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Kleene's Amazing Second Recursion Theorem

Published online by Cambridge University Press:  15 January 2014

Yiannis N. Moschovakis
Yiannis N. Moschovakis*
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA Department of Mathematics, University of Athens, Greece, E-mail:, ynm@math.ucla.edu
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Extract

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:

Second Recursion Theorem (SRT). Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+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 : Nm+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)).

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 16 , Issue 2 , June 2010 , pp. 189 - 239
Copyright
Copyright © Association for Symbolic Logic 2010

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