Definition:Universal Class

Definition
The Universal Class is the class of which all sets are members. The fundamental law of the universal class is that:


 * $\forall x: x \in U$

where $U$ denotes the universal class. This is not quite a rigorous definition, even though it specifies a unique class. The universal class is defined most commonly in literature as:


 * $U = \{ x | x = x \}$

From this, it follows immediately that:


 * $\forall x: ( x \in U \iff x = x )$

From which we derive the fundamental law, since $x = x$.

Zermelo-Fraenkel Theory
If the universal class is allowed to be a set in ZF(C) set theory, then a contradiction results. One equivalent of the Axiom of Subsets states that:


 * $\forall z: \forall A: \left({A \subseteq z \implies A \in U}\right)$

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 Quine's New Foundations, allow the universal set to be a value of a variable, and reject certain instances of the Axiom of Subsets.

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.

Also See

 * Definition:Universal Set

Source

 * : $\S 2.7$