A model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Journal of Mathematical Logic 14(2), 2014]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel’s Lemma for local finite algebras over differentially closed fields.

It is proved in this article that the theory of separably closed nontrivially valued fields of characteristic p > 0 and imperfection degree e > 0 (e ≤ ∞) has quantifier elimination in the language ${{\cal L}_{p,{\rm{div}}}} = \{ + , - , \times ,0,1\} \cup {\{ {\lambda _{n,j}}(x;{y_1}, \ldots ,{y_n})\} _{0 \le n < \omega ,0 \le j < {p^n}}} \cup \{ |\}$; in particular, when e is finite, the corresponding theory has quantifier elimination in the language ${\cal L} = \{ + , - , \times ,0,1\} \cup \{ {b_1}, \ldots ,{b_e}\} \cup {\{ {\lambda _{e,j}}(x;{b_1}, \ldots ,{b_e})\} _{0 \le j < {p^e}}} \cup \{ |\}$.

We consider a group G that does not have the independence property and study the definability of certain subgroups of G, using parameters from a fixed elementary extension G of G. If X is a definable subset of G, its trace on G is called an externally definable subset. If H is a definable subgroup of G, we call its trace on G an external subgroup. We show the following. For any subset A of G and any external subgroup H of G, the centraliser of A, the A-core of H and the iterated centres of H are external subgroups. The normaliser of H and the iterated centralisers of A are externally definable. A soluble subgroup S of derived length ℓ is contained in an S-invariant externally definable soluble subgroup of G of derived length ℓ. The subgroup S is also contained in an externally definable subgroup X ∩ G of G such that X generates a soluble subgroup of G of derived length ℓ. Analogue results are discussed when G is merely a type definable group in a structure that does not have the independence property.

In this paper, we consider Foreman’s maximality principle, which says that any nontrivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We observe that it is consistent that every c.c.c. forcing adds a real and that for every uncountable regular cardinal κ, every κ-closed forcing of size 2<κ collapses some cardinal.

We prove that if the linear-time and polynomial-time hierarchies coincide, then every model of Π1(ℕ) + ¬Ω1 has a proper end-extension to a model of Π1(ℕ), and so Π1(ℕ) + ¬Ω ⊢ BΣ1. Under an even stronger complexity-theoretic assumption which nevertheless seems hard to disprove using present-day methods, Π1(ℕ) + ¬Exp ⊢ BΣ1. Both assumptions can be modified to versions which make it possible to replace Π1(ℕ) by IΔ0 as the base theory.

We also show that any proof that IΔ0 + ¬Exp does not prove a given finite fragment of BΣ1 has to be “nonrelativizing”, in the sense that it will not work in the presence of an arbitrary oracle.

We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

This paper uses the framework of reverse mathematics to investigate the strength of two recurrence theorems of topological dynamics. It establishes that one of these theorems, the existence of an almost periodic point, lies strictly between WKL and ACA (working over RCA0). This is the first example of a theorem with this property. It also shows the existence of an almost periodic point is conservative over RCA0 for ${\rm{\Pi }}_1^1$ -sentences.

We say that a theory T is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence T is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle in a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle in a cone.

Let $ \le _c $ be computable the reducibility on computably enumerable equivalence relations (or ceers). We show that for every ceer R with infinitely many equivalence classes, the index sets $\left\{ {i:R_i \le _c R} \right\}$ (with R nonuniversal), $\left\{ {i:R_i \ge _c R} \right\}$, and $\left\{ {i:R_i \equiv _c R} \right\}$ are ${\rm{\Sigma }}_3^0$ complete, whereas in case R has only finitely many equivalence classes, we have that $\left\{ {i:R_i \le _c R} \right\}$ is ${\rm{\Pi }}_2^0$ complete, and $\left\{ {i:R \ge _c R} \right\}$ (with R having at least two distinct equivalence classes) is ${\rm{\Sigma }}_2^0$ complete. Next, solving an open problem from [1], we prove that the index set of the effectively inseparable ceers is ${\rm{\Pi }}_4^0$ complete. Finally, we prove that the 1-reducibility preordering on c.e. sets is a ${\rm{\Sigma }}_3^0$ complete preordering relation, a fact that is used to show that the preordering relation $ \le _c $ on ceers is a ${\rm{\Sigma }}_3^0$ complete preordering relation.

We show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht–Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.

We prove that the set of formulae provable in the full Lambek calculus with the structural rule of contraction is undecidable. In fact, we show that the positive fragment of this logic is undecidable.

The objective of this article is to characterise elimination of finite generalised imaginaries as defined in [9] in terms of group cohomology. As an application, I consider series of Zariski geometries constructed [10, 23, 24] by Hrushovski and Zilber and indicate how their nondefinability in algebraically closed fields is connected to eliminability of certain generalised imaginaries.

Assume V ≠ L. Then L is not E-Recursively Enumerable in any member of L via a forcing argument inspired by Friedberg [1]. This approach leads to an extension of Levy-Shoenfield absoluteness.

We prove that it is consistent that $\aleph _\omega $ is strong limit, $2^{\aleph _\omega } $ is large and the universality number for graphs on $\aleph _{\omega + 1} $ is small. The proof uses Prikry forcing with interleaved collapsing.

We show that the class of graded c-nilpotent Lie algebras over a fixed field K is closed under free amalgamation. In [1] this result was applied, but its proof was incorrect. In case of a finite field K we obtain a Fraïssé limit of all finite graded c-nilpotent Lie algebras over K. This gives an example for the following more general considerations. The existence of free amalgamation for the age of a Fraïssé limit implies the universality of its automorphism group for all automorphism groups of substructures of that Fraïssé limit. We use [6] and [5].

All spaces are assumed to be separable and metrizable. Our main result is that the statement “For every space X, every closed subset of X has the perfect set property if and only if every analytic subset of X has the perfect set property” is equivalent to b > ω1 (hence, in particular, it is independent of ZFC). This, together with a theorem of Solecki and an example of Miller, will allow us to determine the status of the statement “For every space X, if every Γ subset of X has the perfect set property then every Γ′ subset of X has the perfect set property” as Γ, Γ′ range over all pointclasses of complexity at most analytic or coanalytic.

Along the way, we define and investigate a property of independent interest. We will say that a subset W of 2ω has the Grinzing property if it is uncountable and for every uncountable Y ⊆ W there exists an uncountable collection consisting of uncountable subsets of Y with pairwise disjoint closures in 2ω. The following theorems hold.

1. (1) There exists a subset of 2ω with the Grinzing property.

2. (2) Assume MA + ¬CH. Then 2ω has the Grinzing property.

3. (3) Assume CH. Then 2ω does not have the Grinzing property.

We study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in [3]. We establish that the amalgamation property is equivalent to the disjoint amalgamation property in all coloring classes; find the Hanf number for the amalgamation property for coloring classes; and improve the results of [3] by showing, in ZFC, that the (disjoint) amalgamation property for classes Kα studied in that paper must hold up to ℶα (only a consistency result was previously known).

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map (a Tukey quotient) $\phi :P \to Q$ carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients of each other are said to be Tukey equivalent. Let ${\cal D}_{\rm{}} $ be the partially ordered set of Tukey equivalence classes of directed sets of size $ \le {\rm{}}$. It is shown that ${\cal D}_{\rm{}} $ contains an antichain of size $2^{\rm{}} $, and so has size $2^{\rm{}} $. The elements of the antichain are of the form ${\cal K}\left( M \right)$, the set of compact subsets of a separable metrizable space M, ordered by inclusion. The order structure of such ${\cal K}\left( M \right)$’s under Tukey quotients is investigated. Relative Tukey quotients are introduced. Applications are given to function spaces and to the complexity of weakly countably determined Banach spaces and Gul’ko compacta.

This paper is a contribution to the growing investigation of strong reducibilities between ${\rm{\Pi }}_2^1$ statements of second-order arithmetic, viewed as an extension of the traditional analysis of reverse mathematics. We answer several questions of Hirschfeldt and Jockusch [13] about Weihrauch (uniform) and strong computable reductions between various combinatorial principles related to Ramsey’s theorem for pairs. Among other results, we establish that the principle $SRT_2^2$ is not Weihrauch or strongly computably reducible to $D_{ < \infty }^2$, and that COH is not Weihrauch reducible to $SRT_{ < \infty }^2$, or strongly computably reducible to $SRT_2^2$. The last result also extends a prior result of Dzhafarov [9]. We introduce a number of new techniques for controlling the combinatorial and computability-theoretic properties of the problems and solutions we construct in our arguments.

We prove that the system of strong impredicative NF is not a part of a consistent 3-stratified extension of NF3.