Strong representability of partial functions in arithmetic theories

Abstract

The following is a generalization of a definition due to Mendelson. definition. A partial function ϑ(x 1, …,x n) is said to be strongly representable in a first-order theory K of arithmetic if there is a well-formed formula A(x 1,…,x n,x n+1) with free variables x 1,…,x n, x n+1 satisfying 1. 1. ⊢ K ( x 1)…( x n )(∃! x n+1 ) A( x 1,…, x n , x n+1 ), and 2. 2. for all integers m 1,…, m n, m n+1 ϑ(m 1,…,m n) = m n+1 iff ⊢ KA( m 1 ,…, m n , m n+1) . theorem. Let K be any consistent r.e. extension of Robinson's system, R. Then the class of partial functions strongly representable in K is exactly the class of all partial recursive functions. The key to the proof is the following Exact Separation Theorem due to Putnam and Smullyan and to Shepherdson: Let K be any consistent r.e. extension of Robinson's system, R, and let α and β be any two disjoint recursively enumerable sets. Then there is a well-formed formula B( x) with free variable x satisfying, for all integers n, 1. 1. n ϵ α iff ⊢ KB( n) and 2. 2. n ϵ β iff ⊢ K ~ B ( n) . If the requirement that K be r.e. is replaced by the requirement that K be ω-consistent and incomplete, one can prove, without appealing to the exact separation theorem, that all total recursive functions and all partial recursive functions having at least two integers not in their range are strongly representable in K. Classifying the partial recursive functions by the alternation of quantifiers preceding primitive recursive predicates in the predicates used to strongly represent the function leads, for each theory K, to a classification which has at most four classes. In particular, for (ω-consistent theories K, the ∑ 1 class is a r.e. class of total recursive functions which contains the K-provable recursive functions, although we do not know if the containment is proper.