kardiinaliteetti

Kardiinaliteetti, often spelled cardinality, is a fundamental concept in set theory. It refers to the number of elements in a set. For finite sets, this is a straightforward count. For example, the set {apple, banana, cherry} has a cardinality of 3. We often denote the cardinality of a set A as |A|.

The concept becomes more nuanced when dealing with infinite sets. Georg Cantor introduced the idea of comparing the sizes of infinite sets using one-to-one correspondence. If a one-to-one correspondence can be established between two sets, they are said to have the same cardinality, even if they are infinite.

Cantor demonstrated that there are different "sizes" of infinity. The cardinality of the set of natural numbers (1, 2, 3, ...) is called countably infinite, denoted by $\aleph_0$ (aleph-null). Any set that can be put into a one-to-one correspondence with the natural numbers is also countably infinite.

However, the set of real numbers has a larger cardinality, known as the cardinality of the continuum, denoted by $c$ or $2^{\aleph_0}$. Cantor's diagonal argument proved that there is no one-to-one correspondence between the natural numbers and the real numbers, meaning the set of real numbers is "uncountably infinite" and larger than the set of natural numbers. The relationship between $\aleph_0$ and $c$ is a central topic in set theory, and the continuum hypothesis posits that there is no cardinality between $\aleph_0$ and $c$.