HOD has been found to be useful in that it is an inner model that can accommodate essentially all known large cardinals.
However, there is a definable well ordering over HOD, so it cannot satisfy large cardinal axioms known to negate choice. On the other hand, there seems to be a hierarchy of choiceless large cardinals that appears to be highly ordered and amenable to systematic investigation.
Recall cardinal definable sets, so here we can take the class of all hereditarily cardinal definable sets "HCD", this would be an inner model of ZF.
Can HCD accommodate ALL known large cardinal axioms?
