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

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

Examples and properties

Gödel’s constructible universe is a transitive big class.