This paper studies the capacity of fast-fading channels from an algorithmic perspective, examining whether the channel capacity can be computed algorithmically or not. To address this question, the concept of Turing machines is used, which provides fundamental performance limits of digital computers. It is shown that certain computable continuous fading probability distribution functions yield capacities that are non-computable. Furthermore, the implications of this non-computability in information theory and coding are discussed, particularly the impossibility of designing universal algorithms that, given the fast-fading channel parameters and a predefined decoding error ϵ, can compute codes operating at the maximum rate with a decoding error probability no higher than ϵ.