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A syntactic approach to continuity of T-definable functionalsApr 22 2019We give a new proof of the well-known fact that all functions $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$ which are definable in G\"odel's System T are continuous via a syntactic approach. Differing from the usual syntactic method, we firstly perform ... More Compositionality of Rewriting Rules with ConditionsApr 19 2019We extend the notion of compositional associative rewriting as recently studied in the rule algebra framework literature to the setting of rewriting rules with conditions. Our methodology is category-theoretical in nature, where the definition of rule ... More Cantor-Bernstein implies Excluded MiddleApr 19 2019We prove in constructive logic that the statement of the Cantor-Bernstein theorem implies excluded middle. This establishes that the Cantor-Bernstein theorem can only be proven assuming the full power of classical logic. The key ingredient is a theorem ... More The incompleteness of an incompleteness argumentApr 18 2019G\"odel's argument for the First Incompleteness Theorem is, structurally, a proof by contradiction. This article intends to reframe the argument by, first, isolating an additional assumption the argument relies on, and then, second, arguing that the contradiction ... More Logics for first-order team propertiesApr 18 2019In this paper, we introduce a logic based on team semantics, called FOT, whose expressive power coincides with first-order logic both on the level of sentences and (open) formulas, and we also show that a sublogic of FOT, called FOT${}^\downarrow$, captures ... More Cubical Syntax for Reflection-Free Extensional EqualityApr 18 2019We contribute XTT, a cubical reconstruction of Observational Type Theory which extends Martin-L\"of's intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of identity types ... More A unified framework for notions of algebraic theoryApr 18 2019Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, ... More Re-pairing bracketsApr 17 2019Consider the following one-player game. Take a well-formed sequence of opening and closing brackets. As a move, the player can pair any opening bracket with any closing bracket to its right, erasing them. The goal is to re-pair (erase) the entire sequence, ... More The model theory of Cohen ringsApr 17 2019The aim of this article is to give an self-contained account of the algebra and model theory of Cohen rings, a natural generalization of Witt rings. Witt rings are only valuation rings in case the residue field is perfect, and Cohen rings arise as the ... More A Comparison of Pair and Triple Partition RelationsApr 16 2019This paper considers three different partition relations from partition calculus, two of which are pair relations and one of which is a triple relation. An examination of the first partition relation and the ramification argument used to prove it will ... More Completeness for Game LogicApr 16 2019Game logic was introduced by Rohit Parikh in the 1980s as a generalisation of propositional dynamic logic (PDL) for reasoning about outcomes that players can force in determined 2-player games. Semantically, the generalisation from programs to games is ... More A short proof of Thoma's theorem on type I groupsApr 16 2019In the theory of unitary group representations, a group is called type I if all factor representations are of type I, and by a celebrated theorem of James Glimm [Gli61b], the type I groups are precisely those groups for which the irreducible unitary representations ... More Separation of bounded arithmetic using a consistency statementApr 14 2019This paper proves Buss's hierarchy of bounded arithmetics $S^1_2 \subseteq S^2_2 \subseteq \cdots \subseteq S^i_2 \subseteq \cdots$ does not entirely collapse. More precisely, we prove that, for a certain $D$, $S^1_2 \subsetneq S^{2D+5}_2$ holds. Further, ... More On Loeb and sequential spaces in $\mathbf{ZF}$Apr 14 2019A topological space is called Loeb if the collection of all its non-empty closed sets has a choice function. In this article, in the absence of the axiom of choice, connections between Loeb and sequential spaces are investigated. Among other results, ... More Axiomatizing first-order consequences in inclusion logicApr 12 2019Inclusion logic is a variant of dependence logic that was shown to have the same expressive power as positive greatest fixed-point logic. Inclusion logic is not axiomatizable in full, but its first-order consequences can be axiomatized. In this paper, ... More Generalized topological semantics for weak modal logicsApr 12 2019The notion of \textit{general} or \textit{generalized} topology can be interpreted in many ways. In this paper, we shall adhere to the concept introduced by Cs\'{a}sz\'{a}r in \cite{csaszar}. Hence, we assume that generalized topology contains empty set ... More Uniform Interpolation and Compact CongruencesApr 12 2019Uniform interpolation properties are defined for equational consequence in a variety of algebras and related to properties of compact congruences on first the free and then the finitely presented algebras of the variety. It is also shown, following related ... More Implications of positive formulas in modules (RIMS)Apr 12 2019In this survey the role of implications of positive formulas -- finitary and infinitary -- is dicussed, in general and in module categories, where they seem of particular importance. A list of algebraic examples is given, some old, some rather new, and ... More The Ramsey property implies no mad familiesApr 11 2019We show that if all collections of infinite subsets of $\N$ have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. This solves a long-standing problem going back to Mathias \cite{mathias}. The proof exploits an idea ... More The Ramsey property implies no mad familiesApr 11 2019Apr 15 2019We show that if all collections of infinite subsets of $\N$ have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. This solves a long-standing problem going back to Mathias \cite{mathias}. The proof exploits an idea ... More Definable maximal cofinitary groups of intermediate sizeApr 11 2019Using almost disjoint coding, we show that for each $1<M<N<\omega$ consistently $\mathfrak{d}=\mathfrak{a}_g=\aleph_M<\mathfrak{c}=\aleph_N$, where $\mathfrak{a}_g=\aleph_M$ is witnessed by a $\Pi^1_2$ maximal cofinitary group. Weak factorization systems and stable independenceApr 11 2019We exhibit a bridge between the theory of weak factorization systems, a categorical concept used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the cofibrantly generated ... More O-minimal de Rham cohomologyApr 11 2019In the present paper we elaborate an o-minimal de Rham cohomology theory for abstract-definable $\mathcal{C}^p$ manifolds with $1\leq p\leq \infty$ in an o-minimal expansion of the real field which admits smooth cell decomposition and defines the exponential ... More On Whitney embedding of o-minimal manifoldsApr 10 2019We prove a definable version of the Whitney embedding theorem for abstract-definable $\mathcal{C}^p$ manifolds with $1\leq p<\infty$, namely: every abstract-definable $\mathcal{C}^p$ manifold is abstract-definable $C^p$ embedded into $R^N$, for some positive ... More The Complexity of Definability by Open First-Order FormulasApr 09 2019In this article we formally define and investigate the computational complexity of the Definability Problem for open first-order formulas (i.e., quantifier free first-order formulas) with equality. Given a logic $\mathbf{\mathcal{L}}$, the $\mathbf{\mathcal{L}}$-Definability ... More Derivatives of normal functions in reverse mathematicsApr 09 2019Consider a normal function $f$ on the ordinals (i. e. a function $f$ that is strictly increasing and continuous at limit stages). By enumerating the fixed points of $f$ we obtain a faster normal function $f'$, called the derivative of $f$. The present ... More Higher random indestructibility of MAD familiesApr 09 2019We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $\kappa$ is indestructible by the higher random forcing $\mathbb Q_\kappa$. We then use this characterisation to show that $\mathrm{add}(\mathbf{null}_\kappa) ... More Learning the undecidable from networked systemsApr 08 2019This article presents a theoretical investigation of computation beyond the Turing barrier from emergent behavior in distributed (or parallel) systems. In particular, we present an algorithmic network that is a mathematical model of a networked population ... Morecs.DCcs.MAcs.SImath.LO68Q30, 03D32, 68R10, 05C30, 05C78, 05C75, 05C60, 05C80, 05C82,
94A15, 68Q01, 03D10, 03D32, 03D35, 03D80 Kelley-Morse set theory does not prove the class Fodor principleApr 08 2019We show that Kelley-Morse set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\text{Ord}$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent ... More Enumeration degrees and non-metrizable topologyApr 08 2019The enumeration degrees of sets of natural numbers can be identified with the degrees of difficulty of enumerating neighborhood bases of points in a universal second-countable $T_0$-space (e.g. the $\omega$-power of the Sierpi\'nski space). Hence, every ... More A General Framework for the Semantics of Type TheoryApr 08 2019We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin-L\"{o}f type theory, two-level type theory and cubical type theory. We establish basic results in the semantics of type theory: every type ... More Unilateral weighted shifts on $\ell^2$Apr 08 2019Given a unilateral shift $B_w$ (determined by a bounded sequence $w$), a sequence $x \in \ell^2$ is "hypercyclic" for $w$ iff the forward iterates of $x$ under $B_w$ are dense in $\ell^2$. We show that it is possible to make the set of $x \in \ell^2$ ... More The complexity of 3-colouring $H$-colourable graphsApr 05 2019We study the complexity of approximation on satisfiable instances for graph homomorphism problems. For a fixed graph $H$, the $H$-colouring problem is to decide whether a given graph has a homomorphism to $H$. By a result of Hell and Ne\v{s}et\v{r}il, ... More Automating Resolution is NP-HardApr 05 2019We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatizable unless P = NP. Indeed, we show it is NP-hard to distinguish ... More Conceptual proofs of the Menger and Rothberger gamesApr 04 2019We provide conceptual proofs of the two most fundamental theorems concerning topological games and open covers: Hurewicz's Theorem concerning the Menger game, and Pawlikowski's Theorem concerning the Rothberger game. Controlling cardinal characteristics without adding realsApr 04 2019We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new ${<}\kappa$-sequences (for some regular $\kappa$). As an application, we present models (assuming the consistency of three strongly compact cardinals) ... More A small ultrafilter number at smaller cardinalsApr 04 2019It is proved to be consistent relative to a measurable cardinal that there is a uniform ultrafilter on the real numbers which is generated by fewer than the maximum possible number of sets. It is also shown to be consistent relative to a supercompact ... More On supercompactness of $ω_1$Apr 03 2019This paper studies structural consequences of supercompactness of $\omega_1$ under $\sf{ZF}$. We show that the Axiom of Dependent Choice $(\sf{DC})$ follows from "$\omega_1$ is supercompact". "$\omega_1$ is supercompact" also implies that $\sf{AD}^+$, ... More Inversion, Iteration, and the Art of Dual WieldingApr 02 2019The humble $\dagger$ ("dagger") is used to denote two different operations in category theory: Taking the adjoint of a morphism (in dagger categories) and finding the least fixed point of a functional (in categories enriched in domains). While these two ... More Temporal Landscapes: A Graphical Temporal Logic for ReasoningApr 01 2019We present an elementary introduction to a new logic for reasoning about behaviors that occur over time. This logic is based on temporal type theory. The syntax of the logic is similar to the usual first-order logic; what differs is the notion of truth ... More Hardy fields, the intermediate value property, and $ω$-freenessApr 01 2019We discuss the conjecture that every maximal Hardy field has the Intermediate Value Property for differential polynomials, and its equivalence to the statement that all maximal Hardy field are elementarily equivalent to the differential field of transseries. ... More Antichains of Copies of Ultrahomogeneous StructuresApr 01 2019We investigate possible cardinalities of maximal antichains in the poset of copies $\langle \mathbb P(\mathbb X),\subset \rangle$ of a countable ultrahomogeneous relational structure $\mathbb X$. It turns out that if the age of $\mathbb X$ has the strong ... More Aristotelian assertoric syllogisticMar 31 2019Aristotelian assertoric syllogistic, which is currently of growing interest, has attracted the attention of the founders of modern logic, who approached it in several (semantical and syntactical) ways. Further approaches were introduced later on. These ... More Notes on the tightness of $G_δ$-modificationsMar 31 2019We construct a countably tight normal $T_1$ space $X$ with $t(X_\delta) >2^\omega$. This is an answer to the question posed by Dow-Juh\'asz-Soukup-Szentmikl\'ossy-Weiss. We also show that if the continuum is not so large, then the tightness of $G_\delta$-modifications ... More Borel sets of Rado graphs and Ramsey's TheoremMar 30 2019The well-known Galvin-Prikry Theorem states that Borel subsets of the Baire space are Ramsey: Given any Borel subset $\mathcal{X}\subseteq [\omega]^{\omega}$, where $[\omega]^{\omega}$ is endowed with the metric topology, each infinite subset $X\subseteq ... More Modelling informational entropyMar 29 2019By 'informational entropy', we understand an inherent boundary to knowability, due e.g. to perceptual, theoretical, evidential or linguistic limits. In this paper, we discuss a logical framework in which this boundary is incorporated into the semantic ... More Sequent calculi and interpolation for non-normal logicsMar 27 2019G3-style Sequent calculi for the logics in the cube of non-normal modal logics and for their deontic extensions are introduced. For each of the calculi considered, we prove that weakening and contraction are height-preserving admissible, and we give a ... More Sequent calculi and interpolation for non-normal logicsMar 27 2019Apr 01 2019G3-style Sequent calculi for the logics in the cube of non-normal modal logics and for their deontic extensions are introduced. For each of the calculi considered, we prove that weakening and contraction are height-preserving admissible, and we give a ... More Partial results on dp-finite fieldsMar 27 2019We prove that NIP valued fields of positive characteristic are henselian. Furthermore, we partially generalize the known results on dp-minimal fields to dp-finite fields. We prove a dichotomy: if K is a sufficiently saturated dp-finite expansion of a ... More Guessing models imply the singular cardinal hypothesisMar 25 2019In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $\kappa \ge \omega_2$, $\textsf{ISP}(\kappa)$ implies that $\textsf{SCH}$ holds above $\kappa$, and (3) forcing posets which have ... More Infinite forcing and the generic multiverseMar 24 2019In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique ... More Degree spectra for transcendence in fieldsMar 23 2019We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e. degrees above ... More On extensions of partial isometriesMar 22 2019In this paper we define a notion of S-extension for a metric space and study minimality and coherence of S-extensions. We give a complete characterization of all finite minimal S-extensions of a given finite metric space. We also define a notion of ultraextensive ... More Effective Aspects of Bernoulli RandomnessMar 22 2019In this paper, we study Bernoulli random sequences, i.e., sequences that are Martin-L\"of random with respect to a Bernoulli measure $\mu_p$ for some $p\in[0,1]$, where we allow for the possibility that $p$ is noncomputable. We focus in particular on ... More A Note on OTM-Realizability and Constructive Set TheoriesMar 21 2019We define an ordinalized version of Kleene's realizability interpretation of intuitionistic logic by replacing Turing machines with Koepke's ordinal Turing machines (OTMs), thus obtaining a notion of realizability applying to arbitrary statements in the ... More Coalgebraic Geometric LogicMar 21 2019Mar 22 2019Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor $\mathsf{T}$ on some full subcategory of the category ... More Coalgebraic Geometric LogicMar 21 2019Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor $\mathsf{T}$ on some full subcategory of the category ... More Solovay reduction and continuityMar 20 2019The objective of this study is a better understanding of the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction by the existence ... More Oligomorphic groups are essentially countableMar 20 2019We study the complexity of the isomorphism relation on classes of closed subgroups of $S_\infty$, the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. A closed subgroup ... More Sets and ProbabilityMar 20 2019In this article the idea of random variables over the set theoretic universe is investigated. We explore what it can mean for a random set to have a specific probability of belonging to an antecedently given class of sets. Mekler's construction and tree propertiesMar 17 2019Mekler constructed a way to produce a pure group from any given structure where the construction preserves $\kappa$-stability for any cardinal $\kappa$. Not only the stability, it is known that his construction preserves various model-theoretic properties ... More Coding in graphs and linear orderingsMar 16 2019There is a Turing computable embedding $\Phi$ of directed graphs $A$ in undirected graphs. Moreover, there is a fixed tuple of formulas that give a uniform interpretation; i.e., for all directed graphs $A$, these formulas interpret $A$ in $\Phi(G)$. It ... More Multi-topological semantics for intuitionistic modal logicMar 16 2019We present three examples of \textit{multi-topological} semantics for intuitionistic modal logic with one modal operator $\Box$ (which behaves in some sense like necessity). We show that it is possible to treat neighborhood models, introduced earlier, ... More Uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sumsMar 15 2019This thesis contains some new results on the uniform rationality of Poincar\'e series of p-adic equivalence relations and Igusa's conjecture on exponential sumsmath.NTmath.AGmath.LO11L07, 03C98, 14E18, 11L05, 11S40, 11U09, 12L12, 11F85, 11T23,
14M25, 03C60, 03C10, 11M41 Dp-minimal expansions of discrete ordered abelian groupsMar 14 2019If $\mathscr{Z}$ is a dp-minimal expansion of a discrete ordered abelian group $(Z,<,+)$ and $\mathscr{Z}$ does not admit a nontrivial definable convex subgroup then $\mathscr{Z}$ is interdefinable with $(Z,<,+)$ and $(Z,<,+)$ is elementarily equivalent ... More Mutual CoinductionMar 14 2019Mar 21 2019In this paper we present mutual coinduction as a dual of mutual induction and also as a generalization of standard coinduction. In particular, we present a precise formal definition of mutual induction and mutual coinduction. In the process we present ... More Mutual CoinductionMar 14 2019Mar 19 2019In this paper we present mutual coinduction as a dual of mutual induction and as a generalization of standard coinduction. In particular, we present a formal definition of mutual induction and mutual coinduction, and in the process we present, and prove, ... More Mutual CoinductionMar 14 2019In this paper we present mutual coinduction as a dual of mutual induction and as a generalization of standard coinduction. In particular, we present a formal definition of mutual induction and mutual coinduction, and in the process we present, and prove, ... More Mutual CoinductionMar 14 2019Apr 08 2019In this paper we present mutual coinduction as a dual of mutual induction and also as a generalization of standard coinduction. In particular, we present a precise formal definition of mutual induction and mutual coinduction. In the process we present ... More A two-dimensional metric temporal logicMar 14 2019We introduce a two-dimensional metric (interval) temporal logic whose internal and external time flows are dense linear orderings. We provide a suitable semantics and a sequent calculus with axioms for equality and extralogical axioms. Then we prove completeness ... More