[FOM] role of large cardinals

[meskew at math.uci.edu]

> There are some choice-violating versions of embedding hypotheses that
> transcend the apparently-choice-compatible embedding (large cardinal)
> hypotheses. So the conventional set theorist would not consider, e.g.,
> j:V into V over NBG to be a "large cardinal hypothesis" since it is
> incompatible with AxC. And it is stronger than, e.g., j:V(kappa + 1)
> into V(kappa + 1). These choice-violating versions are considered to
> be "possibly consistent".

It is my understanding that this is usually considered as Axiom I0: There
is a nontrivial elementary embedding of L(Vλ+1) into itself with the
critical point below λ.  I0 does not directly contradict Choice, and
it is stronger than ZF+Reinhardt simply because it says something about
the structure of this self-embeddable class.