逻辑

For every natural number $n$ we introduce a new weak choice principle $\mathrm{nRC_{fin}}$: Given any infinite set $x$, there is an infinite subset $y\subseteq x$ and a selection function $f$ that chooses an $n$-element subset from every…

We explore the expressive power of HOL, a system of higher-order logic, and its relationship to the simply-typed lambda calculus and Church's simple theory of types, arguing for the potential of HOL as a unifying logical framework, capable…

We give a complete first-order axiomatization of the structure $(\mathbb{Z},+,(\ell^{\mathbb{N}})_{\ell\in L})$, where $L \subseteq \mathbb{Z}_{\ge 2}$ is a set of pairwise multiplicatively independent integers and $\ell^{\mathbb{N}} =…

Affine continuous logic is extended to affine integration logic. Affine compactness theorem is proved by both the ultramean construction and Henkin's method. Also, a proof system and a completeness theorem are given. An appropriate variant…

In this paper we present a simple approach to big Ramsey combinatorics of the Cantor set $2^\omega$. Using Infinite Dual Ramsey Theorem of Carlson and Simpson, we show that $2^\omega$, viewed as a topological space, has finite big Ramsey…

This paper introduces an alternative approach to proving the existence of choice functions for specific families of sets within Zermelo-Fraenkel set theory (ZF) without assuming any form on the Axiom of Choice (AC). Traditional methods of…

We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some…

We exhibit a geometric morphism from the Grothendieck topos representing the Solovay model to the $\kappa$-pyknotic sets of Barwick--Haine and Clausen--Scholze. We then use the properties of this morphism and automatic continuity in the…

If $T$ has dependent dividing, then the burden agrees with the dp-rank witnessed by NIP formulas. We use this observation to prove that if $T$ has dependent dividing, then the burden is sub-additive. We also state a connection between the…

We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global…

We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type $p\in S(A)$ is weakly o-minimal if for some relatively $A$-definable linear order, $<$, on $p(\mathfrak{C})$ every…

Let $T$ be a countable complete first-order theory with a definable, infinite, discrete linear order. We prove that $T$ has continuum-many countable models. The proof is purely first-order, but raises the question of Borel completeness of…

Given a Borel class of trees, we show that there is a tree in that class whose Scott sentence is not too much more complicated than the definition of the class. In particular, if the class is definable by a $\Pi_\alpha$ sentence, then there…

Every bounded definable open set is a union of finitely many open strong cells in a weakly o-minimal expansion of a real closed field. We prove this fact and another theorem similar to it.

As a topological generalization of the notion of a multiset, a boolean multispace is a boolean space $X$ with a continuous function $u\colon X\to \mathbb Z_{>0}$, where $\mathbb Z_{>0}=\{1,2,\dots\}$ has the discrete topology. In this paper…

The technique of symmetric extensions is derived from forcing and it is one of the most important tools for studying models without the Axiom of Choice. Despite being incredibly successful since the 1960s, our understanding of the technique…

This paper makes significant progress towards resolving a conjecture relating strong forcing axioms like $PFA$ and the derived model at a limit of Woodin cardinals $\kappa$. In particular, using a concept called Covering Matrices, we show…

We develop a framework, in the style of Adler, for interpreting the notion of "witnessing" that has appeared (usually as a variant of Kim's Lemma) in different areas of neostability theory as a binary relation between abstract independence…

We investigate forms of filter extension properties in the two-cardinal setting involving filters on $P_\kappa(\lambda)$. We generalize the filter games introduced by Holy and Schlicht in \cite{HolySchlicht:HierarchyRamseyLikeCardinals} to…

We introduce the notion of Ramsey partition regularity, a generalisation of partition regularity involving infinitary configurations. We provide characterisations of this notion in terms of certain ultrafilters related to tensor products…