Infinities

Infinities are concepts describing unboundedness or unlimited quantity. They are not numbers in the ordinary sense, and can be approached in two ways: as potential infinity, a process that can continue without bound, and as actual infinity, a completed infinite object such as the set of all natural numbers.

Cardinality measures the size of sets. Finite sets have a finite number of elements; infinite sets do not. The natural numbers are countably infinite, denoted aleph-null, and so are the integers and the rationals. The real numbers form an uncountable infinity, proven by Cantor's diagonal argument; their cardinality is 2^aleph-null, often called the continuum.

Ordinals describe order types. The first infinite ordinal is omega; larger ordinals such as omega-one exist, with omega-one being the first uncountable ordinal. Cardinals and ordinals are related but distinct: cardinals measure size, ordinals measure order and enable transfinite induction.

In analysis, infinity appears as a limit. Functions may diverge to infinity, while infinite series can converge to finite values. The extended real line adds +∞ and −∞, but arithmetic with infinities follows specific rules to avoid contradictions.

Beyond mathematics, infinity features in physics and philosophy in discussions of spacetime, singularities, and the nature of actual infinity. Some questions about infinity, such as the continuum hypothesis, are independent of the standard axioms of set theory, illustrating that certain properties of infinity cannot be settled within a single formal framework.