This paper is organized as follows. After the preliminaries, we investigate in Section 2 the subject of dominating functions in models of # 2 bounding. We show that there is a family of total recursive functions indexed by a proper # 2 cut such that no total recursive function eventually dominates every function in the family. This result is optimal in the sense that there is no bounded # 2 family of total recursive functions such that any total recursive function is eventually dominated by one in the family. Apart from the intrinsic interest provided by such combinatorial properties, the method used in the proof is later adapted to show that no minimal pairs exist in any model of # 2 bounding without # 2 induction. In Section 3 we show that minimal pairs exist in every model of # 2 induction. In Section 4 we show that there is no minimal pair in any model of # 2 bounding in which # 2 induction fails. In the final section, we return to the problem of Shoenfield's Conjecture, and show that even with the failure of the minimal pair theorem, the conjecture is still refuted within the theory of # 2 bounding. We provide two examples, one involves only meet operator, the other only join operator. We end by posing a number of open problems.

We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than Π1 1 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore [2007] that are not Π1 1 conservative over RCA0. A prior-ity argument with Shore blocking shows that it is also Π 1 1-conservative over BΣ2. We also provide a theorem provable by a finite injury priority argument that is conservative over IΣ1 but implies IΣ2 over BΣ2, and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the ω-model consisting of the recursive sets.

Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK

Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of α-finiteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the α-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of α-r.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the α-r.e. degrees is complicated, we get that for every admissible ordinal