相关论文: Life without "Choice"
We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak…
The Axiom of Choice (AC for short) is the most (in)famous axiom of the usual foundations of mathematics, ZFC set theory. The (non-)essential use of AC in mathematics has been well-studied and thoroughly classified. Now, fragments of…
The Axiom of Dependent Choice $\mathsf{DC}$ and the Axiom of Countable Choice $\mathsf{AC}_\omega$ are two weak forms of the Axiom of Choice that can be stated for a specific set: $\mathsf{DC}(X)$ asserts that any total binary relation on…
Individual choices often depend on the order in which the decisions are made. In this paper, we expose a general theory of measurable systems (an example of which is an individual's preferences) allowing for incompatible (non-commuting)…
Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice DC rather than the…
The goal of this paper is twofold. In addition to the results stated in the next paragraph, we present some classical results on absoluteness relevant to functional analysis that are well known to logicians but not nearly as well advertised…
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…
This article examines Hilbert spaces constructed from sets whose existence is incompatible with the Countable Axiom of Choice (CC). Our point of view is twofold: (1) We examine what can and cannot be said about Hilbert spaces and operators…
In this paper we present another proof of the analytic version of the Hahn-Banach theorem in terms of convex functionals.
This paper proves the existence of nonmeasurable dense sets with additional properties using combinatorial techniques.
The main aim of the article is to show, in the absence of the Axiom of Choice, relationships between the following, independent of $\mathbf{ZF}$, statements: "Every countable product of compact metrizable spaces is separable (respectively,…
We show that the axiom of choice, a basic yet controversial postulate of set theory, leads to revise the standard understanding of one of the pillars of our best physical theories, namely the no-signaling principle. While it is well known…
In this paper I introduce a new and intuitive first-order foundational theory (where the concept of set is not primitive) and use it to show that the power set of an infinite set does not exist. In particular, proofs of uncountability of a…
Weak Feller property of controlled and control-free Markov chains lead to many desirable properties. In control-free setups this leads to the existence of invariant probability measures for compact spaces and applicability of numerical…
We prove (ZF+DC) e.g. : if mu =|H(mu)| then mu^+ is regular non measurable. This is in contrast with the results for mu = aleph_{omega} on measurability see Apter Magidor [ApMg]
Let X be a finite set of alternatives. A choice function c is a mapping which assigns to nonempty subsets S of X an element c(S) of S. A rational choice function is one for which there is a linear ordering on the alternatives such that c(S)…
As a cornerstone of functional analysis, Hahn Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics,…
Beiglboeck, Bergelson and Fish proved that if subsets A,B of a countable discrete amenable group G have positive Banach densities a and b respectively, then the product set AB is piecewise syndetic, i.e. there exists k such that the union…
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…
The average result of a weak measurement of some observable $A$ can, under post-selection of the measured quantum system, exceed the largest eigenvalue of $A$. The nature of weak measurements, as well as the presence of post-selection and…