Infinity

Infinity is a concept used to describe a quantity that grows without bound or to indicate processes that never end. It is not a specific number, but rather an idea that helps mathematics and science model unbounded quantities. Philosophers distinguish between potential infinity, an unending process (for example, counting 1, 2, 3, …), and actual infinity, a completed infinite set.

In mathematics, infinity is formalized via set theory and analysis. Cantor showed that infinite sets can have different sizes: the set of natural numbers is countably infinite with cardinality aleph-null (ℵ0), while the real numbers are uncountably infinite and have cardinality greater than ℵ0 (often described as the cardinality of the continuum, 2^ℵ0). The theory provides a hierarchy of infinities and foundations for limits and convergence in calculus.

In calculus and analysis, infinity appears in limits, infinite series, and improper integrals. A sequence or function can grow without bound or approach a finite limit as its input tends to infinity. Infinite series may converge to finite values or diverge. The notion of infinity is also used to describe unbounded domains, such as intervals extending without end.

In geometry, infinity appears in projective geometry by adding points at infinity to make lines intersect and to simplify angle relations. In physics, infinities arise as idealizations, such as singularities in general relativity or divergences in quantum theories, and often signal a limitation of a model.

Philosophically, debates distinguish potential infinity from actual infinity, and whether infinity can exist as a completed quantity. In practice, infinity is a powerful abstraction enabling precise formulation of limits, probabilities, and measurements across mathematics and science.