Who first discovered that some R.E. sets are not recursive?

Question

Who first discovered that some recursively enumerable sets are not recursive, or equivalently that some semidecidable sets are undecidable? And in what context? Was the earliest formulation of this idea phrased in terms of partial recursive functions? Or something else?

Answer 1

According to the paper of Emil Post "Recursively Enumerable Sets of Positive Integers and Their Decision Problems" (1944):

Basic to the entire theory is the following result we must credit to Church, Rosser, Kleene, jointly.

THEOREM. There exists a recursively enumerable set of positive integers which is not recursive.

Cited are three papers all from 1936:

• "An unsolvable problem of elementary number theory" (Church)
• "General recursive functions of natural numbers" (Kleene)
• "Extensions.of some theorems of Gödel and Church"(Rosser)

The Post paper and its three predecessors are all collected in Martin Davis's anthology The Undecidable.

Answer 2

Emil Post, in his paper "Recursively Enumerable Sets of Positive Integers and Their Decision Problems" (1944). This can be found in the book "The Undecidable: Basic Papers On Undecidable Propositions, Unsolvable Problems, and Computable Functions", Martin Davis ed., pp. 305-37.