TITLE: Is there a known closed form sequence for algebraic number one-to-one with the natural numbers?
QUESTION [5 upvotes]: We have a closed form recursive sequence for the rational numbers, where they appear only once. It is called the Calkin-Wilf sequence. Namely,
$q_{i+1}=\frac{1}{2⌊q_i⌋-q_i+1},q_1=1$.
Has someone found out an analog for algebraic numbers? I couldn't find any when searching but would be very surprised if one wasn't discovered already or serious public attempts were made to find one.

REPLY [3 votes]: There are several problems with trying to find an explicit list.

Firstly, the minimal irreducible polynomials, though clearly countable, can't be explicitly identified.

Secondly, even if you somehow managed to bypass that obstacle, the next issue is how to distinguish the conjugates.

Bottom line: It's a hopeless task, so I would just settle for a proof of countability (which is routine).