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This entry is about the concept in mathematics, specifically in set theory/type theory/category theory. For the concept in chemistry see at chemical element.

Contents

Idea

An element of a set is a thing which “belongs to,” or “is an element of,” that set.

The circularity of this definition is unavoidable in foundational set theories in which “set” is an undefined term. In “definitional” set theories, where “set” is defined in terms of something else, elements are likewise defined in terms of the same “something else.”

If sets (or setoids) are regarded as the semantics of some type theory, then an element of a set is the interpretation of a term of some type.

Generalisations

• A term of some type.
• a global element of any object in any category or higher category with a terminal object.
• a generalised element of any object in any category or higher category;
• a morphism in a category, a different sort of element from an object;
• An object in a category;
• in general, $k$-morphisms or $k$-cells in an $\infty$-category;