Non-finite Axiomatizability and Undecidability of $\mathsf{C}\mathsf{h}\mathsf{e}\mathsf{q}$

We prove that $\mathsf{C}\mathsf{h}\mathsf{e}\mathsf{q}$ is not finitely axiomatizable, resolving a longstanding open problem in intermediate and modal logics. We further prove that the undecidability of Medvedev logic implies the undecidability of $\mathsf{C}\mathsf{h}\mathsf{e}\mathsf{q}$.