Kleene's Amazing Second Recursion Theorem

This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene [1938]. In modern notation, with all the hypotheses stated explicitly and in a strong (uniform) form, it reads as follows: S econd R ecursion T heorem (SRT). Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φ n : ℕ 1+ n ⇀ V is a recursive partial function of (1 + n) arguments with values in V so that the standard assumptions (a) and (b) hold with (a) Every n-ary recursive partial function with values in V is for some e . (b) For all m, n, there is a recursive function : N m+1 → ℕ such that Then, for every recursive, partial function f of (1+m+n) arguments with values in V, there is a total recursive function of m arguments such that Proof. Fix e ϵ ℕ such that and let . We will abuse notation and write ž; rather than ž() when m = 0, so that (1) takes the simpler form in this case (and the proof sets ž = S ( e, e )).