相关论文: Several amazing discoveries about compact metrizab…

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…

Given a compact space in a fixed universe of set theory, one can naturally define its interpretation in any ZFC extension of the universe. We investigate the stability of some classes of compact spaces with respect to extensions of this…

According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of…

Let $\mathcal{E}$ denote the space of entire functions with the topology of uniform convergence on compact sets. The action of $\mathbb C$ by translations on $\mathcal E$ is defined by $T_zf(w) = f(w+z)$. Let $\mathcal{U}$ denote the set of…

This article concerns the Herrlich-Chew theorem stating that a Hausdorff zero-dimensional space is $\mathbb{N}$-compact if and only if every clopen ultrafilter with the countable intersection property in this space is fixed. It also…

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…

In the context of $\mathsf{ZF}$, we analyze a version of Hindman's finite unions theorem on infinite sets, which normally requires the Axiom of Choice to be proved. We establish the implication relations between this statement and various…

This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on…

This paper provides a systematic exposition of Lyapunov stability for compact sets in locally compact metric spaces. We explore foundational concepts, including neighborhoods of compact sets, invariant sets, and the properties of dynamical…

Using iterated Sacks forcing and topological games, we prove that the existence of a totally imperfect Menger set in the Cantor cube with cardinality continuum is independent from ZFC. We also analyze the structure of Hurewicz and consonant…

We introduce a new topological generalization of the $\sigma$-projective hierarchy, not limited to Polish spaces. Earlier attempts have replaced $^{\omega}\omega$ by $^{\kappa}\kappa$, for $\kappa$ regular uncountable, or replaced countable…

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…

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to $\omega_1$-sequences of the selection principle and…

The theory ZFC implies the scheme that for every cardinal $\delta$ we can make $\delta$ many dependent choices over any definable relation without terminal nodes. Friedman, the first author, and Kanovei constructed a model of ZFC$^-$ (ZFC…

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…

In this paper, we investigate the existence and uniqueness of fixed points for self-mappings defined on bipolar metric spaces using a new class of contractive conditions, namely polynomial-type contractions. Our main results establish…

We introduce Z-stability, a notion capturing the intuition that if a function f maps a metric space into a normed space and if the norm of f(x) is small, then x is close to a zero of f. Working in Bishop's constructive setting, we first…

We introduce the notion of compactifiable classes -- these are classes of metrizable compact spaces that can be up to homeomorphic copies ``disjointly combined'' into one metrizable compact space. This is witnessed by so-called compact…

Consider the self-map F of the space of real-valued test functions on the line which takes a test function f to the test function sending a real number x to f(f(x))-f(0). We show that F is discontinuous, although its restriction to the…

Although Zermelo-Fraenkel set theory (ZFC) is generally accepted as the appropriate foundation for modern mathematics, proof theorists have known for decades that virtually all mainstream mathematics can actually be formalized in much…