Interpreting true arithmetic in the local structure of the enumeration degrees

Degree theory studies mathematical structures, which arise from a formal notion of reducibility between sets of natural numbers based on their computational strength. In the past years many reducibilities, together with their induced degree structures have been investigated. One of the aspects in these investigations is always the characterization of the theory of the studied structure. One distinguishes between global structures, ones containing all possible degrees, and local structures, containing all degrees bounded by a fixed element, usually the degree which contains the halting set. It has become apparent that modifying the underlying reducibility does not influence the strength of the first order theory of either the induced global or the induced local structure. The global first order theories of the Turing degrees [15], of the many-one degrees [11], of the 1-degrees [11] are computably isomorphic to second order arithmetic. The local first order theories of the computably enumerable degrees , of the many-one degrees [12], of the ∆2 Turing degrees [14] are computably isomorphic to the theory of first order arithmetic. In this article we consider enumeration reducibility, and the induced structure of the enumeration degrees. Enumeration reducibility introduced by Friedberg and Rogers [4] arises as a way to compare the computational strength of the positive information contained in sets of naturals numbers. A set A is enumeration reducible to a set B if given any enumeration of the set B, one can effectively compute an enumeration of the set A. The induced structure of the enumeration degrees De is an upper semilattice with least element and jump operation. This structure raises particular interest as it can be viewed as an extension of the structure of the Turing degrees. There is an isomorphic copy of the the Turing degrees in De. The elements of this copy are called the total enumeration degrees. The enumeration jump operation gives rise to a local substructure, Ge,consisting of all degrees in the interval enclosed by the least degree and its first jump. Cooper [1] shows that the elements of Ge are precisely the enumeration degrees which contain Σ2 sets, or equivalently are made up entirely of Σ 0 2 sets, which we call Σ 0 2 degrees. This structure can in turn be viewed as an extension of the structure of the ∆2 Turing degrees, which is isomorphic to the Σ 0 2 total degrees. Slaman and Woodin [16] prove that the theory of the global structure of the enumeration degrees, De, is computably isomorphic to the theory of second order arithmetic and show that the local theory is undecidable. In the same article