The question is exactly that of the title: what are Moschovakis cardinals?
Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency proofs came before their proofs?," user14111 posted (Are there examples of statements that have been proven whose consistency proofs came before their proofs?) an answer involving "Moschovakis cardinals," a large cardinal notion which was shown to be inconsistent at some point in time. Now, googling for Moschovakis cardinals reveals nothing besides that answer and this (http://mathforum.org/kb/thread.jspa?forumID=13&threadID=22263&messageID=59655#59655) Math Forum Discussion post, which seems(?) to be responding to a post which was then deleted.
According to user14111, the notion of a Moschovakis cardinal arose in an unpublished manuscript circulated around the late 60s; given the timing, my current guess is that "Moschovakis cardinal" is just a synonym for "Reinhardt cardinal," but I'll admit there is no real basis for my guess.
Why I'm interested. (Assuming these aren't just Reinhardts in disguise) I'm always interested in large cardinal axioms inconsistent with ZFC; in particular, can Moschovakis cardinals survive in ZF? Also, on a purely historical level, it would be interesting to know about.
Even if Moschovakis=Reinhardt, I'm still intrigued: why would that name be used? I've heard Reinhardt cardinals called Kunen cardinals before, since Kunen proved their inconsistency; but Moschovakis seems to have no relation to the subject that I'm aware of.