Consider the set $\mathcal{P}(\mathbb{R})$ of all subsets of $\mathbb{R}$, the set of real numbers. It has a natural partial order: $A \leq B$ iff $A \subseteq B$.

Can one extend this order to a total order?

(I was discussing this with a friend and we didn't know if this is possible. If we replace $\mathbb{R}$ by any finite set, this is possible. We were not sure even when we replace $\mathbb{R}$ by $\mathbb{N}$.)