We define Quillen model structures on a family of presheaf toposes arising from tree unravellings of Kripke models, leading to a homotopy theory for modal logic. Modal preservation theorems and the Hennessy-Milner property are revisited from a homotopical perspective.
The aim of this note is to show that many papers on various kinds of filters (and related concepts) in (subreducts of) residuated structures are in fact easy consequences of more general results that have been known for a long time.
We investigate the expressive power of a Turing-complete logic based on game-theoretic semantics. By defining suitable fragments and variants of the logic, we obtain a range of natural characterizations for some fundamental families of model classes.
Abstract Active matter with continuous energy injection that exhibits various nonequilibrium emergent behaviors, such as swarming and motility induced phase separation, has been extensively studied in the past decades. Achieving desired patterns and phases in fabricating functional materials by assembling active matter is a rising and challenging direction. Nevertheless, the ossibility of a stably ordered structure of active matter remains elusive. Toward this goal, the interplay between the active force and the volume exclusion provides a new way to manipulate mechanical stability. Here, we demonstrate a new type of active, two dimensional (2D) pseudocrystal system consisting of arrays of active rods. The pseudocrystals with tetratic array demonstrate robust stability against the thermal noise. Increasing the active force leads to a phase transition from pseudocrystal to swarming via shear melting. In the traveling pseudocrystals, on the contrary, the synchronized movement of active rods reduces internal stresses and enhances the stability of pseudocrystals. Topological defects quickly propagate in the traveling pseudocrystals and assist the stability. The present framework provides innovative insights into potentially new designs and manipulations of active materials. PACS numbers:
Answering a question of Hrušák, we show that every analytic tall ideal on $ω$ contains an $F_σ$ tall ideal. We also give an example of an $F_σ$ tall ideal without a Borel selector.
We investigate the logical structure of intuitionistic Kripke-Platek set theory IKP, and show that the first-order logic of IKP is intuitionistic first-order logic IQC.
We show that it is consistent relative to a huge cardinal that for all infinite cardinals $κ$, $\square_κ$ holds and there is a stationary $S \subseteq κ^+$ such that $\mathrm{NS}_{κ^+} \restriction S$ is $κ^{++}$-saturated.
We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [8] holds while the classical Borel Conjecture fails.
Let $G$ be a model of Presburger arithmetic. Let $\mathcal{L}$ be an expansion of the language of Presburger $\mathcal{L}_{Pres}$. In this paper we prove that the $\mathcal{L}$-theory of $G$ is $\mathcal{L}_{Pres}$-minimal iff it has the exchange property and any bounded definable set has a maximum.
We construct new models of $ZF$ with an uncountable set of reals that has a unique condensation point. This addresses a question by Sierpiński from 1918.
There is no bad group of Morley rank 2n+1 with an abelian Borel subgroup of Morley rank n. In particular, there is no bad group of Morley rank 3 (O. Fr{é}con).
There exists no bad group (in the sense of Gregory Cherlin), namely any simple group of Morley rank 3 is isomorphic to PSL2(K) for an algebraically closed field K.
Building on recent work of Philip Welch, we prove that (lightface) $Σ^0_3$ determinacy is equivalent to the existence of a wellfounded model satisfying the axiom scheme of (boldface) $\mathbfΠ^1_2$ monotone induction.
We give an affirmative answer to a question of Gorelic \cite{Gorelic}, by showing it is consistent, relative to the existence of large cardinals, that there is a proper class of cardinals $α$ with $cf(α)=ω_1$ and $α^ω> α.$
We give an example of a definable set in every free or torsion-free (non-elementary) hyperbolic group that is not in the Boolean algebra of equational sets. Hence, the theories of free and torsion-free (non-elementary) hyperbolic groups are not equational in the sense of G. Srour.
In this note we prove and disprove some chain conditions in type definable and definable groups in dependent, strongly dependent and strongly^{2} dependent theories.
We prove in this paper that the types of system F inhabited uniquely by ?I-terms (the I-types) have a positive quantifier. We give also consequences of this result and some examples.