On the lattice of recursively enumerable sets

Abstract

This paper presents some new theorems concerning recursively enumerable (r.e.) sets. The aim of the paper advance the search a decision procedure the elementary theory of r.e. sets. More precisely, an effective method sought deciding whether or not an arbitrary sentence formulated in the lower predicate calculus with sole relative symbol c true of the r.e. sets. The main achievement of the paper the characterisation of the hh-simple sets as those coinfinite r.e. sets whose r.e. supersets form a Boolean algebra. The reader referred Davis's book [1] basic information about the partial recursive (p.r.) functions and about r.e. sets. Other background material required a proper understanding of the present paper consists of [8], [3, Theorem 2], [10, Introduction and ?4], and [5] where the contributions have been listed in their natural order. We take the formulation of the lower predicate calculus given in Abraham Robinson [9]. Natural numbers are denoted by lower case Roman letters and sets of them by lower case Greek letters. The empty set denoted by 0 and the set of all natural numbers by v. The complement of any set a denoted by a'; a called cofinite or coinfinite just if a' finite or infinite respectively. For sets a, / we write aoc3 just if the set (a -3 u (3 a) finite; otherwise we write aZ 3. By function we mean a map of some subset of v x v x **. x v into v; functions will be denoted by upper case Roman letters as will relations on the natural numbers. The informal logical signs used are v &, ->, (x), (Ex), -,> which are be read as or , and , implies , not, for all x, exists x, is equivalent to respectively. Let ,' be a finite class of propositions; then s otherwise sup a be oo. The plan of the paper as follows. In the first section we give a brief discussion of the elementary theory of r.e. sets and prove that its decision problem of the same degree as that of the elementary theory of the lattice obtained by taking the equivalence classes of r.e. sets with respect -. In ?2 we prove the main theorem which states: if a an r.e. subset of an r.e. set / then either there exists a recursive subset 8 of:/ such that a u 8 = 3 or there exists a recursive sequence {8il of disjoint finite subsets of / such that 8i a nonempty all i. This theorem was inspired by