**October 18**

**Hanul Jeon**,
Cornell University

**On a cofinal Reinhardt embedding without Powerset**

Reinhardt embedding is an elementary embedding from $V$ to $V$ itself, whose existence was refuted under the Axiom of Choice by Kunen's famous theorem. There were attempts to get a consistent version of a Reinhardt embedding, and dropping the Axiom of Powerset is one possibility. Richard Matthews showed that $\mathsf{ZFC} + \mathrm{I}_1$ proves $\mathsf{ZFC}$ without Powerset is consistent with a Reinhardt embedding, but the embedding $j\colon V\to V$ in the Matthews' model does not satisfy the cofinality (i.e., for every set $a$ there is $b$ such that $a\in j(b)$). In this talk, I will show from $\mathsf{ZFC} + \mathrm{I}_0$ that $\mathsf{ZFC}$ without Powerset is consistent with a cofinal Reinhardt embedding.