Why the name Recursively Enumerable and Recursive?

Question

Why did the sets of languages accepted/decided by a TM get the name Recursively Enumerable and Recursive, respectively?

Answer 1

A recursive set is synonymous with computable set. In the beginning of computer science, the notion of computer had not been completely fleshed out, and there were various ideas on the matter. These ran from Turing's machines, Church's Lambda calculus, and Kleene-Gödels recursive function theory. The idea is a set is computable if its elements can be decided by a (terminating) algorithm.

Recursively enumerable sets are more general. It turns out that not every subset of the natural numbers is decidable by an algorithm. However, for some sets, we can give a procedure (possibly non-terminating) that will enumerate the elements of the set (enumerate is used because each number appears exactly once thus a bijection from the natural numbers to the set in question is defined). This is equivalent to saying that the set is semi-decidable in more modern terms.