We will define these notions and discuss some of the partial results mentioned above. This is ongoing joint work with W. Chan and S. Jackson.