1,800 papers
2604.04492An effective version of the Stone duality
The paper studies computability-theoretic aspects of topological $T_0$-spaces. We introduce effective versions of the notions of a countable $c$-poset and a (second-countable) topological space with base. Based on this, we prove an effective version of the known Stone-type duality between the category $\mathbf{AS}$ (whose objects are almost semispectral spaces with base and whose morphisms are spectral mappings) and the category $\mathbf{DP}$ (whose objects are distributive $c$-posets and whose morphisms are strict mappings). Namely, we show that for an arbitrary set $Z\subseteq ω$, this duality is preserved when one restricts to objects which have $Z$-computably enumerable presentations only. Following this approach, we establish several results in computable topology. We prove that every degree spectrum of a countable algebraic structure can be realized as the degree spectrum of a topological space with base. We show that for any non-zero natural number $N$, there is a computable topological space with base that has precisely $N$-many computable copies, up to effective spectral homeomorphisms.
2604.04492Apr 2026
View2604.03932Cyclic group representations for relation algebras $57_{65}$ and $63_{65}$
We exhibit finite cyclic group representations for relation algebras $57_{65}$ and $63_{65}$. As a consequence, of the ten symmetric integral RAs on four atoms having at least one flexible atom, all are now known to have a representation over a finite cyclic group except for $33_{65}$, which is not even known to be finitely representable.
2604.03932Apr 2026
View2604.03825Tarskian truth theories over set theory
This work uses mostly model-theoretic methods to establish new proof-theoretic theorems about several axiomatic theories of truth over KP (Kripke-Platek set theory) and stronger theories, especially ZF (Zermelo-Fraenkel set theory).
2604.03825Apr 2026
View2604.03477Towards Trans-Exponential O-minimal Expansion of $(\mathbb{R},+,\cdot, 0, 1 <)$
We add an analytic trans-exponential function $\varphi$ to $\mathbb{R}_{an,\exp}$. We reduce the o-minimality of $\mathbb{R}_{an,\exp,\varphi}$ to the existence of "many" regular values for some definable systems of functions, which is a necessary condition for the o-minimality of $\mathbb{R}_{an,\exp,\varphi}$.
2604.03477Apr 2026
View2604.03080An unstable abstract elementary class of modules: A variation of Paolini-Shelah's example
We construct a class $\hat{K}$ of torsion-free abelian groups such that $\hat{\mathbf{K}}=(\hat{K}, \leq_p)$ is an abstract elementary class with $\operatorname{LS}(\hat{\mathbf{K}})=\aleph_0$ such that: $(\cdot)$ $\hat{\mathbf{K}}$ is not stable; $(\cdot)$ $\hat{\mathbf{K}}$ has the joint embedding property and no maximal models, but does not have the amalgamation property; $(\cdot)$ $\hat{\mathbf{K}}$ is $(<\aleph_0)$-tame. The class we construct is a variation of [PaSh, Section 4] which isolates the core mechanism of the Paolini-Shelah construction.
2604.03080Apr 2026
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A Concise Proof of the $L_0$ Dichotomy
Carroy, Miller, Schrittesser, and Vidnyánszky established the $L_0$ dichotomy: there is a Borel graph of Borel chromatic number three that admits a continuous homomorphism to every analytic graph of Borel chromatic number at least three. Their proof relies on a transfinite analysis of terminal approximations over a decreasing $ω_1$-sequence of analytic sets. I give a new, substantially shorter proof of this result by adapting the graph-theoretic framework recently introduced by Bernshteyn for the $G_0$ dichotomy. The central device is a $σ$-ideal of \emph{small} sets of homomorphisms from finite path approximations into the target graph, where smallness is witnessed by a bounded odd-walk condition on vertex projections. The key lemma that largeness is preserved under the doubling operation is established via the First Reflection Theorem, replacing the original transfinite construction with a single Borel reflection argument. The continuous homomorphism from the canonical graph $\mathbb{L}_c$ into the target is then obtained as a limit of shrinking families of copies, in direct analogy with Bernshteyn's proof for $G_0$.
2604.02589Apr 2026
View2604.03324The first fatal axiom for weakened sequential products on finite MV-effect algebras: Local obstruction, exact low-rank classification, and the rank-one boundary case
We study total binary operations on effect algebras obtained by truncating the Gudder-Greechie axiom package for a sequential product. The point is not to reprove the known nonexistence of non-Boolean full sequential products on finite chains, but to determine, axiom by axiom, where finite MV-effect algebras first fail. We prove two structural facts valid on every effect algebra. First, the operation σ_E(a,b) = 0 if a=0, and b if a \neq 0, satisfies (S1)-(S3), so (S3) is never fatal by itself. Second, any operation satisfying (S1)-(S4) already has the right-unit property a \circ 1 = a, even without (S5). From this we derive a local obstruction theorem: if an effect algebra contains an atom of finite isotropic index at least 2, then it admits no (S1)-(S4) operation. Consequently, a finite MV-effect algebra admits such an operation if and only if it is Boolean. In this precise sense, (S4) is the first fatal axiom on finite MV-effect algebras. On the constructive side, let E_u = [0,u] \subseteq Z^r be the simplicial interval representation of a finite MV-effect algebra. We show that additive maps E_u \to E_v are exactly the restrictions of positive group homomorphisms Z^r \to Z^s, equivalently maps x \mapsto Mx given by nonnegative integer matrices with Mu \le v. This yields a complete classification of (S1)+(S2) operations by row-wise subunital matrices. We then solve the first genuinely higher-rank (S1)-(S3) problem: on the rank-two Boolean algebra B_2 = E_{(1,1)} \cong C_1^2, all such operations are classified and there are exactly 34. Thus the finite-chain collapse at (S3) is a rank-one boundary phenomenon, whereas on finite MV-effect algebras the sharp threshold for nonexistence occurs exactly at (S4).
2604.03324Apr 2026
View2604.02092Bounded Ramsey's theorem for triples in computability theory
We study a restriction of Ramsey's theorem for 2-coloring of triples, in which homogeneous sets for color~1 are of bounded size ($\mathsf{BRT}^3_2$). We prove that the computational content of this statement is very close to Ramsey's theorem for pairs ($\mathsf{RT}^2_2)$, in that it satisfies the same known computability-theoretic upper bounds, but that $\mathsf{BRT}^3_2$ is not computably-reducible to $\mathsf{RT}^2_2$, even when allowing multiple applications of $\mathsf{RT}^2_2$.
2604.02092Apr 2026
View2604.02082Fischer-Servi logic does not have interpolation
We prove that the Fischer-Servi logic $\mathsf{IK}$ does not have the (Craig) interpolation property. This is obtained by showing that the corresponding class of modal Heyting algebras lacks the amalgamation property. We also generalize this result to some extensions of the Fischer-Servi logic such as $\mathsf{IT}$, $\mathsf{IK4}$, $\mathsf{IS4}$, and $\mathsf{IGL}$.
2604.02082Apr 2026
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The cohesive and stable Ramsey theorems and proof size over a weak base theory
We show that over the weak base theory $\mathrm{RCA}_0^*$, cohesive Ramsey's theorem for pairs $\mathrm{CRT}^2_2$ implies exponential closure of the definable cut $\mathrm{I}^0_1$, which is the intersection of all $Σ^0_1$-definable cuts. Consequences include non-elementary proof speedup of $\mathrm{RCA}_0^* + \mathrm{CRT}^2_2$ over $\mathrm{RCA}_0^*$ for $Π_1$ sentences and the unprovability of $\mathrm{CRT}^2_2$ in $\mathrm{RCA}_0^* + \mathrm{CAC}$. On the other hand, we show that $\mathrm{RCA}_0^* + \mathrm{SRT}^2_2$, where $\mathrm{SRT}^2_2$ is stable Ramsey's theorem for pairs, is polynomially simulated by $\mathrm{RCA}_0^*$ with respect to proofs of $\forall Π^0_3$ sentences. Nevertheless, $\mathrm{SRT}^2_2$ also implies a nontrivial property of $\mathrm{I}^0_1$, specifically closure under functions of quasipolynomial growth rate.
2604.01808Apr 2026
View2604.01041Lower Bounds on Inverse Cellular Automata via Proof Complexity
We study the complexity of inverse cellular automata on configurations of bounded size. Deciding injectivity in this setting is co-NP-complete by a theorem of Durand. We give a simpler proof of this theorem by a direct reduction from UNSAT to this problem, avoiding more complicated intermediate constructions. We also show that one direction of the reduction can be formalized in the weak theory of bounded arithmetic $V^0$. Durand's coNP-completeness result allows one to view inverse cellular automata acting on bounded size configurations as propositional proofs, cf. Cavagnetto, and we prove lower bounds on their size. The proof uses known lower bounds for bounded-depth Frege systems together with the Paris--Wilkie translation of arithmetic proofs into propositional proofs, which allows us to transfer proof complexity lower bounds to our setting.
2604.01041Apr 2026
View2604.00771Goodstein at the Second Threshold: An Independence Result for $ID_2$
The classical Goodstein process, defined via hereditary base-$k$ exponential normal form, is a well-known example of a principle unprovable in Peano Arithmetic. In this paper, we generalize this framework by constructing a new Goodstein process based on the Hardy hierarchy. We develop an ordinal notation system utilizing a two-step collapsing procedure, which yields a proof-theoretic ordinal of $ψ_0ψ_1(\varepsilon_{Ω_2+1})$. By defining $k$-normal forms for natural numbers within this system, we introduce a Goodstein-type process and demonstrate that the theory of non-iterated positive inductive definitions for two operators ($ID_2$) cannot prove its termination. This result establishes a new independence result at the second proof-theoretic threshold, further extending the reach of Goodstein-type principles beyond the Bachmann-Howard level.
2604.00771Apr 2026
View2604.00747Grothendieck's Equality vs Voevodsky's Equality
We discuss how canonical and universal constructions, properties and characterizations interact with equality in the framework of Homotopy Type Theory, comparing it with Grothendieck's use of equality and shedding further light on (efficient) formalisation of mathematics. This is achieved by investigating examples that range from monoids, groups, rings and modules to cohomology theories in the category of modules over commutative rings and culminate in a cohomological criterion of flatness.
2604.00747Apr 2026
View2604.00720Approximation of structures:local and global
We provide a mathematically rigorous definition of local approximation and demonstrate its applicability to some interesting classes of structures. In particular, we prove that any compact simple Lie group is locally approximated by finite groups. The definition and main examples are motivated by physics but the techniques are of model theory. Namely, we introduce the ultraproduct of emerging metric structures, which generalises the ultraproduct in metric model theory.
2604.00720Apr 2026
View2604.00194Point-free MV-topologies
We propose a point-free approach to MV-topological spaces in the wake of previous works on both classical and fuzzy topology. In order to do that, we introduce suitable frame-type structures and a class of fuzzy topological spaces which includes and suitably extends the one of MV-topological spaces. Then we show an adjoint situation between such structures, and restrict such an adjointness to a duality between the corresponding classes of ``spatial frames'' and ``sober spaces''. We also use neighbourhood systems to characterize sobriety in this context.
2604.00194Mar 2026
View2604.00123Imaginaries in perfect bounded pseudo algebraically closed fields with finitely many independent valuations
In this paper, we prove weak elimination of imaginaries for perfect bounded pseudo-algebraically closed fields equipped with finitely many independent valuations. Our approach combines an extension result for types to invariant types with an amalgamation theorem. As a special case, we obtain full elimination of imaginaries when the field is equipped with a single valuation.
2604.00123Mar 2026
View2604.00122Definable Functions to Quotients in Ordered Abelian Groups
In this paper we study definable families of functions from an ordered abelian group into various naturally arising definable quotients. We show that for an ordered abelian group $G$ and definable family of convex subgroups $\{D\}_{D\in\mathcal{D}}$, any definable family of functions $\{f_D\} _{D\in\mathcal{D}}$ with $f_D:G^d\rightarrow\frac{G}{D}$ is uniformly piecewise linear; for a prime $p$, integers $s,r\geq 1$, and groups $D^{[p^s]}$ defined later, if $f_D:G^d\rightarrow\frac{G}{D+p^rG}$ or $f_D:G^d\rightarrow\frac{G}{D^{[p^s]}+p^rG}$ we instead obtain that the definable family of functions is uniformly piecewise a boolean combination of linear functions to quotients by subgroups which are uniformly definable from $D$.
2604.00122Mar 2026
View2603.29930Non-Archimedean Analogue of Chase's Lemma
We formulate and verify a non-Archimedean counterpart of Chase's lemma. Following the framework by K.\ Eda removing restriction of cardinality from analogy on direct product between countable cardinal and non-$ω_1$-measurable cardinal, we extend the non-Archimedean counterpart of Chase's lemma to a non-Archimedean counterpart of the extension by K.\ Eda of the extension by M.\ Dugas and B.\ Zimmermann-Huisgen of Chase's lemma.
2603.29930Mar 2026
View2603.29528AKE principles for deeply ramified fields
We study the model theory of deeply ramified fields of positive characteristic. Generalizing the perfect case treated in work by Jahnke and Kartas on the model theory of perfectoid fields, we obtain Ax-Kochen/Ershov principles for certain deeply ramified fields of positive characteristic and fixed degree of imperfection. Our results apply in particular to all deeply ramified henselian valued fields of rank 1.
2603.29528Mar 2026
View2603.29521A General Theory of Class Symmetric Systems
We develop a general theory for class-sized symmetric systems as a natural extension of symmetric systems with respect to class forcing. In particular, adapting the usual notions of pretameness and tameness for class forcing, we present sufficient conditions for the preservation of the axioms of Gödel-Bernays set theory (without the axiom of choice), and for the forcing theorem to hold for class-sized symmetric systems.
2603.29521Mar 2026
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