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 mixed identities for oligomorphic automorphism groups of countable relational structures. Our main result gives sufficient conditions for such a group to not admit a mixed identity without particular constants. We study numerous examples and prove in many cases that there cannot be a non-singular mixed identity.

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.

The Kruskal–Friedman theorem asserts: in any infinite sequence of finite trees with ordinal labels, some tree can be embedded into a later one, by an embedding that respects a certain gap condition. This strengthening of the original Kruskal theorem has been proved by I. Kříž (Ann. Math. 1989), in confirmation of a conjecture due to H. Friedman, who had established the result for finitely many labels. It provides one of the strongest mathematical examples for the independence phenomenon from Gödel’s theorems. The gap condition is particularly relevant due to its connection with the graph minor theorem of N. Robertson and P. Seymour. In the present article, we consider a uniform version of the Kruskal–Friedman theorem, which extends the result from trees to general recursive data types. Our main theorem shows that this uniform version is equivalent both to $\Pi ^1_1$-transfinite recursion and to a minimal bad sequence principle of Kříž, over the base theory $\mathsf {RCA_0}$ from reverse mathematics. This sheds new light on the role of infinity in finite combinatorics.

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.

The classical satisfiability problem (SAT) is used as a natural and general tool to express and solve combinatorial problems that are in NP. We postulate that provability for implicational intuitionistic propositional logic (IIPC) can serve as a similar natural tool to express problems in Pspace. We demonstrate it by proving two essential results concerning the system. One is a natural reduction from full IPC (with all connectives) to implicational formulas of order three. Another result is a convenient interpretation in terms of simple alternating automata. Additionally, we distinguish some natural subclasses of IIPC corresponding to the complexity classes NP and co-NP.

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 $\mathfrak {b}< \mathrm {add}(\mathcal {SN})<\mathrm {cov}(\mathcal {M})<\mathfrak {d}=\mathfrak {c}$.

This paper shows how to set up Fine’s “theory-application” type semantics so as to model the use-unrestricted “Official” consequence relation for a range of relevant logics. The frame condition matching the axiom $(((A \to A) \land (B \to B)) \to C) \to C$—the characteristic axiom of the very first axiomatization of the relevant logic E—is shown forth. It is also shown how to model propositional constants within the semantic framework. Whereas the related Routley–Meyer type frame semantics fails to be strongly complete with regards to certain contractionless logics such as B, the current paper shows that Fine’s weak soundness and completeness result can be extended to a strong one also for logics like B.

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 finite subsets s of the natural numbers such that $|s|=1+\min s$ is known as the Schreier barrier in combinatorics and Banach Space theory, and as the family of exactly $\omega $-large sets in Logic. We formulate and prove the generalizations of Friedman’s Free Set and Thin Set theorems and of Rainbow Ramsey’s theorem to colorings of the Schreier barrier. We analyze the strength of these theorems from the point of view of Computability Theory and Reverse Mathematics. Surprisingly, the exactly $\omega $-large counterparts of the Thin Set and Free Set theorems can code $\emptyset ^{(\omega )}$, while the exactly $\omega $-large Rainbow Ramsey theorem does not code the halting set.

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.