Alternative proof of a theorem of Kleene

Extract

We offer a simple alternative proof of Theorem II, Kleene [1]:

Theorem. For certain primitive recursive -predicates R and S,

We separate the definitions of < o and O:

Definition 1. (1) 1 <_o2; (2) for all y, if 1 <_oy, then y <_o 2^v; (3) for all e, if (n)(Ey)T(e, n_o, y) and (n)(e(n_o) <_oe((n+1)o)), then (n)(e(n_o) <^o 3·5^e); (4) for all x, y, z, if x <_oy, y <_oz, then x <_oz; (5) a <_ob only as required by (1)–(4).

Definition 2. a ϵ O if and only if 1 <_o 2^α.

If we write α(2^a·3^b) = O for a <_ob in Conditions (1) to (4) of Definition 1 and take their conjunction, we get an arithmetic predicate P(α). By the Frege-Dedekind device for handling inductive definitions,

Hence, by Theorem 1 (Part 2) of [2], we have a primitive recursive S such that a <_o b ≡ (α)(Ey)S(a, b, α, y). By Definition 2,

The classical use of the function quantifier (α) in the above proof is conspicuous. It is not clear that the constructive or intuitionistic content of Kleene's proof (cf. [1] Footnote 40) can be obtained by this method.