Uniformly reflexive structures: On the nature of gödelizations and relative computability

Abstract:

In this paper we present an axiomatic theory within which much of the theory of computability can be developed in an abstract manner. The paper is based on the axiomatically defined concept of a Uniformly Reflexive Structure (U.R.S.). The axioms are chosen so as to capture what we view to be the essential properties of a “gödelization” of a set of functions on arbitrary infinite domain. It can be shown that (with a “standard gödelization") both the partial recursive functions and the meta-recursive functions satisfy the axioms of U.R.S. In the first part of this paper, we define U.R.S. and develop the basic working theorems of the subject (e.g., analogues of the Kleene recursion theorems). The greater part of the paper is concerned with applying these basic results to (1) investigating the properties of gödelizations, and (2) developing an intrinsic theory of relative computability. The notion of relative computability which we develop is equivalent to Turing reducibility when applied to the partial recursive functions. Applied to appropriate U.R.S. on arbitrary domains, it provides an upper-semi-lattice ordering on the set of all functions (both total and partial) on that domain.