the underlying set of the object. The extension of a set is the set itself. That a set can capture the notion of the extension of anything is the idea behind the axiom of extensionality in axiomatic set theory. This kind of extension is used so constantly in contemporary mathematics based on set theory that it can be called an implicit assumption. A typical effort in mathematics evolves out of an observed mathematical object requiring description, the challenge being to find a characterization for which the object becomes the extension. Computer science In computer science, some database textbooks use the term 'intension' to refer to the schema of a database, and 'extension' to refer to particular instances of a database. Metaphysical implications There is an ongoing controversy in metaphysics about whether or not there are, in addition to actual, existing things, non-actual or nonexistent things. If there are—if, for instance, there are possible but non-actual dogs (dogs of some non-actual but possible species, perhaps) or nonexistent beings (like Sherlock Holmes, perhaps)—then these things might also figure in the extensions of various concepts and expressions. If not, only existing, actual things can be in the extension of a concept or expression. Note that "actual" may not mean the same as "existing". Perhaps there exist things that are merely possible, but not actual. (Maybe they exist in other universes, and these universes are other "possible worlds"—possible alternatives to the actual world.) Perhaps some actual things are nonexistent. (Sherlock Holmes seems to be an actual example of a fictional character; one might think there are many other characters Arthur Conan Doyle might have invented, though he actually invented Holmes.) A similar problem arises for objects that no longer exist. The extension of the term "Socrates", for example, seems to be a (currently) non-existent object. Free logic is one attempt to avoid some of these problems. General semantics Some fundamental formulations in the field of general semantics rely heavily on a valuation of extension over intension. See for example extension, and the extensional devices. See also Enumerative definition Extensional definition Extensional logic Generalization Sense and reference Semantic property Type–token distinction External links Towards a Reference Terminology for Ontology Research and Development Extensional processes Semantics Set theory Concepts in logic Definition Formal semantics (natural language)