kardinality

Cardinality is a fundamental concept in set theory that refers to the number of elements in a set. It provides a way to measure the size of a set, even for infinite sets. For finite sets, cardinality is simply the count of its distinct members. For example, the set {apple, banana, cherry} has a cardinality of 3.

The cardinality of a set A is often denoted by |A| or card(A). When comparing the cardinalities of two sets, if there exists a bijection (a one-to-one correspondence) between them, then they have the same cardinality. This concept is particularly important when dealing with infinite sets.

There are different sizes of infinity. Countable infinity is the cardinality of the set of natural numbers {1, 2, 3, ...}. Any set that can be put into a one-to-one correspondence with the natural numbers is said to be countably infinite. The set of integers and the set of rational numbers are examples of countably infinite sets.

Uncountable infinity refers to sets with a cardinality greater than that of the natural numbers. The set of real numbers is a classic example of an uncountably infinite set. Georg Cantor's diagonal argument is a famous proof demonstrating that the cardinality of the real numbers is strictly greater than the cardinality of the natural numbers. The smallest uncountable cardinality is denoted by the Hebrew letter aleph-one (ℵ₁), while the cardinality of the natural numbers is denoted by aleph-null (ℵ₀). The continuum hypothesis postulates that the cardinality of the real numbers is equal to ℵ₁, but this hypothesis has been proven to be independent of the standard axioms of set theory.