Algebraic numbers are countable

Here we're defining countable as either finite or countably infinite (so the natural numbers are countable and so are finite sets).

[@chrisclub3185]

You could also put P_N to be the set of integer polynomials of degree N with integer coefficients, which is countable, then do essentially the same argument.

[@coconutmath4928]

Indeed, I wanted to avoid that argument since it relies on Zx...xZ being countable, but it is definitely valid.