infinite

Infinity is a concept describing something without any bound or end. In mathematics, infinity is not a real number but a notion used to describe sizes, limits, and processes that continue without termination. Two broad ideas are used: potential infinity, the idea of growing without bound (for example, the sequence 1, 2, 3, ...), and actual infinity, the notion of a completed infinite total (such as the set of all natural numbers).

In set theory, infinite sets can have different sizes. Cantor showed that not all infinities are equal: the set of natural numbers is countably infinite (cardinality aleph-null), while the set of real numbers is uncountably infinite, with a greater cardinality. The continuum hypothesis posits no set has cardinality strictly between aleph-null and the real numbers, but it remains independent of standard axioms of set theory.

In analysis, infinity appears in limits, infinite series, and improper integrals. The concept allows precise definitions of convergence and divergence, and underpins many calculations in calculus and mathematical analysis.

In physics and cosmology, discussions of the infinite arise in idealizations such as infinite spatial extent or infinite past or future, though physical theories typically impose limits or use models with finite, measurable quantities.

Philosophically, the infinite raises questions about the nature of reality, the divine, and the limits of human knowledge. In culture, it appears in poetry, art, and literature as a symbol of boundlessness.