Summary of cardinality

Finite. $|A|=n$ if a bijection $A\to\{1,\ldots,n\}$ exists.

Countably infinite. $|A|=\aleph_0$ if a bijection $A\to\mathbb{N}$ exists.

Examples.
- $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{N}^2$, countable unions of countable sets: $\aleph_0$.
- $\mathbb{R}$, $\mathcal{P}(\mathbb{N})$, Cantor set: $2^{\aleph_0}$ (uncountable).

Comparison. $|A|\le|B|$ iff injection $A\hookrightarrow B$.

Cantor–Bernstein. Injections both ways give a bijection.

Cantor's theorem. $|A|<|\mathcal{P}(A)|$.

Diagonal argument. Proves uncountability of $\mathbb{R}$.

CH. $2^{\aleph_0}=\aleph_1$ is independent of ZFC.

Conclusion. Cardinality classifies sizes of sets, revealing the hierarchy of infinities.