逻辑

We study the consistency strength of Lebesgue measurability for $\Sigma^1_3$ sets over Zermelo set theory ($Z$) in a completely choiceless context. We establish a result analogous to the Solovay-Shelah theorem.

In this paper, we propose two-sorted modal logics for the representation and reasoning of concepts arising from rough set theory (RST) and formal concept analysis (FCA). These logics are interpreted in two-sorted bidirectional frames, which…

In this paper together with the preceding Part I \cite{CHR}, we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to \cite{CHR} the treatment of the mixed…

We slightly generalize a notion of rank introduced by Glasner and Megrelishvili, which captures the oscillations of elements of Ellis semigroups, so that it can be applied to any compact Hausdorff space instead of being limited to the…

In this paper, we provide a Hilbert-style axiomatisation for the crisp bi-G\"{o}del modal logic $\KbiG$. We prove its completeness w.r.t.\ crisp Kripke models where formulas at each state are evaluated over the standard bi-G\"{o}del algebra…

Walker's Cancellation theorem for abelian groups tells us that if $A$ is finitely generated and $G$ and $H$ are such that $A \oplus G \cong A \oplus H$, then $G \cong H$. Michael Deveau showed that the theorem can be effectivized, but not…

We investigate a notion of inverse for neutrices inspired by Van den Berg and Koudjeti's decomposition of a neutrix as the product of a real number and an idempotent neutrix. We end up with an algebraic structure that can be characterized…

Nadkarni's Theorem asserts that for a countable Borel equivalence relation (CBER) exactly one of the following holds: (1) It has an invariant Borel probability measure or (2) it admits a Borel compression, i.e., a Borel injection that maps…

I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game-theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and…

Let $(\Gamma,+,F)$ be a finitely generated $\mathbb Z[F]$-module where $F$ is an injective endomorphism of the abelian group $\Gamma$. We restrict ourselves to a finite automa presentable subclass, introduced by J. Bell and R. Moosa in…

A partition is finitary if all its members are finite. For a set $A$, $\mathscr{B}(A)$ denotes the set of all finitary partitions of $A$. It is shown consistent with $\mathsf{ZF}$ (without the axiom of choice) that there exist an infinite…

We introduce and study a class of betweenness algebras-Boolean algebras with binary operators, closely related to ternary frames with a betweenness relation. From various axioms for betweenness, we chose those that are most common, which…

We prove a number of results relating the concepts of Keisler measures, generic stability, randomizations, and NIP formulas. Among other things, we do the following: (1) We introduce the notion of a Keisler-Morley measure, which plays the…

We prove that the provability logic of all provability predicates is exactly Fitting, Marek, and Truszczy\'nski's pure logic of necessitation $\mathsf{N}$. Moreover, we introduce three extensions $\mathsf{N4}$, $\mathsf{NR}$, and…

Let $k,\ell\geq 2$ be two multiplicatively independent integers. Cobham's famous theorem states that a set $X\subseteq \mathbb{N}$ is both $k$-recognizable and $\ell$-recognizable if and only if it is definable in Presburger arithmetic.…

This expository article is based on two lectures given by the first author at the Fields Institute in the Fall 2021 Thematic Program on Trends in Pure and Applied Model Theory. We give a detailed proof of a qualitative version of the…

We devise exact conditions under which a join semilattice with a weak contact relation can be semilattice embedded into a Boolean algebra with an overlap contact relation, equivalently, into a distributive lattice with additive contact…

We discuss two two-layered logics formalising reasoning with paraconsistent probabilities that combine the Lukasiewicz $[0,1]$-valued logic with Baaz $\triangle$ operator and the Belnap--Dunn logic.

The aim of Reverse Mathematics(RM for short)is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak…

Let $\mathcal{V}$ be a congruence permutable variety generated by a finite nilpotent algebra $\mathbf{A}$. If $\mathbf{A}$ is a product of algebras of prime power order, then the class $\mathcal{V}_\text{si}$ of subdirectly irreducible…