Kleene ’ s Amazing Second Recursion Theorem Extended

Kleene states the theorem with V = N, relative to specific φ, S n , supplied by his Enumeration Theorem, m = 0 (no parameters ~y) and n ≥ 1, i.e., not allowing nullary partial functions. And most of the time, this is all we need; but there are a few important applications where choosing “the right” φ, S n , restricting the values to a proper V ( N or allowing m > 0 or n = 0 simplifies the proof considerably. With V = { 0} (singleton 0) andm = n = 0, for example, the characteristic equation {z̃}( ) = f(z̃) (2)