Even ordinals and the Kunen inconsistency

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.