The proof of part 1 of the first recursion theorem is obtained by iterating the enumeration operator Φ beginning with the empty set. First, a sequence Fk is constructed, for k = 0, 1, \ldots. Let F0 be the empty set. Proceeding inductively, for each k, let Fk + 1 be F_k \cup \Phi(F_k). Finally, F is taken to be \bigcup F_k. The remainder of the proof consists of a verification that F is recursively enumerable and is the least fixed point of Φ. The sequence Fk used in this proof corresponds to the Kleene chain in the proof of the Kleene fixed-point theorem.
