Definition:Universe (Set Theory)/Zermelo-Fraenkel Theory

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Universal Set in Zermelo-Fraenkel Set Theory

If the universal class is allowed to be a set $\mathbb U$ in ZF(C) set theory, then a contradiction results.

One equivalent of the axiom of specification states that:

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

However, we may conservatively extend the ZFC axioms to incorporate classes, which is done in von Neumann–Bernays–Gödel (NBG) set theory.

The basic gadget we work with is a class, and a set is defined to be an element of a class. Some refer to a proper class to be one which is not contained in any class (so an "improper" class would be one that is contained in some class, thus it is a set).

We avoid the contradiction mentioned by modifying the axiom of specification through restricting quantifiers to range over sets, but not all classes. (We also demand that sets are not bijective to the class of all ordinals.)

Michael Shulman's "Set theory for category theory" (arXiv:0810.1279v2 [math.CT]) studies various esoteric foundational issues relevant for category theory, and gives the definition of a Universe in any set theory as: a model for the ZFC axioms. This covers the von Neumann universe, the Grothendieck universe, the class of all sets in NBG set theory, etc.

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 set are precisely the Universe of Discourse of quantification.