What makes computable, decidable sets recursive?

Question

What does recursive mean in this case and how does recursion apply?

Answer 1

"Computable," "decidable," and "recursive" are all (in this context at least) synonyms by definition, so there's not much to explain mathematically.

Historically, the term "recursive" is used since recursion was the way algorithms were viewed early on; consider e.g. the class of primitive recursive functions, which was isolated significantly before the class of all recursive/computable/decidable functions and which is defined explicitly in terms of a recursion scheme. These days, however, it feels a bit odd since Turing machines don't have any obvious recursion built into their definition, and "computable" is more intuitive. I actually think "decidable" is the most intuitive term, but it's not used nearly as much in this way.