# Definition:Universal Class

## Definition

The **universal class** is the class of which all sets are members.

The **universal class** is defined most commonly in literature as:

- $V = \set {x: x = x}$

where $x$ ranges over all sets.

It can be briefly defined as the **class of all sets**.

## Notation

The use of $V$ as the symbol used to denote the **universal class** follows the presentation by Raymond M. Smullyan and Melvin Fitting, in their *Set Theory and the Continuum Problem*.

Much of the literature uses $U$ and its variants, for example $\Bbb U$.

However, as $U$ is often used to denote the **universal set**, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to use $V$ for the **universal class** in order to reduce confusion between the two.

The symbol $\mathfrak A$ is also seen in older literature; however, the number of those who still like using this awkward and difficult-to-read Germanic font is decreasing.

## Also see

- Results about
**the universal class**can be found here.

## Zermelo-Fraenkel 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:

- $\forall z: \forall A: \paren {A \subseteq z \implies A \in U}$

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 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.

## Sources

- 1963: Willard Van Orman Quine:
*Set Theory and Its Logic*: $\S 2.7$ - 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 5.22$ - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 1$ Extensionality and separation