Definition:Universe (Set Theory)

Definition
Sets are considered to be subsets of some large universal set, also called the universe. Exactly what this universe is will vary depending on the subject and context.

When discussing particular sets, it should be made clear just what that universe is.

The usual symbol used to signify the universe is $\mathfrak A$. However, this is old-fashioned and inconvenient, so some newer texts have taken to using $\mathbb U$ or just $U$ instead.

With this notation, this definition can be put into symbols as:
 * $\forall S: S \subseteq \mathbb U$

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 set 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 set are precisely the Universe of Discourse of quantification.