Cardinalities

Cardinality is the notion that measures the size of a set by counting its elements. The cardinality of a set A is denoted |A|. Two sets have the same cardinality if there exists a bijection between them. Finite sets have cardinalities equal to a natural number; infinite sets have infinite cardinalities.

Common examples: the set of natural numbers N, the set of integers Z, and the set of rationals Q are all countably infinite, with cardinality aleph_0. The set of real numbers R is uncountable and has cardinality c, often written as 2^{aleph_0}.

Cantor's theorem shows that for any set A, its power set P(A) has strictly greater cardinality than A, so no set can be put in bijection with its own power set.

Cardinal numbers describing infinite sizes are called aleph numbers; aleph_0 is the size of the natural numbers, aleph_1 is the first uncountable cardinal, and so on. The size of the real numbers is c = 2^{aleph_0}. The continuum hypothesis asks whether c equals aleph_1; it is independent of the standard axioms of set theory.

Cardinal arithmetic extends finite arithmetic to infinite sizes. Under the usual axioms of set theory, for infinite cardinals κ and λ, κ + λ = max(κ, λ) and κ · λ = max(κ, λ). Exponentiation κ^λ can be larger; in particular 2^{κ} > κ. Finite arithmetic remains the same for finite cardinals.

Cardinality provides a framework for comparing sizes of sets across mathematics, and is central in set theory, combinatorics, and analysis.