 Axiom of limitation of size

Every proper class is in bijection with the universe. Over class-theoretic Zermelo set theory, with class-parameter separation, this is equivalent to replacement plus global choice. For replacement, a proper functional image of a set would give a surjection from that set onto the universe, contradicted by diagonal separation. The proper class of ordinals supplies the global well-order. Conversely rank followed by uniformly chosen rank-segment well-orders is set-like, and every proper class in that order has order type equal to the entire ordinal class.

The Axiom of Limitation of Size is a principle in set theory that addresses the size of sets and prevents certain paradoxes that can arise from considering "too large" sets. It is most commonly associated with the context of higher-order set theories and is used to establish a distinction between small sets (which can be elements of other sets) and large sets (which cannot be elements of any set).