While Kunen's famous inconsistency theorem shows there is no Reinhardt cardinal, there have been various attempts to obtain a consistent variant of a Reinhardt cardinal. One possible approach is working over $\mathsf{CZF}$--Constructive ZF. This theory is a constructive analog of $\mathsf{ZF}$ in the sense that adding the Law of Excluded Middle to $\mathsf{CZF}$ results in $\mathsf{ZF}$.

Since ordinals over the constructive settings are ill-behaved, we characterize large cardinal properties by capturing the structural properties of $V_\kappa$ or $H_\kappa$, which results in large set axioms. In this talk, we will formulate large set axioms for large cardinals characterized by elementary embeddings, and examine their consistency strengths. What makes the examination interesting is the consistency strength of $\mathsf{CZF}$ with a 'small' large set axioms, like inaccessible sets, is far below that of the second-order arithmetic. However, it turns out that large set axioms characterized by elementary embeddings have very high consistency strength, beyond that of ZFC and more.

This is joint work with Richard Matthews.