Coding in the Partial Order of Enumerable Sets

Abstract We develop methods for coding with first-order formulas into the partial order E of enumerable sets under inclusion. First we use them to reprove and generalize the (unpublished) result of the first author that the elementary theory of E has the same computational complexity as the theory of the natural numbers. Relativized versions of the coding methods show that the p.o. of Σ 0 p and Σ 0 q sets are not elementarily equivalent for natural numbers p ≠ q . As a further application, definability of the class of quasimaximal sets in E is obtained. On the other side, we prove theorems limiting coding and definability in E , thereby establishing a sharp contrast between E and other structures occurring in computability theory.