Definition. A universe consists of a type $\mathsf{U}:\mathsf{Set}$ of codes along with a function $\mathsf{El}:\mathsf{U}\to \mathsf{Set}$.

Example. The universe of finite sets has $\mathsf{U}:\equiv \mathbb{N}$, $\mathsf{El}n:\equiv \mathsf{Fin}n$, where $\mathsf{Fin}$ is inductively defined byThat is, $\mathsf{Fin}n\equiv \{FZ,FSFZ,\dots ,{FS}^{n-1}FZ\}\cong \{0,\dots ,n-1\}$.