The set of transcendental but computable numbers is also countable, however: there is a countable number of algorithms. Therefore, the majority of the reals are uncomputable, like Chaintin's constant (the probability that a random algorithm halts). This can in turn be made stronger by noticing that the definition of Chaitin's constant is finite, and by the same argument, definable real numbers are countable. It follows that the majority of the reals are indefinable.

Seems nicely analogous to "regular matter" vs dark matter and dark energy in the Universe.