Papers

Given a complete first-order theory $T$, we characterise elimination of hyperimaginaries in $T$ in terms of the exactness of the pro-completion $\textrm{Pro}(\textrm{def}(T))$ of the syntactic category $\textrm{def}(T)$ of $T$, thus extending Makkai's result connecting elimination of imaginaries with the exactness of $\textrm{def}(T)$. Likewise, we characterise the heq construction in terms of the exact completion of $\textrm{Pro}(\textrm{def}(T))$.

Let $X$ be a Polish space and let $\mathcal K(X)$ be its Vietoris hyperspace. A family $\mathcal I\subseteq\mathcal K(X)$ is hereditary if it is downward closed under inclusion. Matheron and Zelený asked whether every comeager hereditary family in $\mathcal K(X)$ contains a dense hereditary $G_\delta$ subfamily. We give an affirmative answer in ZFC.

The classes of SOP$_2$ and SOP$_3$ first-order theories coincide. This answers a question of Džamonja and Shelah from 2004.

Starting with CH and Hajnal's coloring, we show that a standard finite-support iteration of $\sigma$-centered forcing notions gives a model of $ MA_{\omega_1}(\sigma$-centered$)+2^{\aleph_0}=\aleph_2 +\omega_1^2\nrightarrow(\omega_1^2,3)^2.$ We also isolate a simple forcing-preservation principle: the same ground-model coloring remains a witness after forcing with any poset whose subfamilies of size at most $\omega_1$ are countable unions of linked sets. Under $\text{MA}_{\omega_1}$, every c.c.c. forcing has this local property, so every existing witness is preserved by every c.c.c. forcing over that model.

$\mathrm{PA}_{\mathrm{smu}}^{-}$ is a weak theory of arithmetic obtained by adding two principles concerning powers of two to basic axioms satisfied by the nonnegative part of a discretely ordered ring. We introduce a theory $\mathrm{PA}_{\mathrm{wit}}^{-}$ in which powers of two and the required witnesses are represented by primitive symbols. We show that, along a fixed sequence in the standard model of arithmetic, every one-variable term eventually agrees with a polynomial over the dyadic rationals. A finite avoidance lemma and the Compactness Theorem then yield a model of $\mathrm{PA}_{\mathrm{smu}}^{-}$ in which division by $3$ fails. In this model even the $n=3$ instance of the Standard Euclidean Division Principle pa16 fails. In fact, the model can be chosen to satisfy every universal $\mathcal{L}_0$-sentence true in the standard model. Consequently, neither pa16 nor the Euclidean Division Principle pa17 is derivable even after these universal truths are adjoined to $\mathrm{PA}_{\mathrm{smu}}^{-}$.

Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics of the expected number of adjacent twin pairs in every dimension. In dimension two, more generally, for each origin-symmetric convex connection body $K\subset(-1/2,1/2)^2$, the twin count converges to a Poisson variable with mean $\operatorname{area}(K^\circ)/16$; hence the zero-one law fails for all fixed $0<r<1/2$ in the $L^\infty$ and Euclidean models. At each critical component threshold $n^k r_n^{D(k-1)}\to a\in(0,\infty)$, the numbers of components of the feasible connected $k$-vertex types converge jointly to independent Poisson variables, yielding the complete first-order convergence law. Between consecutive thresholds a zero-one law holds. For $D\ge3$, we also construct a definable common-neighborhood configuration of probability order $1/n$.

We compare several "pathological" notions of continuity that make every additive function continuous. Our goal is to determine which of these notions is best behaved and deserves to be called Cauchy-Hamel continuity. The conclusion is that a bilateral version of a previously introduced notion of Q-continuity seems the most promising candidate. Our work is relevant to the axiomatic foundations of mechanics, specifically to the problem of axiomatically characterizing the parallelogram rule for the composition of forces. It was noted by Darboux that a continuity assumption is needed for such a characterization of the parallelogram rule and that, absent such a continuity axiom, "exotic" physical models based on alternative composition rules may exist. An intriguing question is whether such exotic physical models still possess some weak, residual notion of continuity. The concepts studied in this paper offer a first hint of a response to this question.

We study the abstract elementary class of acts with pure embeddings. In particular, we show that stability and superstability in this class can be characterized in terms of the monoid $S$ being LO (for every $s,t \in S$, we have that $s \in St$ or $t \in Ss$) and weakly noetherian (every ideal is finitely generated), respectively. Moreover, under mild set-theoretic assumptions, we characterize the stability spectrum via the minimal cardinality of a generating set for every ideal. As an application, we obtain a Baer-like criterion for pure injective acts when $S$ is LO. We use this to provide an alternative proof that the class of acts has enough pure injectives when $S$ is LO.

We study the amalgamation decision problem (ADP): given a universal first-order sentence $\Phi$, decide whether the class $\mathrm{fm}(\Phi)$ of its finite models has the amalgamation property. We call $\Phi$ semantic Horn if $\mathrm{fm}(\Phi)$ is closed under binary direct products. By McKinsey's theorem, this is equivalent to $\Phi$ being logically equivalent to a universal Horn sentence. The distinction concerns the input representation: $\Phi$ itself need not be given in Horn form, and converting it into an explicit Horn normal form can incur an exponential blow-up. We prove that the ADP is decidable under this semantic promise. To this end, we associate with $\Phi$ a finite-domain CSP template, called its completion template, and a distinguished infinite family of CSP instances, called its atlas instances. The class $\mathrm{fm}(\Phi)$ has the amalgamation property precisely when all atlas instances have a solution. For semantic Horn inputs, we prove that the completion template admits a semilattice polymorphism. Consequently, its CSP has bounded width and is decided by a fixed level of local consistency. We introduce uniform contextual strategies, finite certificates expressing this local-consistency condition simultaneously for all atlas instances, and give an effective fixed-point procedure for deciding whether such a certificate exists. The resulting algorithm runs in 2ExpTime in general and in ExpTime under any fixed bound on the arities of the input relations. More generally, the construction gives a decision procedure whenever the associated completion template has bounded width. These bounds are optimal: the semantic Horn ADP is 2ExpTime-complete in general and ExpTime-complete for every fixed arity bound of at least three. Both hardness results already hold for syntactic universal Horn sentences.

Answering a question of Todorcevic, we prove higher-dimensional canonization theorems for the topological Ramsey space $\mathrm{FIN}^{[\infty]}_2$. Our results build upon the work of Lopez-Abad and continue the line of research initiated by Erdős and Rado, and further developed by Pudlák and Rödl, Prömel and Voigt, Taylor, and Klein and Spinas. We identify the canonical functions on fronts of $\mathrm{FIN}^{[\infty]}_2$ and develop an extension of the separating-mixing technique of Prömel and Voigt to canonize arbitrary functions $g:\mathcal{F}\to\omega$, where $\mathcal{F}$ is a front of $\mathrm{FIN}^{[\infty]}_2$. We further extend our canonization theorem to arbitrary Borel maps $g:\mathrm{FIN}^{[\infty]}_2\to\mathbb{R}$, establishing new canonical Ramsey theorems for Polish spaces. In particular, we provide a complete classification of the canonical functions on $\mathrm{FIN}^{[n]}_2$, together with an explicit formula for their number as a function of $n$.

Kleene triples, introduced by Jalali, provide a representation of arbitrary Kleene algebras in terms of twist-products. We introduce modal Kleene algebras and modal Kleene triples, and establish a categorical equivalence between the corresponding categories, extending Jalali's duality to the modal setting. We study the centered case and show that Kleene algebras with implication can be naturally described within this framework by viewing implication as a family of modal operators. Finally, we develop a topological duality for modal Kleene triples.

In this paper, a new model for multi-valued logic is presented based on the notion of a bundle of predicates. The degree of truth and various logical operations for predicates in our model of multi-valued logic are rigorously defined and investigated. Furthermore, the bundle of sets as a special type of predicate bundles is thoroughly investigated to provide a rigorous model for fuzzy sets and certain adjectives/adverbs in linguistics. In addition, solutions to necessity/possibility and the sorites paradox in modal logic are given.

Let $\mathcal{H}_{\mathcal{Q}}$ be the field of unary germs at $0^+$ definable in the quasianalytic o-minimal structure $\mathbb{R}_{\mathcal{Q}}$ of Kaiser-Rolin-Speissegger. We construct a canonical ordered differential-field embedding of $\mathcal{H}_{\mathcal{Q}}$ into the field of natural-support generalized Laurent series. Every unary germ has a unique such asymptotic expansion, and bounded germs have expansions with only nonnegative exponents. A germ admits a convergent generalized Laurent-series representation exactly when its canonical expansion converges; in that case, every convergent representation with well-ordered support is canonical. We also prove that $\mathbb{R}_{\mathcal{Q}}$ is branching in the sense of Dembner [1]. A nonresonant hyperbolic saddle with divergent Dulac series yields a bounded positive real-analytic $\mathbb{R}_{\mathcal{Q}}$-definable germ whose canonical generalized power series diverges. This gives a negative answer to a question of Dembner asking whether every unary germ definable in a branching structure admits a convergent generalized power-series representation. Nevertheless, every bounded unary $\mathbb{R}_{\mathcal{Q}}$-germ has a canonical formal generalized power-series expansion. Consequently, the convergent germs form a proper ordered differential subfield of $\mathcal{H}_{\mathcal{Q}}$.

The stability programme initiated by Ulam asks when approximate solutions to algebraic identities must lie near exact ones. For lattices, this leads to the question of when a map that nearly preserves joins and meets can be approximated by a genuine lattice homomorphism. Badora–Kochanek–Przebieracz developed a neighbourhood-based framework for distributive lattices, centred on a separation (sandwich) lemma that constructs an exact join homomorphism between a join-subhomomorphism and a join-superhomomorphism via an order envelope. We revisit this mechanism and identify the single step at which distributivity is used: a decomposition identity for elements lying below a join. Without distributivity, separation can fail already in the modular lattice $M_3$ and in a small finite orthomodular lattice. On the positive side, we show that separation holds in arbitrary lattices whenever the lower bounding map is isotone. For orthomodular lattices—algebraic models of quantum logic—we develop a blockwise stability theory on Boolean blocks. Approximate identities on compatible pairs yield exact homomorphic selections on each block (for joins, for meets, and for both operations under bi-admissibility). We present several gluing criteria for assembling blockwise selections, and we give a concrete finite example showing that gluing can fail when block overlaps are non-trivial. Finally, in the spirit of Kalton–Roberts, we obtain blockwise approximation results for nearly additive functions on orthomodular lattices by finitely additive measures, with an illustration on finite-dimensional projection lattices.

We study the expansion of the real field by the graphs of power functions on the unit circle. Under a natural number-theoretic conjecture, we prove that adding such dense subsets does not increase the topological complexity of definable sets: every open definable set remains semialgebraic. The proof uses a two-sorted structure that separates the linear and algebraic data, inspired by Zilber's raising to powers. Using Hrushovski's amalgamation method, we construct and axiomatize a class of rich structures, and then show that the intended structure is a model. This provides a new example of a tame expansion of the real field by dense trajectories.

Loeb measure theory stands as one of the most influential concepts in nonstandard analysis, underpinning nearly all applications in probability, stochastic processes, and mathematical economics. The paper resolves a fundamental open problem in Loeb measure theory originally posed by Keisler and Sun: let $(\Omega,\mathcal{F},\mu)$ and $(\Omega,\mathcal{G},\nu)$ be two Loeb equivalent internal probability spaces, and $\mathcal H$ be the internal algebra generated from $\mathcal{F}\cup\mathcal{G}$. Does there exist an internal probability measure $P$ on $\mathcal H$ such that $(\Omega,\mathcal{H},P)$ is Loeb equivalent to $(\Omega,\mathcal{F},\mu)$? While arXiv:2112.13955 recently provided a positive answer for hyperfinite probability spaces, the problem remained open for general internal probability spaces. We establish the existence of such an internal probability measure for all internal probability spaces.

We investigate the first-order theories of full \(n\)-branching ordinal trees \(\mathfrak{T}_\alpha^n\). We obtain a complete classification of the standard trees up to elementary equivalence: every \(\mathfrak{T}_\alpha^n\) is elementarily equivalent to one of four canonical types determined by the ordinal \(\alpha\). For each canonical type we establish effective quantifier elimination and prove decidability of the theory. Along the way we develop the realizability theory of colored ordinal characters and obtain a sharp bound on the length of minimal witnesses. We also clarify the relationship between these trees and monadic second-order logic over ordinals, and show that the equal-height relation is not first-order definable in any standard tree.

Tarski's high school algebra problem asks whether every true identity concerning addition, multiplication, and exponentiation of positive integers follows from a list of 11 elementary identities. Surprisingly, Wilkie showed that the following identity is valid over the positive integers and yet does not follow from Tarski's axioms: \begin{align*} &\left((1+x)^y + (1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x + (1+x^2+x^4)^x\right)^y = \\ &\left((1+x)^x + (1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y + (1+x^2+x^4)^y\right)^x. \end{align*} Gurevič gave an algebra on 59 elements that satisfies Tarski's axioms but not Wilkie's identity, and over the years several authors whittled down the size of such a countermodel, culminating in a countermodel of size 12 due to Burris and Yeats. On the other hand, Zhang proved that there is no countermodel with fewer than 11 elements. Using SAT, we prove that the smallest countermodels are of size 12, as conjectured by Burris and Yeats. Moreover, we show that there are exactly 8,957,952 countermodels on 12 elements up to isomorphism and provide a simple classification of them. Our SAT approach outperforms dedicated tools for finding countermodels in equational theories, namely Mace4 and SEM. Furthermore, using autoformalization, we prove the correctness of our main result in Lean.

We completely classify the possible complexity classes of non-closed operator ranges on separable Banach spaces over a Polish non-Archimedean non-trivially valued field. These are precisely $\boldsymbol{\Pi }_{1+\lambda +n+2}^{0}$ for a countable ordinal $\lambda $ that is either zero or a limit ordinal, and a finite ordinal $n$. Considering Fréchet spaces produces the additional complexity classes $\boldsymbol{\Pi }_{\lambda }^{0}$ for a countable limit ordinal $\lambda $.

For a commutative Bezout ring R, we give a criterion, in terms of colon ideals with principal radical, for the lattice $\mathring{\mathcal{K}}(\mathrm{Spec}(R))$ of compact open subsets of $\mathrm{Spec}(R)$ to be a Heyting algebra. Bezhanishvili and Tressl showed that $\mathring{\mathcal{K}}(\mathrm{Spec}(C(T)))$ is pseudocomplemented whenever T is a basically disconnected compact Hausdorff space, and asked whether $\mathrm{Spec}(C(\beta\mathbb{N}))$ is actually an Esakia space. We show it is not: applying our criterion to $C(\beta\mathbb{N}) \cong \ell^\infty(\mathbb{N},\mathbb{R})$ produces a diagonal counterexample, and the same obstruction rules out $\beta D$ for every infinite discrete D. A grid-existence theorem for $\sigma$-complete Boolean algebras lets us push the construction to every basically disconnected compact Hausdorff space, settling the Bezhanishvili-Tressl question completely: for no infinite compact Hausdorff space T is $\mathrm{Spec}(C(T))$ an Esakia space.

We show that, up to isomorphism, the number of $Q$-points is either finite, $2^{\mathfrak{d}}$ or $2^{\mathfrak{c}}$. This answers a question asked by Borodulin-Nadzieja, Martínez-Celis, Morawski and Świerczyńska, and by Halbeisen and the authors. We also show that under mild hypotheses, the existence of infinitely many $Q$-points implies the existence of non-atomic $Q$-measures, and of $2^{\mathfrak{c}}$-many Tukey-top $Q$-points, strengthening results of Raghavan and of Borodulin-Nadzieja et al..

We develop a symbolic elimination theory for finite systems of recursive containment inequalities whose unknowns are convex subsets of a finite-dimensional real vector space. The right-hand sides are formal expressions generated from variables and parameters by convex linear combinations, finite union, and a positive geometric join encoding strict convex combinations. We prove that every parameter assignment has a unique smallest convex-set-valued solution and give a finite Gaussian-elimination-type procedure that eliminates the unknowns while preserving this solution and produces parameter-only expressions for its coordinate sets. More generally, let $\mathcal B$ be a family of subsets containing $\emptyset$ and closed under finite unions, nonnegative dilation, Minkowski sums, positive geometric joins, and convex hulls. If all parameter sets lie in $\mathcal B$, then every coordinate set of the smallest solution lies in $\mathcal B$; when these operations are effective, so is the resulting description. In particular, if the parameters are finite unions of hemihedra—where a hemihedron is a bounded convex semi-linear set, equivalently a convex finite union of relative interiors of polytopes—then each coordinate set is a hemihedron and admits a quantifier-free semi-linear description. We apply this theory to lamination hulls. For \[ V=U\oplus\bigoplus_{i=1}^{k}W_i,\qquad \dim W_i=1,\qquad \Lambda=\bigcup_{i=1}^{k}(U+W_i), \] we prove that the lamination hull $G_{\Lambda}^{(\infty)}(S)$ of every finite $S\subset V$ is semi-algebraic and effectively computable by a quantifier-free formula over the reals.

We study when embeddings lift from structures to their ultrapowers, and when an ultrapower embeds into its direct power. We offer a refinement of Blass's theorem on the Rudin-Keisler order and introduce cardinal invariants $\tau_\mathcal{A}$ which imply the embeddability $\mathcal{A}^I / \mathcal{U} \hookrightarrow \mathcal{A}^I$.

The strong tree property and the super tree property (also called ITP) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa$ is strongly compact if and only if the strong tree property holds at $\kappa$, and supercompact if and only if ITP holds at $\kappa$. Generalizing a result of Golshani and Hayut, we show that from large cardinals it is consistent for ITP to hold simultaneously at any countable initial segment of successors of singular cardinals. More formally, given any countable ordinal $\theta$, we construct a forcing extension in which ITP holds at the first $\theta$ successors of singulars. We then extend this result further to obtain the strong tree property on long segments of successors of singular cardinals of multiple cofinalities simultaneously.

We prove a Bauer-Poulsen dichotomy theorem for simplices of invariant measures associated with permutation groups. More precisely, let $G$ be a transitive group of permutations of a countable set $\mathcal{S}$, and let $H$ be the stabilizer of a point of $\mathcal{S}$. Let $\overline{G}$ and $\overline{H}$ denote their closures in the topology of pointwise convergence. Assume the Polish group $\overline{H}$ has relative Property (T) in $\overline{G}$. Then the simplex $\mathcal{M}_\mathrm{inv}(2^\mathcal{S})$ of invariant probability measures for the induced action $G\curvearrowright 2^\mathcal{S}$ is Bauer if and only if $\overline{G}$ has Property (T), and is Poulsen otherwise. This addresses some examples and questions considered by Austin. We deduce this result from a more general model-theoretic statement of independent interest. To this end, we initiate the study of existentially closed models in affine logic.

The logical problem of conditionals, as conceived by Stalnaker, amounts to axiomatizing a particular semantics for conditionals which utilizes selection functions that take propositions (i.e., sets of worlds) as arguments. While the sentential form of this semantics is recursively axiomatizable, we prove that its enrichment with first-order quantifiers is not—that is, we show that Stalnaker's logical problem of conditionals is unsolvable in the language with first-order quantifiers. We demonstrate this by showing how to interpret arithmetic in the logic. In the conclusion, we discuss the implications of this result for the study of conditional logic.

We present a new proof of the embedding of the classical modal logic S4 into the logic of Boolean Bunched Implications (BBI). While the original proof is semantical, this proof is entirely syntactical. It proceeds via Hilbert-style calculi, and is built by analogy with a recently discovered proof by Godel of the embedding of intuitionistic propositional logic into S4. We present the first full proofs of deduction theorems for BBI, which are used to show that the embedding is preserved by reasoning with assumptions, including where those assumptions are organised into bunches. Unlike the existing proof, our proof is stable under arbitrary axiomatic extensions of S4, and applies to all known axiomatic extensions of BBI in the literature. We observe how this embedding is related to semantical properties of both logics. Moreover, the proof extends gracefully to language extensions of BBI, as we show with hybrid BBI, classical BI, and sub-classical BBI.

We prove that the polymodal logic $\mathsf{J}$ is strongly complete with respect to \textit{$\mathsf{J}$-bouquets}, a topological refinement of its Kripke semantics. In particular, it is strongly topologically complete. This yields the following completeness result for the provability logic $\mathsf{GLP}$: a countable set of formulae $\Gamma$ is consistent with $\mathsf{GLP}$ if and only if there is a $\mathsf{J}$-bouquet $B$ and $r\in B$ such that $B, r\Vdash \mathsf{GLP}$ and $B, r\Vdash\Gamma$. In contrast, we exhibit counterexamples showing that $\mathsf{GLP}$ is not strongly complete with respect to Beklemishev-Gabelaia spaces.

We study a Lévy-Montague reflection scheme $\mathsf{Rfn}$ in second-order arithmetic: for each formula $\varphi$, the scheme asserts that every set belongs to a countable coded $\omega$-model such that $\varphi$ is absolute, at all parameters from the model, between the model and the universe. Our central result is a model extension construction: every countable model of $\mathsf{RCA}_0$ can be extended, without changing its first-order part, to a model of $\mathsf{WKL}_0$ together with the full scheme $\mathsf{Rfn}$. It follows at once that $\mathsf{WKL}_0+\mathsf{Rfn}$ is $\Pi^1_1$-conservative over both $\mathsf{WKL}_0$ and $\mathsf{RCA}_0$, that its first-order part is exactly $\mathrm{I}\Sigma_1$, and that it is $\Pi^0_2$-conservative over $\mathsf{PRA}$. The result opens an avenue for adopting, within a theory conservative over $\mathsf{PRA}$, Feferman's $\mathsf{ZFC}$-formalization of universe-based category-theoretic arguments that was achieved using Lévy-Montague reflection. The conservation proof itself, however, is non-finitary. The extension is the union of an $\omega_1$-tower of forcing extensions, and its uncountable cofinality is what secures reflection. We are only able to prove the conservation in $\mathsf{PRA}+\text{1-Con}(\mathsf{Z}_2)$. The results were obtained with extensive use of Anthropic's large language model Fable 5.

We study a discrete representation of contact relations on bounded distributive lattices and some of their reducts. In the course of the paper, we apply ideas and techniques developed by Ivo Düntsch, Dimiter Vakarelov, and Michael Winter, and we show—among others—that contact relations on a distributive join semi-lattice $L$ with the bottom element correspond to reflexive and symmetric relations which are closed in the product of the Stone space of $L$.