  We will consider a number of new large-cardinal properties, the
$\alpha$-tremendous cardinals for each limit ordinal $\alpha>0$, the
hyper-tremendous cardinals, the $\alpha$-enormous cardinals for each limit
ordinal $\alpha>0$, and the hyper-enormous cardinals. For limit ordinals
$\alpha>0$, the $\alpha$-tremendous cardinals and hyper-tremendous cardinals
have consistency strength between I3 and I2. An $\omega$-enormous cardinal has
consistency strength greater than I0, and also all the large-cardinal axioms
discussed in the second part of Hugh Woodin's paper on suitable extender
models, not known to be inconsistent with ZFC and of greater consistency
strength than I0. Ralf Schindler and Victoria Gitman have developed the notion
of a virtual large-cardinal property, and a clear sense can be given to the
notion of "virtually $\omega$-enormous". A virtually $\omega$-enormous cardinal
can be shown to dominate a Ramsey cardinal.
  It can be shown that a cardinal $\kappa$ which is a critical point of an
elementary embedding $j:V_{\lambda+2} \prec V_{\lambda+2}$, in a context not
assuming choice, is necessarily a hyper-enormous cardinal. Building on this
insight, we can obtain the result that the existence of such an elementary
embedding is in fact outright inconsistent with ZF. The assertion that there is
a proper class of $\alpha$-enormous cardinals for every limit ordinal
$\alpha>0$ can be shown to imply a version of the Ultimate-L Conjecture.
