We give a first-order coding without parameters of a copy of in the computably enumerable weak truth table degrees. As a tool, we develop a theory of parameter definable subsets.

We show that the elementary theory of the recursively enumerable tt-degrees has the same computational complexity as true first-order arithmetic. As auxiliary results, we prove theorems about exact pairs and initial segments in the tt-degrees.

We study connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We say that a class of computably enumerable sets bounded iff there is an m-incomplete computably enumerable set A such that every set in is m-reducible to A. For example, we show that the class of effectively simple sets is bounded; but the class of maximal sets is not. Furthermore, the class of computably enumerable sets Turing reducible to a computably enumerable set B is bounded (...) iff B is low2. For r = bwtt,tt,wtt and T, there is a bounded class intersecting every computably enumerable r-degree; for r = c, d and p, no such class exists. (shrink)

It will be shown that for all numbers n and m with n > m 1 the Boolean pairs have undecidable elementary theories.

A splitting of an r.e. set A is a pair A1, A2 of disjoint r.e. sets such that A1 A2 = A. Theorems about splittings have played an important role in recursion theory. One of the main reasons for this is that a splitting of A is a decomposition of A in both the lattice, , of recursively enumerable sets and in the uppersemilattice, R, of recursively enumerable degrees . Thus splitting theor ems have been used to obtain results about (...) the structure of , the structure of R, and the relationship between the two structures. Furthermore it is fair to say that questions about splittings have often generated important new technical developments in recursion theory. In this article we survey most of the results and techniques associated with splitting properties of r.e. sets in ordinary recursion theory. (shrink)