We study universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example, we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments.

Dedicated to the memory of Alexander Prestel (1941–2024)

We show, assuming PD, that every complete finitely axiomatized second-order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second-order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g., supercompact) cardinals does not imply the categoricity of all finitely axiomatizable complete second-order theories. More exactly, we show that a non-categorical complete finitely axiomatized second-order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second-order theories with a model of a certain singular cardinality $\kappa $ of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming $V=L$, that every complete finitely axiomatized second-order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second-order theory with a countable model which is non-categorical.

Let $\mathsf {KP}$ denote Kripke–Platek Set Theory and let $\mathsf {M}$ be the weak set theory obtained from $\mathsf {ZF}$ by removing the collection scheme, restricting separation to $\Delta _0$-formulae and adding an axiom asserting that every set is contained in a transitive set ($\mathsf {TCo}$). A result due to Kaufmann [9] shows that every countable model, $\mathcal {M}$, of $\mathsf {KP}+\Pi _n\textsf {-Collection}$ has a proper $\Sigma _{n+1}$-elementary end extension. We show that for all $n \geq 1$, there exists an $L_\alpha $ (where $L_\alpha $ is the $\alpha ^{\textrm {th}}$ approximation of the constructible universe L) that satisfies $\textsf {Separation}$, $\textsf {Powerset}$ and $\Pi _n\textsf {-Collection}$, but that has no $\Sigma _{n+1}$-elementary end extension satisfying either $\Pi _n\textsf {-Collection}$ or $\Pi _{n+3}\textsf {-Foundation}$. Thus showing that there are limits to the amount of the theory of $\mathcal {M}$ that can be transferred to the end extensions that are guaranteed by Kaufmann’s theorem. Using admissible covers and the Barwise Compactness theorem, we show that if $\mathcal {M}$ is a countable model $\mathsf {KP}+\Pi _n\textsf {-Collection}+\Sigma _{n+1}\textsf {-Foundation}$ and T is a recursive theory that holds in $\mathcal {M}$, then there exists a proper $\Sigma _n$-elementary end extension of $\mathcal {M}$ that satisfies T. We use this result to show that the theory $\mathsf {M}+\Pi _n\textsf {-Collection}+\Pi _{n+1}\textsf {-Foundation}$ proves $\Sigma _{n+1}\textsf {-Separation}$.

We prove that every $\Sigma ^0_2$ Gale-Stewart game can be won via a winning strategy $\tau $ which is $\Delta _1$-definable over $L_{\delta }$, the $\delta $th stage of Gödel’s constructible universe, where $\delta = \delta _{\sigma ^1_1}$, strengthening a theorem of Solovay from the 1970s. Moreover, the bound is sharp in the sense that there is a $\Sigma ^0_2$ game with no strategy $\tau $ which is witnessed to be winning by an element of $L_{\delta }$.

The Fregean ontology can be naturally interpreted within set theory with urelements, where objects correspond to sets and urelements, and concepts to classes. Consequently, Fregean abstraction principles can be formulated as set-theoretic principles. We investigate how the size of reality—i.e., the number of urelements—interacts with these principles. We show that Basic Law V implies that for some well-ordered cardinal $\kappa $, there is no set of urelements of size $\kappa $. Building on recent work by Hamkins [10], we show that, under certain additional axioms, Basic Law V holds if and only if the urelements form a set. We construct models of urelement set theory in which the Reflection Principle holds while Hume’s Principle fails for sets. Additionally, assuming the consistency of an inaccessible cardinal, we produce a model of Kelley–Morse class theory with urelements that has a global well-ordering but lacks a definable map satisfying Hume’s Principle for classes.

We study the Lyndon interpolation property (LIP) and the uniform LIP (ULIP) for extensions of $\mathbf {S4}$ and intermediate propositional logics. We prove that among the 18 consistent normal modal logics of finite height extending $\mathbf {S4}$ known to have CIP, 11 logics have LIP and 7 logics do not. We also prove that for intermediate propositional logics, the Craig interpolation property, LIP, and ULIP are equivalent.

For each $n\geq 1$, let $FT_n$ be the free tree monoid of rank n and $E_n$ the full extensive transformation monoid over the finite chain $\{1, 2, \ldots , n\}$. It is shown that the monoids $FT_n$ and $E_{n+1}$ satisfy the same identities. Therefore, $FT_n$ is finitely based if and only if $n\leq 3$.

An original family of labelled sequent calculi $\mathsf {G3IL}^{\star }$ for classical interpretability logics is presented, modularly designed on the basis of Verbrugge semantics (a.k.a. generalised Veltman semantics) for those logics. We prove that each of our calculi enjoys excellent structural properties, namely, admissibility of weakening, contraction and, more relevantly, cut. A complexity measure of the cut is defined by extending the notion of range previously introduced by Negri w.r.t. a labelled sequent calculus for Gödel–Löb provability logic, and a cut-elimination algorithm is discussed in detail. To our knowledge, this is the most extensive and structurally well-behaving class of analytic proof systems for modal logics of interpretability currently available in the literature.

Answering a question of Kaye, we show that the compositional truth theory with the full collection scheme is conservative over Peano Arithmetic. We demonstrate it by showing that countable models of compositional truth which satisfy the internal induction or collection axioms can be end-extended to models of the respective theory.

Let $\Gamma $ be a compact Polish group of finite topological dimension. For a countably infinite subset $S\subseteq \Gamma $, a domatic $\aleph _0$-partition (for its Schreier graph on $\Gamma $) is a partial function $f:\Gamma \rightharpoonup \mathbb {N}$ such that for every $x\in \Gamma $, one has $f[S\cdot x]=\mathbb {N}$. We show that a continuous domatic $\aleph _0$-partition exists, if and only if a Baire measurable domatic $\aleph _0$-partition exists, if and only if the topological closure of S is uncountable. A Haar measurable domatic $\aleph _0$-partition exists for all choices of S. We also investigate domatic partitions in the general descriptive graph combinatorial setting.

Denote by $\mathcal {NA}$ and $\mathcal {MA}$ the ideals of null-additive and meager-additive subsets of $2^{\omega }$, respectively. We prove in ZFC that $\mathrm {add}(\mathcal {NA})=\mathrm {non}(\mathcal {NA})$ and introduce a new (Polish) relational system to reformulate Bartoszyński’s and Judah’s characterization of the uniformity of $\mathcal {MA}$, which is helpful to understand the combinatorics of $\mathcal {MA}$ and to prove consistency results. As for the latter, we prove that $\mathrm {cov}(\mathcal {MA})<\mathfrak {c}$ (even $\mathrm {cov}(\mathcal {MA})<\mathrm {non}(\mathcal {N})$) is consistent with ZFC, as well as several constellations of Cichoń’s diagram with $\mathrm {non}(\mathcal {NA})$, $\mathrm {non}(\mathcal {MA}),$ and $\mathrm {add}(\mathcal {SN})$, which include $\mathrm {non}(\mathcal {NA})<\mathfrak {b}< \mathrm {non}(\mathcal {MA})$ and $\mathfrak {b}< \mathrm {add}(\mathcal {SN})<\mathrm {cov}(\mathcal {M})<\mathfrak {d}=\mathfrak {c}$.

We study infinite groups interpretable in power bounded T-convex, V-minimal or p-adically closed fields. We show that if G is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\times G_2$, where $G_1$ is a K-linear group and $G_2$ is a $\mathbf {k}$-linear group. The analysis is carried out by studying the interaction of G with four distinguished sorts: the valued field K, the residue field $\mathbf {k}$, the value group $\Gamma $, and the closed $0$-balls $K/\mathcal {O}$.

We investigate the primitive recursive content of linear orders. We prove that the punctual degrees of rigid linear orders, the order of the integers $\mathbb {Z}$, and the order of the rationals $\mathbb {Q}$ embed the diamond (preserving supremum and infimum). In the cases of rigid orders and the order $\mathbb {Z}$, we further extend the result to embed the atomless Boolean algebra; we leave the case of $\mathbb {Q}$ as an open problem. We also show that our results for the rigid orders, in fact, work for orders having a computable infinite invariant rigid sub-order.

We study a family of variants of Jensen’s subcomplete forcing axiom, $\mathsf {SCFA,}$ and subproper forcing axiom, $\mathsf {SubPFA}$. Using these, we develop a general technique for proving nonimplications of $\mathsf {SCFA}$, $\mathsf {SubPFA}$ and their relatives and give several applications. For instance, we show that $\mathsf {SCFA}$ does not imply $\mathsf {MA}^+(\sigma $-closed) and $\mathsf {SubPFA}$ does not imply Martin’s Maximum.

Following [1], given cardinals $\kappa <\lambda $, we say $\kappa $ is a club $\lambda $-Berkeley cardinal if for every transitive set N of size $<\lambda $ such that $\kappa \subseteq N$, there is a club $C\subseteq \kappa $ with the property that for every $\eta \in C$, there is an elementary embedding $j: N\rightarrow N$ with $\mathrm {crit }(j)=\eta $. We say $\kappa $ is $\nu $-club $\lambda $-Berkeley if $C\subseteq \kappa $ as above is a $\nu $-club. We say $\kappa $ is $\lambda $-Berkeley if C is unbounded in $\kappa $. We show that under $\textsf {AD}^{+}$, (1) every regular Suslin cardinal is $\omega $-club $\Theta $-Berkeley (see Theorem 7.1), (2) $\omega _1$ is club $\Theta $-Berkeley (see Theorem 3.1 and Theorem 7.1), and (3) the ’s are $\Theta $-Berkeley – in particular, $\omega _2$ is $\Theta $-Berkeley (see Remark 7.5).

Along the way, we represent regular Suslin cardinals in direct limits as cutpoint cardinals (see Theorem 5.1). This topic has been studied in [31] and [4], albeit from a different point of view. We also show that, assuming $V=L({\mathbb {R}})+{\textsf {AD}}$, $\omega _1$ is not $\Theta ^+$-Berkeley, so the result stated in the title is optimal (see Theorem 9.14 and Theorem 9.19).

For which choices of $X,Y,Z\in \{\Sigma ^1_1,\Pi ^1_1\}$ does no sufficiently strong X-sound and Y-definable extension theory prove its own Z-soundness? We give a complete answer, thereby delimiting the generalizations of Gödel’s second incompleteness theorem that hold within second-order arithmetic.

We prove the existence of a model companion of the two-sorted theory of c-nilpotent Lie algebras over a field satisfying a given theory of fields. We describe a language in which it admits relative quantifier elimination up to the field sort. Using a new criterion which does not rely on a stationary independence relation, we prove that if the field is NSOP$_1$, then the model companion is NSOP$_4$. We also prove that if the field is algebraically closed, then the model companion is c-NIP.

We prove two compactness theorems for HOD. First, if $\kappa $ is a strong limit singular cardinal with uncountable cofinality and for stationarily many $\delta <\kappa $, $(\delta ^+)^{\mathrm {HOD}}=\delta ^+$, then $(\kappa ^+)^{\mathrm {HOD}}=\kappa ^+$. Second, if $\kappa $ is a singular cardinal with uncountable cofinality and stationarily many $\delta <\kappa $ are singular in $\operatorname {\mathrm {HOD}}$, then $\kappa $ is singular in $\operatorname {\mathrm {HOD}}$. We also discuss the optimality of these results and show that the first theorem does not extend from $\operatorname {\mathrm {HOD}}$ to other $\omega $-club amenable inner models.

The family of relevant logics can be faceted by a hierarchy of increasingly fine-grained variable sharing properties—requiring that in valid entailments $A\to B$, some atom must appear in both A and B with some additional condition (e.g., with the same sign or nested within the same number of conditionals). In this paper, we consider an incredibly strong variable sharing property of lericone relevance that takes into account the path of negations and conditionals in which an atom appears in the parse trees of the antecedent and consequent. We show that this property of lericone relevance holds of the relevant logic $\mathbf {BM}$ (and that a related property of faithful lericone relevance holds of $\mathbf {B}$) and characterize the largest fragments of classical logic with these properties. Along the way, we consider the consequences for lericone relevance for the theory of subject-matter, for Logan’s notion of hyperformalism, and for the very definition of a relevant logic itself.

In previous publications, it was shown that finite non-deterministic matrices are quite powerful in providing semantics for a large class of normal and non-normal modal logics. However, some modal logics, such as those whose axiom systems contained the Löb axiom or the McKinsey formula, were not analyzed via non-deterministic semantics. Furthermore, other modal rules than the rule of necessitation were not yet characterized in the framework.

In this paper, we will overcome this shortcoming and present a novel approach for constructing semantics for normal and non-normal modal logics that is based on restricted non-deterministic matrices. This approach not only offers a uniform semantical framework for modal logics, while keeping the interpretation of the involved modal operators the same, and thus making different systems of modal logic comparable. It might also lead to a new understanding of the concept of modality.