We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th (ℤ , +, 1, 0, Pr) where Pr is a predicate for the prime numbers and their negations is decidable, unstable, and supersimple. This is in contrast with Th (ℤ , +, 0, Pr, <) which is known to be undecidable by the works of Jockusch, Bateman, and Woods.

Assuming that every set is constructible, we find a ${\text{\Pi }}_1^1 $ maximal cofinitary group of permutations of $\mathbb{N}$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a ${\text{\Pi }}_1^1 $ maximal cofinitary group in L.

Suppose that ${\cal M}$ is countable, binary, primitive, homogeneous, and simple. We prove that the SU-rank of the complete theory of ${\cal M}$ is 1 and hence 1-based. It follows that ${\cal M}$ is a random structure. The conclusion that ${\cal M}$ is a random structure does not hold if the binarity condition is removed, as witnessed by the generic tetrahedron-free 3-hypergraph. However, to show that the generic tetrahedron-free 3-hypergraph is 1-based requires some work (it is known that it has the other properties) since this notion is defined in terms of imaginary elements. This is partly why we also characterize equivalence relations which are definable without parameters in the context of ω-categorical structures with degenerate algebraic closure. Another reason is that such characterizations may be useful in future research about simple (nonbinary) homogeneous structures.

We show that there is a strong connection between Weihrauch reducibility on one hand, and provability in EL0, the intuitionistic version of RCA0, on the other hand. More precisely, we show that Weihrauch reducibility to the composition of finitely many instances of a theorem is captured by provability in EL0 together with Markov’s principle, and that Weihrauch reducibility is captured by an affine subsystem of EL0 plus Markov’s principle.

Given a metric space (X , d), equipped with a locally finite Borel measure, a measurable set $A \subseteq X$ is a density set if the points where A has density 1 are exactly the points of A. We study the topological complexity of the density sets of the real line with Lebesgue measure, with the tools—and from the point of view—of descriptive set theory. In this context a density set is always in $\Pi _3^0$. We single out a family of true $\Pi _3^0$ density sets, an example of true $\Sigma _2^0$ density set and finally one of true $\Pi _2^0$ density set.

Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied. In particular, it is shown that all notions coincide for non-multidimensional theories where the dimensions are associated to strongly minimal types.

In this article, a notion of Schauder equivalence relation ℝℕ/L is introduced, where L is a linear subspace of ℝℕ and the unit vectors form a Schauder basis of L. The main theorem is to show that the following conditions are equivalent:

(1) the unit vector basis is boundedly complete;

(2) L is a Fσ in ℝℕ;

(3) ℝℕ/L is Borel reducible to ℓ∞.

We show that any Schauder equivalence relation generalized by a basis of ℓ2 is Borel bireducible to ℝℕ/ℓ2 itself, but it is not true for bases of c0 or ℓ1. Furthermore, among all Schauder equivalence relations generated by sequences in c0, we find the minimum and the maximum elements with respect to Borel reducibility.

We use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures in relational languages, in which the class of algebraically closed substructures is closed under free amalgamation. We show that any free amalgamation theory has elimination of hyperimaginaries and weak elimination of imaginaries. With this result, we use several families of well-known homogeneous structures to give new examples of rosy theories. We then prove that, for free amalgamation theories, simplicity coincides with NTP2 and, assuming modularity, with NSOP3 as well. We also show that any simple free amalgamation theory is 1-based. Finally, we prove a combinatorial characterization of simplicity for Fraïssé limits with free amalgamation, which provides new context for the fact that the generic Kn-free graphs are SOP3, while the higher arity generic $K_n^r$-free r-hypergraphs are simple.

We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to definable families of cuts. Along the way we give some new examples of weakly o-minimal nonvaluational structures.

Measuring says that for every sequence ${\left( {{C_\delta }} \right)_{\delta < {\omega _1}}}$ with each ${C_\delta }$ being a closed subset of δ there is a club $C \subseteq {\omega _1}$ such that for every $\delta \in C$ , a tail of $C\mathop \cap \nolimits \delta$ is either contained in or disjoint from ${C_\delta }$ . We answer a question of Justin Moore by building a forcing extension satisfying measuring together with ${2^{{\aleph _0}}} > {\aleph _2}$ . The construction works over any model of ZFC + CH and can be described as a finite support forcing iteration with systems of countable structures as side conditions and with symmetry constraints imposed on its initial segments. One interesting feature of this iteration is that it adds dominating functions $f:{\omega _1} \to {\omega _1}$ mod. countable at each stage.

We present abstract complexity results about Coquand and Hyland–Ong game semantics, that will lead to new bounds on the length of first-order cut-elimination, normalization, interaction between expansion trees and any other dialogical process game semantics can model and apply to. In particular, we provide a novel method to bound the length of interactions between visible strategies and to measure precisely the tower of exponentials defining the worst-case complexity. Our study improves the old estimates on average by several exponentials.

Starting from large cardinals we construct a model of ZFC in which the GCH fails everywhere, but such that GCH holds in its HOD. The result answers a question of Sy Friedman. Also, relative to the existence of large cardinals, we produce a model of ZFC + GCH such that GCH fails everywhere in its HOD.

For a group G definable in a first order structure M we develop basic topological dynamics in the category of definable G-flows. In particular, we give a description of the universal definable G-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of G, that is to the quotient ${G^{\rm{*}}}/G_M^{{\rm{*}}00}$ (where G* is the interpretation of G in a monster model). More generally, we obtain these results locally, i.e., in the category of Δ-definable G-flows for any fixed set Δ of formulas of an appropriate form. In particular, we define local connected components $G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ and $G_{{\rm{\Delta }},M}^{{\rm{*}}000}$, and show that ${G^{\rm{*}}}/G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ is the Δ-definable Bohr compactification of G. We also note that some deeper arguments from [14] can be adapted to our context, showing for example that our epimorphism from the Ellis group to the Δ-definable Bohr compactification factors naturally yielding a continuous epimorphism from the Δ-definable generalized Bohr compactification to the Δ-definable Bohr compactification of G. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.

We consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes ${\cal D}_2 \left( {{\bf{\Pi }}_3^0 } \right)$ and ${\cal D}_\omega \left( {{\bf{\Pi }}_3^0 } \right)$, that is, the class of sets which are 2-differences (respectively, ω-differences) of ${\bf{\Pi }}_3^0 $ sets.

Graph normal form, GNF, [1], was used in [2, 3] for analyzing paradoxes in propositional discourses, with the semantics—equivalent to the classical one—defined by kernels of digraphs. The paper presents infinitary, resolution-based reasoning with GNF theories, which is refutationally complete for the classical semantics. Used for direct (not refutational) deduction it is not explosive and allows to identify in an inconsistent discourse, a maximal consistent subdiscourse with its classical consequences. Semikernels, generalizing kernels, provide the semantic interpretation.

We establish the decidability of the ${{\rm{\Sigma }}_2}$ theory of both the arithmetic and hyperarithmetic degrees in the language of uppersemilattices, i.e., the language with ≤, 0 , and $\sqcup$. This is achieved by using Kumabe-Slaman forcing, along with other known results, to show given finite uppersemilattices ${\cal M}$ and ${\cal N}$, where ${\cal M}$ is a subuppersemilattice of ${\cal N}$, that every embedding of ${\cal M}$ into either degree structure extends to one of ${\cal N}$ iff ${\cal N}$ is an end-extension of ${\cal M}$.

We prove that WRP and saturation of the ideal NSω1 together imply $\left\{ {a \in [\lambda ]^{\omega _1 } :{\text{cof}}\left( {{\text{sup}}\left( a \right)} \right) = \omega _1 } \right\}$, for every cardinal λ with cof(λ) ≥ω2 .

We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating μ-hyperfinite equivalence relations admit μ-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a counterexample in the locally countable case.

We study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

We introduce a notion of valued module which is suitable to study valued fields of positive characteristic. Then we built-up a robust theory of henselianity in the language of valued modules and prove Ax-Kochen Ershov type results.