As a countable, everywhere-dense, totally ordered set without minimal or maximal elements, $\Bbb{A}$, the set of real algebraic numbers, must be order isomorphic to $\Bbb{Q}$. I'm wondering how "nice" such an isomorphism can b made. Does it admit any reasonably explicit description?
Also, what would the asymptotic properties of an order isomorphism $\Bbb{A} \rightarrow \Bbb{Q}$ be? It seems like it would have to grow quite quickly, but I'm not sure how quickly.
I don't think there's a nice-looking explicit isomorphism. On the other hand, given any $\varepsilon>0$ there exist isomorphisms that move no points by more than $\varepsilon$. Proof: Partition the real line into successive intervals with lengths $<\varepsilon$ and with transcendental endpoints. Then build the isomorphism separately within each of these intervals.