Transactions of the AMS. Year 1969. Volume 144. Number 00.

Uniformly reflexive structures: On the nature of g\"odelizations and relative computability
Eric G. Wagner
1-41

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.

Henselization of a ring with respect to an ideal
Silvio Greco
43-65

Smooth homology spheres and their fundamental groups
Michel A. Kervaire
67-72

Local and global subordination theorems for vector-valued analytic functions
Mark Finkelstein; Robert Whitley
73-76

Two notes on recursive functions and regressive isols
Joseph Barback
77-94

Moduli for special Riemann surfaces of genus $2$
John Schiller
95-113

On the closure of the bicyclic semigroup
Carl Eberhart; John Selden
115-126

Nonsingular sections to Euclidean bundles
Kenneth C. Millett
127-143

Deceptive convergence of Fourier series on ${\rm SU}(2)$
R. A. Mayer
145-172

The Hauptvermutung for $3$-complexes
Edward M. Brown
173-196

Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients
James S. W. Wong
197-215

Lie algebras of characteristic $2$
Ronald C. Hamelink
217-233

\v Ceby\v sev sets in Hilbert space
Edgar Asplund
235-240