metaset

Metaset is a term used in various contexts to denote a collection that is conceptually at a higher level than an ordinary set. In many discussions, a metaset comprises objects that are themselves sets or other meta-objects, emphasizing a distinction between object-level mathematics and meta-level discourse.

In logic and the philosophy of mathematics, metasets arise when distinguishing constructions that exist within a mathematical universe from discussions about that universe. Standard set theory uses a cumulative hierarchy to avoid paradoxes, so there is no single universal metaset within the base theory. Consequently, metasets are often discussed informally or within a higher-type or meta-theoretical framework, such as type theory, where one can talk about collections of objects at a higher level without conflating levels.

Notation for metasets is not standardized. Some authors treat a metaset as a shorthand for a set of sets or as a meta-level collection described in prose, while others formalize the idea within a higher-order or typed setting. The lack of a universal formalization means the term is typically used to clarify discourse rather than to assert a specific primitive notion in a given formal system.

Examples tend to be conceptual. A metaset might be described as the collection of all properties definable about elements of a universe, a notion useful for discussing meta-mathematical questions but not intended as a concrete set within ordinary set theory. In practice, discussions of metasets are part of broader discussions of hierarchy, higher-order logic, and the transition between object-level and meta-level reasoning.

See also: set, higher-order logic, type theory, definable set, power set, mathematical hierarchy.