Axiom of limitation of size

A principle in class theories stating that every proper class is equinumerous with the universe of sets. It characterizes proper classes as collections too large to be sets.

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Von Neumann–Bernays–Gödel set theory: A two-sorted axiomatic theory of sets and classes, commonly abbreviated NBG. NBG uses limitation of size to distinguish proper classes from sets.
Class theory: A foundational framework that treats classes as collections and may distinguish them from sets. Limitation of size is formulated in theories that reason about classes.
Morse–Kelley set theory: A class theory with stronger comprehension principles than NBG, in which classes can be quantified over. Its stronger class axioms provide a different setting for principles about class size.
Burali–Forti paradox: A contradiction arising from treating the order type of all ordinals as an ordinal. The proper class of ordinals exemplifies a collection too large to be a set.
Proper class: A class that is not a set in a class theory. Limitation of size characterizes every proper class by its size relative to all sets.
Equinumerosity: The relation of having the same cardinality, defined by the existence of a bijection. The axiom states that each proper class has the universe's cardinality.
New Foundations: A set theory introduced by W. V. Quine that uses stratified comprehension and does not take classes as a separate sort. Its treatment of collections differs from the set–class distinction presupposed here.
Russell's paradox: A contradiction from considering the set of all sets that do not contain themselves. It illustrates why unrestricted set formation cannot accommodate collections of unrestricted size.
Bijection: A function that is both injective and surjective, pairing each element of one domain with exactly one of another. Equinumerosity in the axiom is expressed by a bijection with the universe of sets.
Axiom: A sentence assumed as a starting principle in a formal theory. Limitation of size is an axiom schema or principle within class theories.