Cardinal number

With the axiom of choice, every set can be well-ordered, so every set has a cardinal. Two sets have the same cardinal if and only if they are equinumerous, and \(|A| \le |B|\) when there is an injection from \(A\) into \(B\). The finite cardinals are the natural numbers, and the first infinite cardinal is \(\aleph_0\).

Let \(A=\{a,b,c\}\) and \(B=\{1,2,3\}\). The function \(f(a)=1\), \(f(b)=2\), \(f(c)=3\) is a bijection, so \(|A|=|B|=3\). In the infinite case, the map \(n\mapsto 2n\) is a bijection between \(\mathbb{N}\) and the even numbers, so the evens have the same cardinal as \(\mathbb{N}\), namely \(\aleph_0\).

A common mistake is to think that all infinite sets have the same size. Cantor proved that \(|\mathbb{R}|>|\mathbb{N}|\) and that \(|A|<|\mathcal{P}(A)|\) for every set \(A\), so there are infinite sets of different sizes. Another mistake is to think the cardinal depends on the order of the elements: it depends only on how many elements there are.