Suppose it should turn out that the “choiceless” large cardinals are consistent. (This is a hierarchy of large cardinals that extends beyond . The first major marker is a Reinhardt cardinal. After that one has Super Reinhardt cardinals and then the hierarchy of Berkeley cardinals. (This is something that Woodin, Bagaria, and myself have been recently investigating.)) Suppose that the principles in this hierarchy are consistent. Then if we are to follow the principle of “maximality” — in the sense of maximizing interpretability power — these principles will lead us upward to theories that violate AC. On this picture, AC would be viewed like V = L, as a limiting principle, a principle that holds up to a certain point in the interpretability hierarchy (while one is following a “natural” path) and then gets turned off past a certain stage.
Thanks for bringing up the choiceless cardinals, Peter. As a strictly amateur kibitzer, my hunch has been that their consistency alone wouldn’t be enough to unseat AC, that they’d have to generate something mathematically attractive (analogous to ordinary LCs generating the #’s, say). In any case, if you (or one of your co-workers) were willing, I suspect more than a few of us would be very interested to hear about the state of play on these cardinals.