• DOI:10.1090/S0002-9947-1968-0227009-1
• Corpus ID: 5622297

ON THE LATTICE OF RECURSIVELY ENUMERABLE SETS

@article{Lachlan1968ONTL,
  title={ON THE LATTICE OF RECURSIVELY ENUMERABLE SETS},
  author={Alistair H. Lachlan},
  journal={Transactions of the American Mathematical Society},
  year={1968},
  volume={130},
  pages={1-37}
}

• A. Lachlan
• Published 1968
• Mathematics
• Transactions of the American Mathematical Society

This paper presents some new theorems concerning recursively enumerable (r.e.) sets. The aim of the paper is to advance the search for a decision procedure for the elementary theory of r.e. sets. More precisely, an effective method is sought for deciding whether or not an arbitrary sentence formulated in the lower predicate calculus with sole relative symbol c is true of the r.e. sets. The main achievement of the paper is the characterisation of the hh-simple sets as those coinfinite r.e. sets whose r.e. supersets form a Boolean algebra. The reader is referred to Davis's book [1] for basic information about the partial recursive (p.r.) functions and about r.e. sets. Other background material required for 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 is denoted by 0 and the set of all natural numbers by v. The complement of any set a is denoted by a'; a is called cofinite or coinfinite just if a' is finite or infinite respectively. For sets a, / we write aoc3 just if the set (a -3 u (3 a) is 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 to be read as " or ", " and ", " implies ", "not", "for all x", "there exists x", "is equivalent to" respectively. Let ,' be a finite class of propositions; then s otherwise sup a is to be oo. The plan of the paper is 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 is 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 to -. In ?2 we prove the main theorem which states: if a is 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 is nonempty for all i. This theorem was inspired by