We establish a coloring theorem for successors of singular cardinals, and use it prove that for any such cardinal $\mu$, we have $\mu^+\nrightarrow[\mu^+]^2_{\mu^+}$ if and only if $\mu^+\nrightarrow[\mu^+]^2_{\theta}$ for arbitrarily large $\theta < \mu$.

We formulate and prove (in ZFC) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle $Pr_1(\mu^+,\mu^+,\mu^+,cf(\mu))$ for singular $\mu$.

We give the natural topological model for $\neg$BD-${\mathbb N}$, and use it to show that the closure of spaces with the anti-Specker property under product does not imply BD-${\mathbb N}$. Also, the natural topological model for $\neg$BD is presented. Finally, for some of the realizability models known indirectly to falsify BD-$\mathbb{N}$, it is brought out in detail how BD-$\mathbb N$ fails.

In this paper we show that there are various pseudo-jump operators definable over inadmissible $J_{\beta}$ that relate to the failure of admissiblity and to non-regularity. We will use these ideas to construct some intermediate degrees.

According to Belegradek, a first order structure is weakly small if there are countably many $1$-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic $2$ is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic $2$ is a field.

Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by $\mathscr{P}(\omega)/\mathrm{finite}$. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are no intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality of the associated measures.

We show that any symmetric, Baire measurable function from the complement of $E_0$ to a finite set is constant on an $E_0$-nonsmooth square. A simultaneous generalization of Galvin's theorem that Baire measurable colorings admit perfect homogeneous sets and the Kanovei-Zapletal theorem canonizing Borel equivalence relations on $E_0$-nonsmooth sets, this result is proved by relating $E_0$-nonsmooth sets to embeddings of the complete binary tree into itself and appealing to a version of Hindman's theorem on the complete binary tree. We also establish several canonization theorems which follow from the main result.

This paper investigates indifferent sets for comeager classes in Cantor space focusing of the class of all 1-generic sets and the class of all weakly 1-generic sets. Jockusch and Posner showed that there exist 1-generic sets that have indifferent sets [10]. Figueira, Miller and Nies have studied indifferent sets for randomness and other notions [7]. We show that any comeager class in Cantor space contains a comeager class with a universal indifferent set. A forcing construction is used to show that any 1-generic set, or weakly 1-generic set, has an indifferent set. Such an indifferent set can by computed by any set in $\overline{\mathbf{GL_2}}$ which bounds the (weakly) 1-generic. We show by approximation arguments that some, but not all, $\Delta^0_2$ 1-generic sets can compute an indifferent set for themselves. We show that all $\Delta^0_2$ weakly 1-generic sets can compute an indifferent set for themselves. Additional results on indifferent sets, including one of Miller, and two of Fitzgerald, are presented.

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens $V =$ HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.

We answer a question of Cummings and Magidor by proving that the P-ideal dichotomy of Todorčević refutes ${\square}_{\kappa, \omega}$ for any uncountable $\kappa$. We also show that the P-ideal dichotomy implies the failure of ${\square}_{\kappa, < \mathfrak{b}}$ provided that $cf(\kappa) > {\omega}_{1}$.

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let ${\sf BC}_{\kappa}$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_{\kappa}$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results:

1. If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$.

2. If it is consistent that ${\sf BC}_{\aleph_1}$, then it is consistent that there is an inaccessible cardinal.

3. If it is consistent that there is a 1-inaccessible cardinal with $\omega$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_{\omega}} + (\forall n < \omega){\sf BC}_{\aleph_n}$ is consistent.

4. If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\sf BC}_{\aleph_{\omega}}$.

5. If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\sf BC}_{\kappa}$ for a proper class of cardinals $\kappa$ of countable cofinality.

We introduce the notions of a mutually algebraic structures and theories and prove many equivalents. A theory $T$ is mutually algebraic if and only if it is weakly minimal and trivial if and only if no model $M$ of $T$ has an expansion $(M,A)$ by a unary predicate with the finite cover property. We show that every structure has a maximal mutually algebraic reduct, and give a strong structure theorem for the class of elementary extensions of a fixed mutually algebraic structure.

We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let 2-$RAN$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $RAN_0$ and so $RAN_0 +$ 2-$RAN$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over $RAN_0$ for arithmetic sentences. Thus, from the Csima—Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2-$RAN$ has non-trivial arithmetic consequences. In Section 4, we show that 2-$RAN$ is conservative over $RCA_0 + B\Sigma_2$ for $\Pi^1_1$-sentences. Thus, the set of first-order consequences of 2-$RAN$ is strictly stronger than $P^-+I\Sigma_1$ and no stronger than $P^- + B\Sigma_2$.

Every definably complete expansion of an ordered field satisfies an analogue of the Baire Category Theorem.

The implicational fragment of the logic of relevant implication, $R_\to$ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, $T_\to$ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of $T_\to$ to the decidability problem of $R_\to$. The decidability of $T_\to$ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over the base $\{\mathsf{B, B', I, W}\}$.

The Axiom of Projective Determinacy implies the existence of a universal $\underset{\tilde{}}{{\Pi}^1_n}\setminus\underset{\tilde{}}{{\Delta}^1_n}$ set for every $n \geq 1$. Assuming $\rm{MA}(\aleph_{1})+\aleph_{1}=\aleph_{1}^{\mathbb{L}}$ there exists a universal $\underset{\tilde{}}{{\Pi}^1_1}\setminus\underset{\tilde{}}{{\Delta}^1_1}$ set. In ZFC there is a universal $\underset{\tilde{}}{{\Pi}^0_\alpha}\setminus\underset{\tilde{}}{{\Delta}^0_\alpha}$ set for every $\alpha$.

We prove that for a finitely generated infinite nilpotent group $G$ with structure $(G,\cdot,\dots)$, the connected component ${G^*}^0$ of a sufficiently saturated extension $G^*$ of $G$ exists and equals \[ \bigcap_{n\in\N} \{g^n\colon g\in G^*\}. \] We construct an expansion of ${\mathbb Z}$ by a predicate $({\mathbb Z},+,P)$ such that the type-connected component ${{\mathbb Z}^*}^{00}_{\emptyset}$ is strictly smaller than ${{\mathbb Z}^*}^0$. 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.

It has been known for six years that the restriction of Girard's polymorphic system $\mathbf{F}$ 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 $\beta$-reduction step of the full intuitionistic propositional calculus translates into one or more $\beta\eta$-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 $\eta$-conversions.

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 comment on an early and inspiring remark of an Omskian mathematician concerning the Cherlin—Zilber Conjecture, meeting in passing some well-known properties of algebraic groups whose generalization to arbitrary groups of finite Morley rank seems to be very uncertain. This paper assumes a familiarity with the model theoretic tools involved in the study of the groups of finite Morley rank.

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 $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa$. 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 $n\geq 2$ and $\mu\geq \aleph_n$, we have $(\aleph_n, \mu)$-ITP.

A seminal theorem due to Weyl [14] states that if $(a_n)$ is any sequence of distinct integers, then, for almost every $x \in \mathbb{R}$, the sequence $(a_n x)$ is uniformly distributed modulo one. In particular, for almost every $x$ in the unit interval, the sequence $(a_n x)$ is uniformly distributed modulo one for every computable sequence $(a_n)$ 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.