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An extensional $λ$-model with $\infty$-grupoid structureJun 13 2019From a topological space, a set with $\infty$-grupoid structure is built and this construction is applied to the case of ordered sets equipped with the Scott topology. The main purpose is to project the $\lambda$-model $D_\infty$ of Dana Scott to an extensional ... More On discrete idempotent pathsJun 13 2019The set of discrete lattice paths from (0, 0) to (n, n) with North and East steps (i.e. words w $\in$ { x, y } * such that |w| x = |w| y = n) has a canonical monoid structure inherited from the bijection with the set of join-continuous maps from the chain ... More Definability, interpretations and étale fundamental groupsJun 12 2019The aim of the paper and of a wider project is to translate main notions of anabelian geometry into the language of model theory. Here we finish with giving the definition of the \'etale fundamental group $\pi^{et}_1(X,x)$ of a non-singular quasiprojective ... More A First-Order Framework for Inquisitive Modal LogicJun 12 2019We present a natural standard translation of inquisitive modal logic InqML into first-order logic over the natural two-sorted relational representations of the intended models, which captures the built-in higher-order features of InqML. This translation ... More Bounds on Scott Ranks of Some Polish Metric SpacesJun 11 2019If $\mathcal{N}$ is a proper Polish metric space and $\mathcal{M}$ is any countable dense submetric space of $\mathcal{N}$, then the Scott rank of $\mathcal{N}$ in the natural first order language of metric spaces is countable and in fact at most $\omega_1^{\mathcal{M}} ... More Big Ramsey degrees of 3-uniform hypergraphsJun 10 2019Given a countably infinite hypergraph $\mathcal R$ and a finite hypergraph $\mathcal A$, the big Ramsey degree of $\mathcal A$ in $\mathcal R$ is the least number $L$ such that, for every finite $k$ and every $k$-colouring of the embeddings of $\mathcal ... More Paraconsistency, resolution and relevanceJun 08 2019Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models for inconsistent ... More Initial self-embeddings of models of set theoryJun 07 2019By a classical theorem of Harvey Friedman (1973), every countable nonstandard model $\mathcal{M}$ of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding $j$, i.e., $j$ is a self-embedding of $\mathcal{M}$ such that $j[\mathcal{M}]\subsetneq\mathcal{M}$, ... More Compactness properties defined by open-point gamesJun 07 2019Let S be a topological property of sequences (such as, for example, "to contain a convergent subsequence" or "to have an accumulation point"). We introduce the following open-point game OP(X,S) on a topological space X. In the n'th move, Player A chooses ... More Bialgebraic Semantics for String DiagramsJun 04 2019Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental ... More Type-theoretic algebraic weak factorisation systemsJun 04 2019Motivated by Homotopy Type Theory, we introduce type-theoretic algebraic weak factorisation systems and show how they give rise to models of Martin-L\"of type theory. This is done by showing that the comprehension category associated to a type-theoretic ... More Towers and gaps at uncountable cardinalsJun 03 2019Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $\mathfrak p(\kappa)=\mathfrak ... More How strong are single fixed points of normal functions?Jun 03 2019In a recent paper by M. Rathjen and the present author it has been shown that the statement ``every normal function has a derivative'' is equivalent to $\Pi^1_1$-bar induction. The equivalence was proved over $\mathbf{ACA_0}$, for a suitable representation ... More Type-theoretic weak factorization systemsJun 01 2019This article presents three characterizations of the weak factorization systems on finitely complete categories that interpret intensional dependent type theory with Sigma-, Pi-, and Id-types. The first characterization is that the weak factorization ... More Analytic P-ideals and Banach spacesMay 31 2019We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of analytic P-ideals ... More Differential Equation Invariance AxiomatizationMay 31 2019This article proves the completeness of an axiomatization for differential equation invariants described by Noetherian functions. First, the differential equation axioms of differential dynamic logic are shown to be complete for reasoning about analytic ... More Rules with parameters in modal logic IIMay 30 2019We analyze the computational complexity of admissibility and unifiability with parameters in transitive modal logics. The class of cluster-extensible (clx) logics was introduced in the first part of this series of papers. We completely classify the complexity ... More Consistency of circuit lower bounds with bounded theoriesMay 30 2019Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the ... More The BF Calculus and the Square Root of NegationMay 30 2019The concept of imaginary logical values was introduced by Spencer-Brown in Laws of Form, in analogy to the square root of -1 in the complex numbers. In this paper, we develop a new approach to representing imaginary values. The resulting system, which ... More Resolution Lower Bounds for Refutation StatementsMay 29 2019For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three applications. (1) An open question in (Atserias, M\"uller ... More On noncommutative generalisations of Boolean algebrasMay 29 2019Skew Boolean algebras (skew BA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed skew BAs, axiomatised here as ... More Identity crises between supercompactness and Vopenka's PrincipleMay 29 2019In this paper we study the notion of $C^{(n)}$-supercompactness introduced by Bagaria in \cite{Bag} and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{(1)}$-supercompact ... More Taking the path computably travelledMay 29 2019We define a real $A$ to be low for paths in Baire space (or Cantor space) if every $\Pi^0_1$ class with an $A$-computable element has a computable element. We prove that lowness for paths in Baire space and lowness for paths in Cantor space are equivalent ... More Remarks on generic stability in independent theoriesMay 28 2019In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of "generic stability" in arbitrary theories. Among other things, we show that the standard definition of generic stability for types ... More Modelling competing theoriesMay 28 2019We introduce a complete many-valued semantics for basic normal lattice-based modal logic. This semantics is based on reflexive many-valued graphs. We discuss an interpretation and possible applications of this logical framework in the context of the formal ... More Modal Logics that Bound the Circumference of Transitive FramesMay 28 2019For each natural number $n$ we study the modal logic determined by the class of transitive Kripke frames in which there are no strictly ascending chains and no cycles of length greater than $n$. The case $n=0$ is the G\"odel-L\"ob provability logic. Each ... More Tame pairs, Definable types and Pro-definabilityMay 27 2019We show (strict) pro-definability of spaces of definable types in various classical first order theories, including o-minimal expansions of divisible abelian groups, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed ... More Epistemic Logic with Partial Dependency OperatorMay 27 2019In this article, we introduce $\textit{partial}$ dependency modality $\mathcal{D}$ into epistemic logic so as to reason about $\textit{partial}$ dependency relationship in Kripke models. The resulting dependence epistemic logic possesses decent expressivity ... More Continuous Sentences Preserved Under Reduced ProductsMay 25 2019Answering a question of Cif\'u Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced ... More Tall cardinals in extender modelsMay 24 2019We obtain a characterization of $\lambda$-tall cardinals in terms of the function $o(\alpha)$ in extender models $L[E]$ which have no inner model with a Woodin caridnal and $L[E] \models \text{``I am iterable''}$. This implies in the equivalence between ... More Dependent products and 1-inaccessible universesMay 24 2019The purpose of this writing is to show that, if we use the definition of elementary $\infty$-topos that has been proposed by Mike Shulman, then the fact that every geometric $\infty$-topos satisfies the required axioms, more specifically the last one ... More Models of set theory in which separation theorem failsMay 24 2019We make use of a finite support product of the Jensen minimal forcing to define a model of set theory in which the separation theorem fails for projective classes $\mathbf\Sigma^1_n$ and $\mathbf\Pi^1_n$, for a given $n\ge3$. The Core Logic ParadoxMay 23 2019This paper provides a proof that Tennant's logical system entails a paradox that is called Core logic paradox, in reference to the new name given by Tennant to his intuitionistic relevant logic. On the Bourbaki's fixed point theorem and the axiom of choiceMay 23 2019In this note we generalize the Moroianu's fixed point theorem. We propose a very elegant common proof of the Bourbaki's fixed point theorem and our result. We apply our result to give a very elegant proof of the fact that, in the Zermelo-Fraenkel system, ... More On the Bourbaki's fixed point theorem and the axiom of choiceMay 23 2019Jun 03 2019In this note we generalize the Moroianu's fixed point theorem. We propose a very elegant common proof of the Bourbaki's fixed point theorem and our result. We apply our result to give a very elegant proof of the fact that, in the Zermelo-Fraenkel system, ... More Co-theory of sorted profinite groups for PAC structuresMay 23 2019We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster ... More PAC structures in nutshellMay 23 2019An expository paper written down after RIMS Model Theory Workshop 2018. To appear in RIMS Kokyuroku. On a Question of JaegersMay 23 2019We show that there exists a positive arithmetical formula $\psi(x,R)$, where $x \in \om$, $R \subseteq \om$, with no hyperarithmetical fixed point. This answers a question of Gerhard Jaeger. As corollaries we obtain results on: (a) the proof-theoretic ... More Rooted Hypersequent Calculus for Modal Logic S5May 22 2019We present a rooted hypersequent calculus for modal propositional logic S5. We show that all rules of this calculus are invertible and that the rules of weakening, contraction, and cut are admissible. Soundness and completeness are established as well. ... More Complete Positivity for Mixed Unitary CategoriesMay 21 2019In this article we generalize the $\CP^\infty$-construction of dagger monoidal categories to mixed unitary categories. Mixed unitary categories provide a setting, which generalizes (compact) dagger monoidal categories and in which one may study quantum ... More Generic planar algebraic vector fields are disintegratedMay 21 2019In this article, we study model-theoretic properties of algebraic differential equations of order $2$, defined over constant differential fields. In particular, we show that the existentially closed theory associated to a general differential equation ... More SRT22 does not imply COH in omega-modelsMay 21 2019In this article, we prove that every $\Delta^0_2$ set has an infinite subset in it or its complement whose Turing jump is not of PA degree relative to $\emptyset'$. We use the relativize statement to build an $\omega$-model of stable Ramsey's theorem ... More SRT22 does not imply COH in omega-modelsMay 21 2019May 24 2019In this article, we prove that every $\Delta^0_2$ set has an infinite subset in it or its complement whose Turing jump is not of PA degree relative to $\emptyset'$. We use the relativize statement to build an $\omega$-model of stable Ramsey's theorem ... More Corson reflectionsMay 20 2019A reflection principle for Corson compacta holds in the forcing extension obtained by Levy-collapsing a supercompact cardinal to~$\aleph_2$. In this model, a compact Hausdorff space is Corson if and only if all of its continuous images of weight~$\aleph_1$ ... More Trois couleurs: A new non-equational theoryMay 20 2019A first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening ... More Realizing realizability results with classical constructionsMay 20 2019J.L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e. symmetric extensions. We also provide a new condition for preserving ... More Realizing realizability results with classical constructionsMay 20 2019Jun 10 2019J.L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e. symmetric extensions. We also provide a new condition for preserving ... More Rough sets and three-valued structuresMay 19 2019In recent years, many papers have been published showing relationships between rough sets and some lattice theoretical structures. We present here some strong relations between rough sets and three-valued {\L}ukasiewicz algebras. Montague Semantics for Lambek PregroupsMay 17 2019Lambek pregroups are algebraic structures modelling natural language syntax, while Montague captures semantics as a homomorphism from syntax to first-order logic. We introduce the variant RelCoCat of the categorical compositional distributional (DisCoCat) ... More Generic derivations on o-minimal structuresMay 17 2019Let $T$ be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language $L$. We study derivations $\delta$ on models $\mathcal{M}\models T$. We introduce the notion of a $T$-derivation: ... More Generic derivations on o-minimal structuresMay 17 2019May 26 2019Let $T$ be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language $L$. We study derivations $\delta$ on models $\mathcal{M}\models T$. We introduce the notion of a $T$-derivation: ... More Embeddings into outer modelsMay 15 2019We explore the possibilities for elementary embeddings $j : M \to N$, where $M$ and $N$ are models of ZFC with the same ordinals, $M \subseteq N$, and $N$ has access to large pieces of $j$. We construct commuting systems of such maps between countable ... More Enriched Lawvere Theories for Operational SemanticsMay 14 2019Enriched Lawvere theories are a generalization of Lawvere theories that allow us to describe the operational semantics of formal systems. For example, a graph-enriched Lawvere theory describes structures that have a graph of operations of each arity, ... More Vaught's Conjecture for Almost Chainable TheoriesMay 14 2019A structure ${\mathbb Y}$ of a relational language $L$ is called almost chainable iff there are a finite set $F \subset Y$ and a linear order $<$ on the set $Y\setminus F$ such that for each partial automorphism $\varphi$ (i.e., local automorphism, in ... More Residuation in lattice effect algebrasMay 14 2019We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such a lattice can be converted into a lattice effect ... More Finite burden in multivalued algebraically closed fieldsMay 13 2019We prove that an expansion of an algebraically closed field by $n$ arbitrary valuation rings is NTP${}_2$, and in fact has finite burden. It fails to be NIP, however, unless the valuation rings form a chain. Moreover, the incomplete theory of algebraically ... More Definable Maximal Independent FamiliesMay 12 2019We study maximal independent families (m.i.f.) in the projective hierarchy. We show that (a) the existence of a $\boldsymbol{\Sigma}^1_2$ m.i.f. is equivalent to the existence of a $\boldsymbol{\Pi}^1_1$ m.i.f., (b) in the Cohen model, there are no projective ... More Sequent-Type Proof Systems for Three-Valued Default LogicMay 12 2019Sequent-type proof systems constitute an important and widely-used class of calculi well-suited for analysing proof search. In my master's thesis, I introduce sequent-type calculi for a variant of default logic employing \Lukasiewicz's three-valued logic ... More Rough Contact in General Rough MereologyMay 12 2019Theories of rough mereology have originated from diverse semantic considerations from contexts relating to study of databases, to human reasoning. These ideas of origin, especially in the latter context, are intensely complex. In this research, concepts ... More Quantifying information flow in interactive systemsMay 10 2019We consider the problem of quantifying information flow in interactive systems, modelled as finite-state transducers in the style of Goguen and Meseguer. Our main result is that if the system is deterministic then the information flow is either logarithmic ... More