We answer a question of Garcia-Ferreira and Hrušák by consistently constructing a MAD family maximal in the Katětov order. We also answer several questions of Garcia-Ferreira.

A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.

The Axiom of Projective Determinacy implies the existence of a universal set for every n ≥ 1. Assuming there exists a universal set. In ZFC there is a universal set for every α.

If G is a group with a supersimple theory having a finite SU-rank, then the subgroup of G generated by all of its normal nilpotent subgroups is definable and nilpotent. This answers a question asked by Elwes, Jaligot, Macpherson and Ryten. If H is any group with a supersimple theory, then the subgroup of H generated by all of its normal soluble subgroups is definable and soluble.

In this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no model-complete theory of II1 factors.

We prove that for a finitely generated infinite nilpotent group G with structure (G, ·, …), the connected component G*0 of a sufficiently saturated extension G* of G exists and equals

We construct an expansion of ℤ by a predicate (ℤ, +, P) such that the type-connected component is strictly smaller than ℤ*0. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for the van der Waerden theorem for finite partitions of groups.

We give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group G(M) on its “external type space” SG.ext(M), can explain, account for, or give rise to, the quotient G/G00, at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or f sg) groups in NIP theories. As part of our analysis we show the existence of “externally definable” generics of G(M) for measure-stable groups. We also point out that for G definably amenable (in a NIP theory) G/G00 can be recovered, via the Ellis theory, from a natural Ellis semigroup structure on the space of global f-generic types.

In contrast with the notion of complexity, a set A is called anti-complex if the Kolmogorov complexity of the initial segments of A chosen by a recursive function is always bounded by the identity function. We show that, as for complexity, the natural arena for examining anti-complexity is the weak-truth table degrees. In this context, we show the equivalence of anti-complexity and other lowness notions such as r.e. traceability or being weak truth-table reducible to a Schnorr trivial set. A set A is anti-complex if and only if it is reducible to another set B with tiny use, whereby we mean that the use function for reducing A to B can be made to grow arbitrarily slowly, as gauged by unbounded nondecreasing recursive functions. This notion of reducibility is then studied in its own right, and we also investigate its range and the range of its uniform counterpart.

We study low level nondefinability in the Turing degrees. We prove a variety of results, including, for example, that being array nonrecursive is not definable by a Σ1 or Π1 formula in the language (≤, REA) where REA stands for the “r.e. in and above” predicate. In contrast, this property is definable by a Π2 formula in this language. We also show that the Σ1-theory of (, ≤, REA) is decidable.

It has been known for six years that the restriction of Girard's polymorphic system F to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait's method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each β-reduction step of the full intuitionistic propositional calculus translates into one or more βη-reduction steps in the restricted Girard system. As a consequence, we obtain a novel and perspicuous proof of the strong normalization property for the full intuitionistic propositional calculus. It is noticed that this novel proof bestows a crucial role to η-conversions.

The following question is open: Does there exist a hyperarithmetic class of computable structures with exactly one non-hyperarithmetic isomorphism-type? Given any oracle α ∈ 2ω, we can ask the same question relativized to α. A negative answer for every α implies Vaught's Conjecture for Lω1ω.

We introduce the notion of skinniness for subsets of and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or 2λ-saturation of NSκλ ∣ X, where NSκλ denotes the non-stationary ideal over , implies the existence of a skinny stationary subset of X. We also show that if λ is a singular cardinal, then there is no skinnier stationary subset of . Furthermore, if λ is a strong limit singular cardinal, there is no skinny stationary subset of . Combining these results, we show that if λ is a strong limit singular cardinal, then NSκλ ∣ X can satisfy neither precipitousness nor 2λ-saturation for every stationary X ⊆ . We also indicate that , where , is equivalent to the existence of a skinnier (or skinniest) stationary subset of under some cardinal arithmetical hypotheses.

We give results exploring the relationship between dominating and unbounded reals in Hechler extensions, as well as the relationships among the extensions themselves. We show that in the standard Hechler extension there is an unbounded real which is dominated by every dominating real, but that this fails to hold in the tree Hechler extension. We prove a representation theorem for dominating reals in the standard Hechler extension: every dominating real eventually dominates a sandwich composition of the Hechler real with two ground model reals that monotonically converge to infinity. We apply our results to negatively settle a conjecture of Brendle and Löwe (Conjecture 15 of [4]). We also answer a question due to Laflamme.

We characterize the expressive power of extensions of Dependence Logic and Independence Logic by monotone generalized quantifiers in terms of quantifier extensions of existential second-order logic.

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every n ≥ 2 and μ ≥ ℕn, we have (ℕn, μ)-ITP.

A seminal theorem due to Weyl [14] states that if (an) is any sequence of distinct integers, then, for almost every x ∈ ℝ, the sequence (anx) is uniformly distributed modulo one. In particular, for almost every x in the unit interval, the sequence (anx) is uniformly distributed modulo one for every computable sequence (an) of distinct integers. Call such an x UD random. Here it is shown that every Schnorr random real is UD random, but there are Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random.