Some Properties of Pseudo-Complements of Recursively Enumerable Sets

Robert A. Di Paola
1966 Transactions of the American Mathematical Society  
Introductory remarks. Those first order systems which exhibit some real mathematical pretensions fall into what is called in [1] the class of arithmetical logics; it is there demonstrated that that any oj-consistent and adequate arithmetical logic is incomplete and brought out that the undecidable sentence can always be taken to be a closed well-formed formula which truly expresses that n0$S where n0 is an integer and S a nonrecursive recursively enumerable set. Thus, we are led to consider
more » ... led to consider those sets of integers whose members are probably (in a system T) in the complement of a given recursively enumerable set S, or, as we shall call them, the pseudo-complements of S, a notion introduced by Davis in [3] . It is to be observed that being a pseudo-complement of S is not a purely extensional property; that is to say, the pseudo-complement of an re (recursively enumerable) set S is not simply a function of S as a set, but also of the particular representation of S in the system T. Different representations of the one set S may give rise to markedly different pseudo-complements even with respect to the same theory T. In this paper we shall explore some of the properties of pseudo-complements of re sets in re consistent extensions of Peano arithmetic. Also, since our definition of a pseudo-complement function provides a natural setting for Davis' theorems, we state his results to achieve comprehensiveness. We prove a separation theorem, Theorem 6, to the effect that if A and B are disjoint re sets, they can be so represented that B is the pseudo-complement of A. From this it easily follows that all re sets are pseudo-complements. The fact that the pseudo-complement of the pseudo-complement is always empty [3] distinguishes sharply the enumeration of the re sets given by a pseudocomplement function from the standard enumerations. Given two numbers, one occurring in a standard enumeration and the other produced by a pseudocomplement function, the problem arises of deciding if these numbers represent
doi:10.2307/1994480 fatcat:dlo6ssfwzbendnzqjzd6f4775i