  Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses
so strong as to contradict the Axiom of Choice -- we solve several variants of
the generalized continuum problem and identify structural features of the
levels $V_\alpha$ of the cumulative hierarchy of sets that are eventually
periodic, alternating according to the parity of the ordinal $\alpha$. For
example, if there is an elementary embedding from the universe of sets to
itself, then for sufficiently large ordinals $\alpha$, the supremum of the
lengths of all wellfounded relations on $V_\alpha$ is a strong limit cardinal
if and only if $\alpha$ is odd.
