finiteness

Finiteness is the property of having only a finite number of elements or a finite extent. In mathematics, an object is finite if its cardinality is a nonnegative integer n. A set is finite when it has n elements for some n; otherwise it is infinite. For example, {1, 2, 3} is finite, while the natural numbers are infinite.

Many core results depend on finiteness: any subset of a finite set is finite; the union of finitely many finite sets is finite; and the product of finitely many finite sets is finite. Finiteness also applies to algebraic structures: a finite group has finitely many elements; a finite-dimensional vector space is finite in size. In combinatorics and formal languages, finite structures underpin counting, enumeration, and constructions.

In computer science, finiteness underpins termination and decidability. Finite automata, which have a finite set of states, recognize regular languages, and many algorithms operate on finite inputs to guarantee eventual completion. Complexity classes often rely on the assumption that resources are bounded by finite measures.

Philosophically and foundationally, finiteness contrasts with infinity. Cantor showed that infinite sets vary in size, but finiteness remains the default notion for counting and many practical contexts, such as measuring a finite volume or a finite data set. Some theories in physics and cosmology explore whether the physical universe is finite or infinite, though this remains an open question in many frameworks.

Related notions include countable versus uncountable sets, finite generation, and finite presentations. The term is widely used across mathematics, computer science, and logic to identify objects whose size is bounded and not infinite.