Cardinal number
A number measuring the size of a set.
A cardinal number, or cardinal for short, is a kind of number that measures the cardinality of a set, i.e., how many elements there are in a set. Cardinality is defined in terms of bijective functions: two sets have the same cardinality if and only if there is a one-to-one correspondence between their elements. For finite sets, cardinal numbers correspond to natural numbers, but for infinite sets, cardinal numbers exhibit more complex behaviors, such as the possibility that a proper subset can have the same cardinality as the whole set.
- field
- Mathematics
- known_for
- Measuring the size of sets, including infinite sets; introducing aleph numbers such as ℵ₀
Lore & Background
The concept of cardinal numbers arises from the need to describe the size of a set, abstracting away the identities of individual members. Two sets have the same cardinality if a bijection exists between them, a relation called equipotence, equipollence, or equinumerosity. For finite sets, this agrees with counting, but for infinite sets, surprising results occur: for example, the set of natural numbers and the set of rational numbers have the same cardinality, even though the natural numbers are a proper subset of the rationals.
Reader's Guide
Cardinal numbers are fundamental to set theory and have implications across mathematics. The cardinality of the natural numbers is denoted ℵ₀ (aleph-null), the smallest aleph number. Georg Cantor showed that infinite sets can have different cardinalities; for instance, the set of real numbers has a strictly greater cardinality than the natural numbers. The properties of infinite cardinal numbers depend on statements independent of Zermelo–Fraenkel set theory, such as the axiom of choice and the continuum hypothesis. For example, all infinite cardinal numbers are aleph numbers if and only if the axiom of choice is true. Cardinality is studied in set theory and used in model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, cardinal numbers form a skeleton of the category of sets.
Did You Know?
- The cardinality of a set A is denoted |A|, card(A), or #A.
- The set of natural numbers and the set of rational numbers have the same cardinality, ℵ₀.
- Cantor's diagonal argument shows that the power set operation always yields a strictly greater cardinality.
- The cardinality of the real numbers is equal to 2^{ℵ₀}.
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