Definition:Universal Class/Zermelo-Fraenkel Theory

Universal Class in Zermelo-Fraenkel Set Theory
If the universal class is allowed to be a set in ZF(C) set theory, then a contradiction can be derived.

One equivalent of the axiom of specification states that:


 * $\forall z: \forall A: \paren {A \subseteq z \implies A \in V}$

Since the universal class contains all classes, then if we assume that it is the value of some variable $z$, then all classes become elements of the universe.

However, due to Russell's Paradox, this cannot be the case.

Therefore, comprehension of the universal set leads to a contradiction and cannot be a value of a variable in ZF set theory.

However, some alternative set theories, such as 's New Foundations, allow the universal set to be a value of a variable, and reject certain instances of the axiom of specification.

All the elements of the universal class are precisely the Universe of Discourse of quantification.

In fact, membership of the universal class distinguishes sets from proper classes, providing a basis for comprehension of certain statements.