  This paper contributes to the theory of large cardinals beyond the Kunen
inconsistency, or choiceless large cardinal axioms, in the context where the
Axiom of Choice is not assumed. The first part of the paper investigates a
periodicity phenomenon: assuming choiceless large cardinal axioms, the
properties of the cumulative hierarchy turn out to alternate between even and
odd ranks. The second part of the paper explores the structure of ultrafilters
under choiceless large cardinal axioms, exploiting the fact that these axioms
imply a weak form of the author's Ultrapower Axiom. The third and final part of
the paper examines the consistency strength of choiceless large cardinals,
including a proof that assuming DC, the existence of an elementary embedding
from $V_{\lambda+3}$ to $V_{\lambda+3}$ implies the consistency of ZFC + $I_0$.
By a recent result of Schlutzenberg, an elementary embedding from
$V_{\lambda+2}$ to $V_{\lambda+2}$ does not suffice.
