Countable sets

Definitions.
- $A$ is *countable* if there is an injection $A\hookrightarrow\mathbb{N}$ (equivalently $|A|\le|\mathbb{N}|$).
- $A$ is *countably infinite* (denumerable) if $|A|=|\mathbb{N}|$.
- $A$ is *uncountable* if $|A|>|\mathbb{N}|$.

Examples.
- Finite sets: countable.
- $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$: countably infinite.
- $\mathbb{R}$: uncountable.

Conclusion. Countability is the first level of infinity.