We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SP-generics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SP-generics, ULIN ∩ co-ULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ co-NP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩co-NP/1 ̸ ⊆ (NP∩co-NP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.

We introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the Berman-Hartmanis isomorphism conjecture [BH77] holds relative to any sp-generic oracle, i.e., for any symmetric perfect generic set A, all NP^A-complete sets are polynomial-time isomorphic relative to A. Prior to this work there were no known oracles relative to which the isomorphism conjecture held. As part of our proof that the isomorphism conjecture holds relative to symmetric perfect generic sets we also show that P A = FewP A for any symmetric perfect generic A.

. A major focus of research in computability theory in recent years has involved denability issues in degree structures. There has been much success in getting general results by coding methods that translate rst or second order arithmetic into the structures. In this paper we concentrate on the issues of getting denitions of interesting, apparently external, relations on degrees that are order-theoretically natural in the structures D and R of all the Turing degrees and of the r.e. Turing degrees, respectively. Of course, we have no formal denition of natural but we oer some guidelines, examples and suggestions for further research. 1. Introduction A major focus of research in computability theory in recent years has involved denability issues in degree structures. The basic question is, which interesting apparently external relations on degrees can actually be dened in the structures themselves, that is, in the rst order language with the single fundamental relation...

By RCA0, we denote the system of second order arithmetic based on recursive com-prehension axioms and Σ 0 1 induction. WKL0 is defined to be RCA0 plus weak König’s lemma: every infinite tree of sequences of 0’s and 1’s has an infinite path. In this paper, we first show that for any countable model M of RCA0, there exists a countable model M ′ of WKL0 whose first order part is the same as that of M, and whose second order part consists of the M-recursive sets and sets not in the second order part of M. By combining this fact with a certain forcing argument over universal trees, we obtain the following result (which has been called Tanaka’s conjecture): if WKL0 proves ∀X∃!Yϕ(X, Y) with ϕ arithmetical, so does RCA0. We also discuss several improvements of this results.

We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, particularly recursion theory, over the past thirty years through his research, writing and teaching. Here, I would like to concentrate on just one instance of that influence that I feel has been of special significance to the study of the degrees of unsolvability in general and on my own work in particular--- the conjectures and questions posed at the end of the two editions of Sacks's first book, the classic monograph Degrees of Unsolvability (Annals

Assuming that all co-analytic games on integers are determined (or equivalently that all reals have "sharps"), we prove (Theorem 1.3) in ZFC that either \Delta 1 2 -Determinacy holds, or that there is a real d 0 , so that if D = fdng is a countable collection of \Delta 1 3 degrees above that of d 0 , then D has a \Delta 1 3 -minimal upper bound. 1 1 Introduction The main theorem of [12] is the following: Theorem 1.1 (ZFC+8r 2 R (r # exists) +: 0 y ) Every countable set of \Delta 1 3 degrees has a minimal upper bound. We say that for reals f; g 2 ! ! f is \Delta 1 3 in g, or f 3 g, if there are \Sigma 1 3 relations \Phi; \Psi expressible in second order number theory so that f(n) = m , \Phi(n; m; g) , :\Psi(n; m; g). This is a reducibility ordering, and setting f = 3 g , f 3 g & g 3 f we have that = 3 is an equivalence relation on ! ! . We let f denote the equivalence class [f ] =3 . In general, when r is a reducibility ordering (meaning r is a transitive, reflexive...

This is a set of lecture notes from a 15-week graduate course at the Penn-sylvania State University taught as Math 574 by Stephen G. Simpson in Spring 2004. The course was intended for students already familiar with the basicsof mathematical logic. The course covered some topics which are important in contemporary mathematical logic and foundations but usually omitted fromintroductory courses at Penn State.