Cardinality

In mathematics, cardinality is an intrinsic property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The concept is understood through one-to-one correspondences between sets. That is, if their objects can be paired such that each object has a pair, and no object is paired more than once.

The basic concepts of cardinality go back as early as the 6th century BCE, and there are several close encounters with it throughout history. However, it is considered to have been first introduced formally to mathematics by Georg Cantor at the turn of the 20th century. Cantor's theory of cardinality was then formalized, popularized, and explored by many influential mathematicians of the time, and has since become a fundamental concept of mathematics.

Two sets are said to be equinumerous or have the same cardinality if there exists a one-to-one correspondence between them. Otherwise, one is said to be strictly larger or strictly smaller than the other. A set is countably infinite if it can be placed in one-to-one correspondence with the set of natural numbers ${\displaystyle \{1,2,3,4,\cdots \}.}$ For example, the set of even numbers ${\displaystyle \{2,4,6,..\}}$ and the set of rational numbers are countable. Uncountable sets are those strictly larger than the set of natural numbers—for example, the set of all real numbers or the powerset of the set of natural numbers.

For finite sets, cardinality coincides with the natural number found by counting their elements. However, it is more often difficult to ascribe "sizes" to infinite sets. Thus, a system of cardinal numbers can be developed to extend the role of natural numbers as abstractions of the sizes of sets. The cardinal number corresponding to a set ${\displaystyle A}$ is written as ${\displaystyle |A|}$ between two vertical bars. Most commonly, the Aleph numbers ${\displaystyle \aleph _{0},\aleph _{1},\aleph _{2},...\aleph _{\omega },\aleph _{\omega +1}...}$ are used, since it can be shown every infinite set has cardinality equivalent to some Aleph.

The set of natural numbers has cardinality ${\displaystyle \aleph _{0}}$. The question of whether the real numbers have cardinality ${\displaystyle \aleph _{1}}$ is known as the continuum hypothesis, which has been shown to be unprovable in standard set theories such as Zermelo–Fraenkel set theory. Alternative set theories and additional axioms give rise do different properties and have often strange or unintuitive consequences. However, every theory of cardinality using standard logical foundations of mathematics admits Skolem's paradox, which roughly asserts that basic properties of cardinality are not absolute, but relative to the model in which the cardinality is measured.