The Theory of the Recursively Enumerable Weak Truth-Table Degrees Is Undecidability

We show that the partial order of Zo-sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi. The upper semilattice R.,, of r.e. weak truth-table (wtt) degrees has been investigated recently by several authors. Yet the question whether its elementary theory is undecidable, as posed first in [Od81], remained open. The first undecidability proof for the theory of the r.e. Turing degrees was announced in [Ha,Sh82]; a simpler one is presented in [ASp,Sh?]. The undecidability of the theory of the r.e. tt-degrees is proved in [Ht,S90]. However, the methods used in these proofs cannot be applied to establish the undecidability of Th(R.tt), since the r.e. wtt-degrees form a distributive semilattice. In this paper, we will show that the partial order &3 of Z?-sets under inclusion is elementarily definable with parameters (e.d.p.) in R.,, using distributivity in an essential way. The idea is to let Z0-sets correspond to certain ideals of Rwtt. These ideals can be represented by pairs of wtt-degrees, a fact which makes it possible to talk about them in the language of Rwtt. After this, the undecidability of Th(Rwtt) can be deduced, using a general model theoretic theorem and E. Herrmann's result that all recursive Boolean pairs are, in a uniform way, e.d.p. in g3. Outline of the paper. The undecidability proof is split up into its model theoretic, algebraic and recursion theoretic components. In the first section, we show that the elementary definability with parameters of S3 in Rwtt implies the undecidability of Th(Rwtt). The next section provides algebraic lemmas about upper semilattices. In ?3, we prove two recursion theoretic results about the r.e. wtt-degrees. Finally, in ?4, all this material is combined. ?0. Definitions, notation and conventions. An upper semilattice with least element (u.s.l.) is a structure P = (P; <, v, 0) such that 0 is the least element of the partial Received March 5, 1991; revised June 24, 1991. Part of this research was done while the authors visited the Mathematical Sciences Research Institute, in Berkeley, California. The first and the third author were supported by MSRI under NSF grant DMS8505550, the third author in addition by NSF grant DMS-8912797. The second author was supported by the DAAD (Deutscher Akademischer Austauschdienst). ?1992, Association for Symbolic Logic 0022-4812/92/5703-0008/$02.10