Basis for $\mathbb R$ over $\mathbb Q$

Give me some examples of basis for $\mathbb R$ (as vector space over field $\mathbb F=\mathbb Q$).

Thanks.

While it is perfectly reasonable to write down a basis for a finitely dimensional vector space, it is not always possible to write one for infinitely dimensional vector spaces.

In fact the assertion that every vector space has a basis is equivalent to the axiom of choice. This does not mean that every infinitely dimensional space has no basis. For example $\mathbb R[x]$ as a vector space over $\mathbb R$ is infinitely dimensional, but it has a basis - $\{x^n\mid n\in\mathbb N\}$.

There are models of set theory without the axiom of choice in which there is no basis for $\mathbb R$ over $\mathbb Q$, which means that one cannot just "write down" such basis, but rather that one can prove the existence of a basis in a non-constructive manner such as Zorn's lemma.