NBG set theory (von Neumann-Bernays-Gödel set theory)

NBG set theory, named after von Neumann, Bernays, and Gödel, extends the standard set theory (like ZFC) by including both sets and classes—collections that can be too large to be sets, such as all sets. While sets can be members of other sets, classes cannot be members of anything but are considered collections used to describe properties or collections that are too big to be sets. NBG formalizes this hierarchy, allowing for a more flexible framework to discuss collections of objects, particularly facilitating the treatment of large collections like "the class of all sets," within a consistent logical system.