A mass problem is a set of Turing oracles. If $P$ and $Q$ are mass problems, we say that P is weakly reducible to $Q$ if every member of $Q$ Turing computes a member of $P$. We say that $P$. is strongly reducible to $Q$ if every member of $Q$ Turing computes a member of $P$ via a fixed Turing functional. The weak degrees and strong degrees are the equivalence classes of mass problems under weak and strong reducibility, respectively. We focus on the countable distributive lattices $\mathcal{P}_{\omega}$and $\mathcal{P}_{\mathcal{s}}$of weak and strong degrees of mass problems given by nonempty $\Pi_{1}^{0}$ subsets of $2^{\omega}$. Using an abstract Gödel/Rosser incompleteness property, we characterize the $\Pi_{1}^{0}$ subsets of $2^{\omega}$ whose associated mass problems are of top degree in $\mathcal{P}_{\omega}$ and $\mathcal{P}_{\mathcal{s}}$, respectively. Let R be the set of Turing oracles which are random in the sense of Martin-Löf, and let $\mathbf{r}$ be the weak degree of $R$. We show that $\mathbf{r}$ is a natural intermediate degree within $\mathcal{P}_{\omega}$. Namely, we characterize $\mathbf{r}$as the unique largest weak degree of a $\Pi_{1}^{0}$ subsets of $2^{\omega}$ of positive measure. Within $\mathcal{P}_{\omega}$ we show that $\mathbb{r}$ is meet irreducible, does not join to $\mathbb{1}$, and is incomparable with all weak degrees of nonempty thin perfect $\Pi_{1}^{0}$ subsets of $2^{\omega}$. In addition, we present other natural examples of intermediate degrees in $\mathcal{P}_{\omega}$. We relate these examples to reverse mathematics, computational complexity, and Gentzen-style proof theory.

This communication deals with positive model theory, a non first order model theoretic setting which preserves compactness at the cost of giving up negation. Positive model theory deals transparently with hyperimaginaries, and accommodates various analytic structures which defy direct first order treatment. We describe the development of simplicity theory in this setting, and an application to the lovely pairs of models of simple theories without the weak non finite cover property.

In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of ℝ which is Σ1-definable in (H(ω2),∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω1, ω1*, C, C* where X is any suborder of the reals of size ω1 and C is any Countryman line. Third, the Proper Forcing Axiom implies the Singular Cardinals Hypothesis at κ unless stationary subsets of Sκ+ω reflect. The techniques are expected to be applicable to other open problems concerning the theory of H(ω2).