Related papers: A Gentle Introduction to the Axiom of Choice
This tutorial deal with the Axiom of Choice and some of its applications to topics related to Computer Science. We will see that the Axiom of Choice is equivalent to some well-known proof principles like Zorn's Lemma or Tuckey's Maximality…
We introduce neutrosophic choice functions, the neutrosophic counterpart of the Axiom of Choice, prove some results, and discuss how it effects the foundations of mathematics in a neutrosophic setting.
We discuss the connections between the failure of the axiom of choice in set theory, and certain model-theoretic structures with enough symmetry.
This text is meant for analysis students who want to learn more about the effects of the axiom of choice on functional analysis, and the things that may go wrong in its absence. As this is a text aimed for analysis students, we will not…
This article presents an elementary proof of Zorn's Lemma under the Axiom of Choice, simplifying and supplying necessary details in the original proof by Paul R. Halmos in his book, Naive Set Theory. Also provided, is a preamble to Zorn's…
People care about decision outcomes and how decisions get made, both when making decisions and reflecting on decisions. But formalizing the full range of normative concerns that drive decisions is an open challenge. We introduce Axiomatic…
This paper provides a complete suite of axioms for a version of set theory that I call Explication. Explication borrows from the two most prominent existing systems of set theory. Explication starts with class variables. After several…
The article considers the Choice Axiom.
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…
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that typically arise from applying decision rules…
If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible…
Zorn's Lemma is a well-known equivalent of the Axiom of Choice. It is usually regarded as a topic in axiomatic set theory, and its historically standard proof (from the Axiom of Choice) relies on transfinite recursion, a non-elementary…
We study a well-known technique of using absoluteness for giving choice-free proofs to some statements which are known to be provable with the axiom of choice. The idea is to reduce the problem to an inner model where the axiom of choice…
Mathematicians manipulate sets with confidence almost every day, rarely making mistakes. Few of us, however, could accurately quote what are often referred to as "the" axioms of set theory. This suggests that we all carry around with us,…
Methods for choosing from a set of options are often based on a strict partial order on these options, or on a set of such partial orders. I here provide a very general axiomatic characterisation for choice functions of this form. It…
Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well…
We introduce a novel notion of probability within quantum history theories and give a Gleasonesque proof for these assignments. This involves introducing a tentative novel axiom of probability. We also discuss how we are to interpret these…
The purpose of this paper is to outline a simple set of axioms for basic set theory from which most fundamental facts can be derived. The key to the whole project is a new axiom of set theory which I dubbed "The Law of Extremes". It allows…
Recent work on the logical structure of non-locality has constructed scenarios where observations of multi-partite systems cannot be adequately described by compositions of non-signaling subsystems. In this paper we apply these frameworks…
We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside…