75 papers
This paper characterizes the gauge profiles of sets of reals defined by effective dimension and demonstrates a separation between these sets and -well approximable numbers using Hausdorff measure.
This paper revisits and improves upon Erdős and Rado's proof regarding well-quasi-ordered finite-image sequences of length to establish that the maximum linearization of such sequences is bounded by a function roughly -times exponential in the maximum linearization of the underlying well-quasi-order, a bound shown to be nearly tight for .
The paper demonstrates that every Hardy field can be extended to an -free Hardy field, a result that connects to classical oscillation criteria and is used to resolve questions posed by Boshernitzan and generalize one of his theorems.
This paper reinterprets Gödel's second incompleteness theorem through proof-theoretic semantics by demonstrating that while certain arithmetical theories cannot prove their own consistency, they nonetheless semantically support it via a compositional notion of consequence based on inferential roles, thereby reframing incompleteness as a divergence between derivability and internal semantic support rather than a gap between syntax and external truth.
This paper extends the correspondence between automorphism groups and -categorical structures to the locally Roelcke precompact and locally -categorical settings by defining the latter, proving a Ryll-Nardzewski theorem, characterizing the associated groups via isometric actions, and establishing that bi-interpretability of structures is equivalent to the isomorphism of their automorphism groups.
This paper proposes a topic-sensitive hyperintensional semantics to model ignorance as a content-dependent attitude, offering sound and complete logical systems for three distinct forms of ignorance while resolving the problem of logical omniscience by accounting for an agent's capacity to grasp proposition content.
This paper establishes that for an inaccessible cardinal , the isomorphism relation of any theory with fewer than non-isomorphic models of size is continuously reducible to the isomorphism relation of any unstable or superstable non-classifiable theory within the space of regressive functions on .
This paper introduces a specific multi-sorted expansion of abelian lattice-ordered groups, equivalent to equipping them with a spectral subspace, and proves that this structure admits a model companion that is complete and possesses quantifier elimination.
This paper characterizes the conditions under which sets of -Cohen, -Mathias, and -Sacks generics possess positive Hausdorff measure, establishing that the measure is positive if and only if the gauge function is not dominated by (for Cohen generics) or eventually dominates (for Mathias and Sacks generics) every element in the Turing ideal .
The paper establishes that the consistency of with being measurable implies the consistency of where is simultaneously the least strongly regular and least measurable cardinal, while all uncountable cardinals below it have countable cofinality.
This paper establishes a non-standard analysis framework that unifies coherent risk measures and their finite-sample estimators through hyperfinite representations, yielding discrete Kusuoka formulae, plug-in consistency results, and asymptotic normality via a transparent probability-to-statistics dictionary.
This paper investigates the number of isomorphism types of Banach spaces for compact spaces of a given weight, proving that there are exactly $2^\kappa\kappa\omega_12^{\omega_1}$) depends on additional set-theoretic axioms such as the Continuum Hypothesis or Baumgartner's axiom.
This paper establishes connections between the infinite and finite sunflower properties and canonical Ramsey properties for countable ultrahomogeneous relational structures, proving their equivalence or implication under specific conditions and demonstrating that these properties hold for various classes including free amalgamation classes and finite metric spaces.
This paper establishes a framework for limit-stage filter constructions in countable-support symmetric iterations that ensures the resulting models satisfy ZF and Dependent Choice, enabling the controlled construction of models where the Axiom of Choice fails for specific families of sets while preserving DC.
This paper establishes that a field is large if and only if it has an elementary extension that is the fraction field of a non-field henselian local domain, a result derived from new findings showing that the étale-open topology refines or is refined by the newly introduced finite-closed topology depending on whether the field is perfect or bounded.
This paper characterizes Martin-Löf and Schnorr randomness by establishing that these notions correspond to the weak merging of computable probability measures under the summable Kullback-Leibler divergence, a result derived by linking the divergence to the incremental growth of the Doob decomposition of a specific submartingale.
This paper establishes that finite chains within the two-branching countable ultrahomogeneous pseudotree possess finite big Ramsey degrees, specifically determining the degree for chains of length two to be seven, thereby presenting the first example of a countable ultrahomogeneous structure in a finite language where some finite substructures have finite big Ramsey degrees while others have infinite ones.
This paper presents a formalization of Borel determinacy in the Lean 4 theorem prover, including definitions of Gale-Stewart games and a proof of Martin's theorem that closely follows his original "purely inductive" argument.
This paper establishes the decidability of the monadic theory of the real numbers with order when quantification is restricted to Borel sets, demonstrating that Boolean combinations of sets form an elementary substructure and that the result extends to larger classes under determinacy hypotheses.
This paper investigates the iteration of the stationary-logic constructible model by proving distributivity and stationary-set preservation for countable iterations of club-shooting forcings using mutually stationary sets, and introducing mutually fat sets to achieve stronger results for uncountable iterations, thereby demonstrating the ability to force models where and where iterated sequences decrease with arbitrarily large order types.