This chapter discusses the notion of α-c.a. functions. The main issue is to properly define what is meant by a computable function o from N to α, which is required for the definition of α-computable approximations. Naturally, to deal with an ordinal α computably, one needs a notation for this ordinal, or more generally, a computable well-ordering of order-type α. To form the basis of a solid hierarchy, the notion of α-c.a. should not depend on which well-ordering one takes, rather it should only depend on its order-type. Thus, one cannot consider all computable copies of α. Rather, one restricts one's self to a class of particularly well-behaved well-orderings, in a way that ensures that they are all computably isomorphic. Having defined α-c.a. functions, the chapter relates these functions to iterations of the bounded jump (the jump inside the weak truth-table degrees).