Cardinality

Cardinality refers to the number of elements in a set, serving as a measure of the set’s size in terms of its elements. The concept is straightforward for finite sets, but for infinite sets, cardinality is defined in terms of one-to-one correspondences (bijections) between sets.

• Finite Sets: The cardinality is simply the count of elements. For example, the set $A=\{1,2,3\}$ has a cardinality of 3.
• Infinite Sets: Two sets are said to have the same cardinality if there exists a bijection between them. For instance, the set of natural numbers $\mathbb{N}$ and the set of integers $\mathbb{Z}$ have the same cardinality, despite $\mathbb{Z}$ seemingly being “larger” because a bijection can be established between them.

Levels of Infinity: Cantor’s theorem shows that not all infinite sets have the same cardinality. For example, the set of real numbers $\mathbb{R}$ has a strictly greater cardinality than the set of natural numbers $\mathbb{N}$, indicating different “sizes” of infinity.