Related papers: On $\Sigma_1$-Definable Closed Unbounded Sets

For each countable ordinal $\alpha \ge 2$, the ideals $\mathsf{conv}_\alpha$ were introduced in ``Critical ideals for countable compact spaces'' (to appear in Fund. Math., see also arXiv:2503.12571) to characterize compact countable spaces…

Given a strong limit cardinal $\lambda$ of countable cofinality, we show that if every $\lambda$-coanalytic subset of the generalised Cantor space ${}^{\lambda}2$ has the $\lambda$-$\mathsf{PSP}$, then there is an inner model with…

The assertion that every definable set has a definable element is equivalent over ZF to the principle $V=\text{HOD}$, and indeed, we prove, so is the assertion merely that every $\Pi_2$-definable set has an ordinal-definable element.…

We show that certain families of sets in $\mathbb{R}^2$ (or $\mathbb{R}^n$) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure.

We show that the tree property, stationary reflection and the failure of approachability at $\kappa^{++}$ are consistent with $\mathfrak{u}(\kappa) = \kappa^+ < 2^\kappa$, where $\kappa$ is a singular strong limit cardinal with the…

We study pushdown systems where control states, stack alphabet, and transition relation, instead of being finite, are first-order definable in a fixed countably-infinite structure. We show that the reachability analysis can be addressed…

We investigate states on von Neumann algebras which are not normal but enjoy various forms of infinite additivity, and show that these exist on $B(H)$ if and only if the cardinality of an orthonormal basis of $H$ satisfies various large…

In the context of large cardinals, the classical diamond principle Diamond_kappa is easily strengthened in natural ways. When kappa is a measurable cardinal, for example, one might ask that a Diamond_kappa sequence anticipate every subset…

We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity of these sets for the standard notions of finite learning,…

This paper presents a theory of systemic undecidability, reframing incomputability as a structural property of systems rather than a localized feature of specific functions or problems. We define a notion of causal embedding and prove a…

Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We…

Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single Set Theory formula with parameters unrelated to other formulas. Exotic expressions involving…

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal…

We show that the order dimension of the partial order of all finite subsets of $\kappa$ under set inclusion is ${\log}_{2}({\log}_{2}(\kappa))$ whenever $\kappa$ is an infinite cardinal. We also show that the order dimension of any locally…

We study the strength of well-founded ultrafilters on ordinals above choiceless large cardinals and their associated Prikry forcings. Gabriel Goldberg showed that all but boundedly many regular cardinals above a rank Berkeley cardinal carry…

We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality $\kappa$, where $\kappa$ is a regular cardinal. The corresponding new notion is…

We explore different generalizations of the classical concept of independent families on $\omega$ following the study initiated by Fisher and Montoya. We show that under $\diamondsuit^*(\kappa)$ we can get strongly independent families on…

Suppose L = {<, . . .} is any countable first order language in which < is interpreted as a linear order. Let T be any complete first order theory in the language L such that T has a kappa-like model where kappa is an inaccessible cardinal.…

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are…

V\"a\"an\"anen and Welch asked in the paper "When cardinals determine the power set: inner models and H\"artig quantifier logic" which large cardinals are consistent with the power set operation $x\mapsto P(x)$ being $\Sigma_1$-definable in…