a1 Institute for Advanced Study and Stanford University, Ohio State University
We deal in the following with certain theories S, by which we mean sets of sentences closed under logical deduction. The basic logic is understood to be the classical one, but we place no restriction on the orders of the variables to be used. However, we do assume that we can at least express certain notions from classical first-order number theory within these theories. In particular, there should correspond to each primitive recursive function ξ a formula φ(χ), where ‘x’ is a variable ranging over natural numbers, such that for each numeral ñ, φ(ñ) expresses in the language of S that ξ(η) = 0. Such formulas, when obtained say by the Gödel method of eliminating primitive recursive definitions in favor of arithmetical definitions in +. ·. are called PR-formulas (cf.  §2 (C)).
(Received August 08 1960)
1 The main results of this paper were communicated to the International Congress for Logic, Methodology and Philosophy of Science at Stanford University, August 24-September 2, 1960.
We wish to thank Professors K. Gödel and G. Kreisel for helpful comments on a draft of this paper.
2 National Science Foundation Fellow at Institute for Advanced Study, 1959–60. The research of this author was sponsored in part by the Office of Ordnance Research under Contract DA-04-200-ORD-997 at Stanford University.
3 The research of the second author (who died July 29, 1961) was supported by a grant from the National Science Foundation of the U.S.