Related papers: Geometric condition for Dependent Choice
Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and G\"odel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed…
Within the framework of Zermelo-Fraenkel set theory without the Axiom of Choice, we establish equivalents to the assertion "the union of a countable collection of finite sets is countable" in the context of metric spaces, probability…
A condition of geometric modular action is proposed as a selection principle for physically interesting states on general space-times. This condition is naturally associated with transformation groups of partially ordered sets and provides…
It is known that first-order logic with some counting extensions can be efficiently evaluated on graph classes with bounded expansion, where depth-$r$ minors have constant density. More precisely, the formulas are $\exists x_1 ... x_k \#y…
Learning a distribution conditional on a set of discrete-valued features is a commonly encountered task. This becomes more challenging with a high-dimensional feature set when there is the possibility of interaction between the features. In…
Deformation quantization produces families of mathematically equivalent quantization procedures from which one must select the physically meaningful ones. As a selection principle we propose that the procedure must allow enough `observable'…
In this work, we propose a global model selection criterion to estimate the graph of conditional dependencies of a random vector based on a finite sample. By global criterion, we mean optimizing a function over the entire set of possible…
This article explores the model-dependent nature of set cardinality, emphasizing that cardinality is not absolute but varies across different axiomatic frameworks. Although Cantor's diagonal argument shows the real numbers are…
In the Zermelo--Fraenkel set theory with the Axiom of Choice a forcing notion is "$\kappa$-distributive" if and only if it is "$\kappa$-sequential". We show that without the Axiom of Choice this equivalence fails, even if we include a weak…
We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can…
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…
Stringent restrictions for model building are imposed by a no-go theorem in noncommutative gauge field theory. Circumventing this theorem is crucial for the construction of realistic models of particle interactions. To this end, the…
We investigate Hindman- and Owings-type Ramsey-theoretic statements in Zermelo-Fraenkel set theory without the Axiom of Choice, with some occasional extra assumptions (such as the Axiom of Dependent Choice and/or the Axiom of Determinacy).…
While hidden class models of various types arise in many statistical applications, it is often difficult to establish the identifiability of their parameters. Focusing on models in which there is some structure of independence of some of…
We obtain a necessary and sufficient condition under which random-coefficient discrete choice models, such as mixed-logit models, are rich enough to approximate any nonparametric random utility models arbitrarily well across choice sets.…
A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…
We study new relations of the following statements with weak choice principles in ZF and ZFA. 1. There does not exist an infinite Hausdorff space X such that every infinite subset of X contains an infinite compact subset. 2. If a field has…
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…
Given finite configurations $P_1, \dots, P_n \subset \mathbb{R}^d$, let us denote by $\mathbf{m}_{\mathbb{R}^d}(P_1, \dots, P_n)$ the maximum density a set $A \subseteq \mathbb{R}^d$ can have without containing congruent copies of any…
We propose a new type of SPDEs, singular or with regularized noises, motivated by a study of the fluctuation of the density field in a microscopic interacting particle system. They include a large scaling parameter $N$, which is the ratio…