Endlichkeit

Endlichkeit denotes the property of being finite, i.e., having a limited extent, duration, number, or amount. It is typically contrasted with Unendlichkeit (infinity). The notion appears across disciplines and is used to describe both qualitative and quantitative limits.

In mathematics, Endlichkeit refers to finite cardinality. A set A is finite if there exists a natural number n and a bijection between A and the set {1, 2, ..., n}. Equivalently, every injective function A → A is surjective, or A can be expressed as a finite union of singletons. Basic consequences include that subsets of finite sets are finite and the union of finitely many finite sets is finite. By contrast, the set of natural numbers N is infinite. Finite structures also appear in algorithms and proofs, where finitary methods are sought.

In philosophy, finitude is used to describe the limits of beings, knowledge and time. Human beings are typically treated as finite creatures, bounded by mortality, sensory capacity and cognitive reach. The discussion contrasts human finitude with divine infinitude or the notion of the infinite in ethics and metaphysics. The distinction between potential and actual infinity has been influential since antiquity.

In science, finiteness can refer to finite energy, finite age of a system, or finite information content. Cosmology debates whether the universe is spatially finite or infinite, with the observable universe itself having finite age and size. In theology and existential thought, finitude underscores the dependence of creatures on a higher or absolute being.

Endlichkeit thus denotes a broad, cross-disciplinary idea of limit and boundedness.