Uniformly reflexive structures: An axiomatic approach to computability

The concept of a Uniformly Reflexive Structure (U.R.S.) is developed as an abstract approach to computability. An axiomatic characterization of a Godelization is given such that any structure satisfying these axioms is a U.R.S. The fundamental lemmata of the subject and the relationships between U.R.S. and the set of computable functions is explored. Theorems analogous to the basic theorems of computability such as the iteration theorem, Kleene's second recursion theorem are established for U.R.S. Each U.R.S. contains a model N of the non-negative integers such that the restriction to N of the functions of U.R.S. contains all recursive functions. From the proof that the partial recursive functions form a U.R.S. we obtain a new Godelization of these functions.