Definition:Universal Class/Zermelo-Fraenkel Theory

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Universal Class in Zermelo-Fraenkel Set Theory

If the universal class is allowed to be a set in ZFC, 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 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.


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.


Sources


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