I was reading this article where the author explains that there are numbers outside the complex set and that you can arbitrarily generate new types using the same method as he described to generate the quaternions.
My question is: Is the set of all mathematical number types countable? IE (1,2,3,...) -> (0, integers, rationals, reals, complex, quaternions, ...)
Using the way he generates to count, you might be able to set up an injection with elements used to generate the set (1 then 1, i then 1, i, j, k, and so on) and there are finite number of operations used to generate the other numbers (or at least in his article he mentioned a finite set of operations.).
Is this right or is there a quick counter example?