Scanlon [5] proves Ax-Kochen-Ershov type results for differential-henselian monotone valued differential fields with many constants. We show how to get rid of the condition with many constants.

Using convex subrings of *$\mathbb{C}$, a nonstandard extension of $\mathbb{C}$, we define several kinds of complex bounded polynomials and we provide their associated analytic functions obtained by taking the quasistandard part.

We present a much-shortened proof of a major result (originally due to Vorob’ev) about intuitionistic propositional logic: in essence, a correction of our 1992 article, avoiding several unnecessary definitions.

We modify the definable ultrapower construction of Kanovei and Shelah (2004) to develop a ZF-definable extension of the continuum with transfer provable using countable choice only, with an additional mild hypothesis on well-ordering implying properness. Under the same assumptions, we also prove the existence of a definable, proper elementary extension of the standard superstructure over the reals.

In a previous article, the author introduced the idea of intrinsic density—a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (${\bf{a}}\prime { \ge _{\rm{T}}}\emptyset \prime \prime$) or compute a diagonally noncomputable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every noncomputable degree.

We also show that the former result holds in the sense of reverse mathematics, in that (over RCA0) the existence of a dominating or diagonally noncomputable function is equivalent to the existence of a set with intrinsic density 0.

We consider two generalizations of the recursion theorem, namely Visser’s ADN theorem and Arslanov’s completeness criterion, and we prove a joint generalization of these theorems.

Let ${\cal S}$ be a Scott set, or even an ω-model of WWKL. Then for each A ε S, either there is X ε S that is weakly 2-random relative to A, or there is X ε S that is 1-generic relative to A. It follows that if A1,…,An ε S are noncomputable, there is X ε S such that each Ai is Turing incomparable with X, answering a question of Kučera and Slaman. More generally, any ∀∃ sentence in the language of partial orders that holds in ${\cal D}$ also holds in ${{\cal D}^{\cal S}}$, where ${{\cal D}^{\cal S}}$ is the partial order of Turing degrees of elements of ${\cal S}$.

Let ${\cal M}$ be ternary, homogeneous and simple. We prove that if ${\cal M}$ is finitely constrained, then it is supersimple with finite SU-rank and dependence is k-trivial for some k < ω and for finite sets of real elements. Now suppose that, in addition, ${\cal M}$ is supersimple with SU-rank 1. If ${\cal M}$ is finitely constrained then algebraic closure in ${\cal M}$ is trivial. We also find connections between the nature of the constraints of ${\cal M}$, the nature of the amalgamations allowed by the age of ${\cal M}$, and the nature of definable equivalence relations. A key method of proof is to “extract” constraints (of ${\cal M}$) from instances of dividing and from definable equivalence relations. Finally, we give new examples, including an uncountable family, of ternary homogeneous supersimple structures of SU-rank 1.