Naming and Diagonalization, from Cantor to Godel to Kleene

Abstract

We trace self-reference phenomena to the possibility of naming functions by names that belong to the domain over which the functions are defined. A naming system is a structure of the form (D, type( ),{ }), where D is a non-empty set; for every a∈ D, which is a name of a k-ary function, {a}: Dk → D is the function named by a, and type(a) is the type of a, which tells us if a is a name and, if it is, the arity of the named function. Under quite general conditions we get a fixed point theorem, whose special cases include the fixed point theorem underlying Gödel's proof, Kleene's recursion theorem and many other theorems of this nature, including the solution to simultaneous fixed point equations. Partial functions are accommodated by including “undefined” values; we investigate different systems arising out of different ways of dealing with them. Many-sorted naming systems are suggested as a natural approach to general computatability with many data types over arbitrary structures. The first part of the paper is a historical reconstruction of the way Gödel probably derived his proof from Cantor's diagonalization, through the semantic version of Richard. The incompleteness proof–including the fixed point construction–result from a natural line of thought, thereby dispelling the appearance of a “magic trick”. The analysis goes on to show how Kleene's recursion theorem is obtained along the same lines.