Editing Extension (semantics) (section)

== Mathematics ==
{{main|Extension (predicate logic)}}

In [[mathematics]], the 'extension' of a mathematical concept <math>C</math> is the [[set (mathematics)|set]] that is specified by <math>C</math>. (That set might be [[#Metaphysical implications|empty, currently.]])

For example, the extension of a [[function (mathematics)|function]] is a set of [[ordered pair]]s that pair up the arguments and values of the function; in other words, the function's graph.  The extension of an object in [[abstract algebra]], such as a [[Group (mathematics)|group]], is 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 (mathematics)|characterization]] for which the object becomes the extension.