How strong is a Reinhardt set over extensions of CZF?

Abstract

We investigate the lower bound of the consistency strength of $\mathsf{CZF}$ with Full Separation $\mathsf{Sep}$ and a Reinhardt set, a constructive analogue of Reinhardt cardinals. We show that $\mathsf{CZF+Sep}$ with a Reinhardt set interprets $\mathsf{ZF^-}$ with a cofinal elementary embedding $j\colon V\prec V$. We also see that $\mathsf{CZF+Sep}$ with a Reinhardt set interprets $\mathsf{ZF^-}$ with a model of $\mathsf{ZF+WA_0}$, the Wholeness axiom for bounded formulas.