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This paper contributes to the theory of large cardinals beyond the Kuneninconsistency, or choiceless large cardinal axioms, in the context where theAxiom of Choice is not assumed. The first part of the paper investigates aperiodicity phenomenon: assuming choiceless large cardinal axioms, theproperties of the cumulative hierarchy turn out to alternate between even andodd ranks. The second part of the paper explores the structure of ultrafiltersunder choiceless large cardinal axioms, exploiting the fact that these axiomsimply a weak form of the author's Ultrapower Axiom. The third and final part ofthe paper examines the consistency strength of choiceless large cardinals,including a proof that assuming DC, the existence of an elementary embeddingfrom $V_{\lambda+3}$ 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.

Even ordinals and the Kunen inconsistency