How to define a well-order on $\mathbb R$? I would like to define a well-order on $\mathbb R$. My first thought was, of course, to use $\leq$. Unfortunately, the result isn't well-founded, since $(-\infty,0)$ is an example of a subset that doesn't have a minimal element.
My next thought was to use that $P(\mathbb N)$ is in bijection with $\mathbb R$ and then to use $\subseteq$. Unfortunately, this is not a total (=linear) order.
Now I'm stuck. Could someone show me how to define a well-order on $\mathbb R$? (Using the axiom of choice is permitted.) Many thanks.
Context: This is an exercise in a book I'm currently reading: 

You can’t: it’s consistent with ZF that $\Bbb R$ not be well-orderable. See this answer for starters.
