nLab big class

Idea

A class $L$ in material set theory is big if for any set $X \in L$ there exists a set $Y \in L$ such that $X \in Y$.

A metalanguage formulation

Consider a class $L$ as a formula $\phi(z)$ with a free variable $z$; intuitively $L$ is the collection of all sets such that $\phi(z)$ is true. Then, in the metalanguage, $L$ is big (i.e., the formula $\phi(x)$ exhibits a big class) if

$$\phi(X) \implies (\exists Y)(\phi(Y) \wedge (X\in Y))$$

Examples and properties

Gödel’s constructible universe is a transitive big class.