The $\Pi _3$-theory of the computably enumerable Turing degrees is undecidable

Abstract

We show the undecidability of the Π3-theory of the partial order of enumerable Turing degrees. 0. Introduction. Recursively enumerable (henceforth called enumerable) sets arise naturally in many areas of mathematics, for instance in the study of elementary theories, as solution sets of polynomials or as the word problems of finitely generated subgroups of finitely presented groups. Putting the enumerable sets into context with each other in various ways yields structures whose study has for long been a mainstay of computability theory. If the sets are related in the most elementary way, namely by inclusion, one obtains a distributive lattice E with very complex algebraic properties. Another way to compare sets is to look at the information content. Turing reducibility is a very general, but the most widely accepted concept of relative computability: a setX of natural numbers is Turing-reducible to Y iff the answer to “n ∈ X?” can be determined by a Turing machine computation which can use answers to oracle questions “y ∈ Y ?” during the computation. (For more restricted notions of relative computability one would for instance place a priori 1991 Mathematics Subject Classification. 03D25,03D35.