Kardinalitesi

Kardinalitesi, or cardinality, is a concept in set theory that measures the size of a set. For finite sets, the cardinality is simply the number of elements. For infinite sets, cardinality is defined by the existence of a bijection: two sets have the same cardinality if there is a bijection between them. The cardinality of a set A is denoted |A|.

Finite and infinite sizes illustrate the idea. The set of natural numbers N has cardinality aleph-null (ℵ0). The set of even numbers has the same cardinality as N. The set of real numbers R has a strictly larger cardinality—the cardinality of the continuum, 2^{ℵ0}—and no bijection exists between N and R (Cantor’s diagonal argument).

Power sets and cardinal arithmetic provide further tools. For any set A, the power set P(A) has cardinality 2^{|A|}. In general, for infinite cardinals κ and λ, κ+λ = max(κ,λ) and κ·λ = max(κ,λ) when at least one is infinite and nonzero. Cardinal exponentiation κ^λ counts functions from λ to κ.

Historically, Cantor introduced the concept in the late 19th century. The continuum hypothesis (CH) posits that there is no cardinal strictly between ℵ0 and 2^{ℵ0}; CH is independent of the standard axioms of set theory.

Cardinality is distinct from ordinal numbers: cardinality concerns size regardless of order, while ordinals describe order types. The concept underpins much of set theory, logic, and theoretical computer science.