On first-order expressibility of satisfiability in submodels

Let $\kappa,\lambda$ be regular cardinals, $\lambda\le\kappa$, let $\varphi$ be a sentence of the language $\mathcal L_{\kappa,\lambda}$ in a given signature, and let $\vartheta(\varphi)$ express the fact that $\varphi$ holds in a submodel, i.e., any model $\mathfrak A$ in the signature satisfies $\vartheta(\varphi)$ if and only if some submodel $\mathfrak B$ of $\mathfrak A$ satisfies $\varphi$. It was shown in [1] that, whenever $\varphi$ is in $\mathcal L_{\kappa,\omega}$ in the signature having less than $\kappa$ functional symbols (and arbitrarily many predicate symbols), then $\vartheta(\varphi)$ is equivalent to a monadic existential sentence in the second-order language $\mathcal L^{2}_{\kappa,\omega}$, and that for any signature having at least one binary predicate symbol there exists $\varphi$ in $\mathcal L_{\omega,\omega}$ such that $\vartheta(\varphi)$ is not equivalent to any (first-order) sentence in $\mathcal L_{\infty,\omega}$. Nevertheless, in certain cases $\vartheta(\varphi)$ are first-order expressible. In this note, we provide several (syntactical and semantical) characterizations of the case when $\vartheta(\varphi)$ is in $\mathcal L_{\kappa,\kappa}$ and $\kappa$ is $\omega$ or a certain large cardinal

The umbilical cord of finite model theory

Model theory was born and developed as a part of mathematical logic. It has various application domains but is not beholden to any of them. A priori, the research area known as finite model theory would be just a part of model theory but didn't turn out that way. There is one application domain -- relational database management -- that finite model theory had been beholden to during a substantial early period when databases provided the motivation and were the main application target for finite model theory. Arguably, finite model theory was motivated even more by complexity theory. But the subject of this paper is how relational database theory influenced finite model theory. This is NOT a scholarly history of the subject with proper credits to all participants. My original intent was to cover just the developments that I witnessed or participated in. The need to make the story coherent forced me to cover some additional developments.Comment: To be published in the Logic in Computer Science column of the February 2023 issue of the Bulletin of the European Association for Theoretical Computer Scienc

The real numbers in inner models of set theory

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Joan BagariaWe study the structural regularities and irregularities of the reals in inner models of set theory. Starting with $L$, Gödel's constructible universe, our study of the reals is thus two-fold. On the one hand, we study how their generation process is linked to the properties of $L$ and its levels, mainly referring to [18]. We provide detailed proofs for the results of that paper, generalize them in some directions hinted at by the authors, and present a generalization of our own by introducing the concept of an infinite order gap, which is natural and yields some new insights. On the other hand, we present and prove some well-known results that build pathological sets of reals. We generalize this study to $L\left[\#_1\right]$ (the smallest inner model closed under the sharp operation for reals) and $L[\#]$ (the smallest inner model closed under all sharps), for which we provide some introduction and basic facts which are not easily available in the literature. We also discuss some relevant modern results for bigger inner models

Verifying Mutable Systems

Model checking has had much success in the verification of single-process and multi-process programs. However, model checkers assume an immutable topology which limits the verification in several areas. Consider the security domain, model checkers have had success in the verification of unicast structurally static protocols, but struggle to verify dynamic multicast cryptographic protocols. We give a formulation of dynamic model checking which extends traditional model checking by allowing structural changes, mutations, to the topology of multi-process network models. We introduce new mutation models when the structural mutations take either a primitive, non-primitive, or a non-deterministic form, and analyze the general complexities of each. This extends traditional model checking and allows analysis in new areas. We provide a set of proof rules to verify safety properties on dynamic models and outline its automizability. We relate dynamic models to compositional reasoning, dynamic cutoffs, parametrized analysis, and previously established parametric assertions.We provide a proof of concept by analyzing a dynamic mutual exclusion protocol and a multicast cryptography protocol

Two first-order logics of permutations

We consider two orthogonal points of view on finite permutations, seen as pairs of linear orders (corresponding to the usual one line representation of permutations as words) or seen as bijections (corresponding to the algebraic point of view). For each of them, we define a corresponding first-order logical theory, that we call $\mathsf{TOTO}$ (Theory Of Two Orders) and $\mathsf{TOOB}$ (Theory Of One Bijection) respectively. We consider various expressibility questions in these theories. Our main results go in three different direction. First, we prove that, for all $k \ge 1$, the set of $k$-stack sortable permutations in the sense of West is expressible in $\mathsf{TOTO}$, and that a logical sentence describing this set can be obtained automatically. Previously, descriptions of this set were only known for $k \le 3$. Next, we characterize permutation classes inside which it is possible to express in $\mathsf{TOTO}$ that some given points form a cycle. Lastly, we show that sets of permutations that can be described both in $\mathsf{TOOB}$ and $\mathsf{TOTO}$ are in some sense trivial. This gives a mathematical evidence that permutations-as-bijections and permutations-as-words are somewhat different objects.Comment: v2: minor changes, following a referee repor

Universism and extensions of V

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist