Choiceless cardinals and the continuum problem

TL;DR

This paper explores the effects of extremely strong large cardinal hypotheses, beyond the Kunen inconsistency and contradicting the Axiom of Choice, on the continuum problem and the structure of the set-theoretic universe.

Contribution

It demonstrates how certain large cardinal assumptions influence the continuum problem and reveals periodic structural features of the cumulative hierarchy levels.

Findings

Under strong hypotheses, the continuum problem is solved for various cases.

Structural features of $V_\alpha$ levels are shown to be eventually periodic.

The parity of $\alpha$ determines the properties of wellfounded relations on $V_\alpha$.

Abstract

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.