Why is $\aleph_0$ the smallest cardinal number?

Question

It is a well-known fact that $\aleph_0 = \vert \mathbf{N} \vert$ is the smallest infinite cardinal number. But I'm wondering why; does anyone know a proof?

Thanks!

Answer 1

Every cardinal is an ordinal, and $ω$ is the first ordinal after the finite ordinals, hence $ω$ is the first infinite ordinal and hence the first infinite cardinal since it is a cardinal.

Answer 2

You can show show the following:

Every infinite set $\mathcal{A}$ has a subset $\mathcal{B}$, so $\mathcal{B}\subseteq\mathcal{A}$, which has the same cardinality as $\mathbb{N}$, which is $\aleph_0 = \vert \mathbf{N} \vert=\vert \mathcal{B} \vert$.

This proves that $\mathbb{N}$ has the smallest infinite cardinal number.