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Understanding the axioms: the interactions of the Axiom of Choice with large cardinal axioms

理解公理：选择公理与大基本公理的相互作用

基本信息

批准号：
MR/T021705/1
负责人：
Asaf Karagila
金额：
$ 147.18万
依托单位：
University of East Anglia
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2020
资助国家：
英国
起止时间：
2020 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=MR%2FT021705%2F1
关键词：
Understanding axioms interactions Axiom Choice

项目摘要

Much of the research in pure mathematics is concerned with proving the existence of abstract mathematical objects from a certain set of initial assumptions, or axioms. Set theory is a branch of mathematical logic, and it serves as the mainstream foundation of mathematics. More precisely, the axioms of set theory function as a "universal interpreter" for all pure mathematical research.One of the standard axioms of set theory is the Axiom of Choice. This axiom relates to choosing an object from a collection, given that the collection is non-empty. We can easily choose an object from a single non-empty collection of objects, and inductively we can choose from any finite number of non-empty collections. However, it is not always possible to coherently describe a way to choose from infinitely many collections at once, even if we know that none of them is empty. For example, if we have finitely many pairs of ants, we can go one pair at a time and choose an ant from each pair. If we have infinitely many pairs, then there are no obvious discerning properties that let us specify, with a finite algorithm, a means of choosing a single ant from each pair. If, however, we are given infinitely many pairs, each consisting of one ant and one wasp, we can always choose the wasp.The Axiom of Choice asserts that there is always a way to make a coherent choice, but it does not provide us with a description of what this choice is. Indeed, it often doesn't even matter in proofs of existence what the exact choice of objects is. Nevertheless, we are often interested in the question of whether or not we can find a way to construct the objects whose existence we proved, since having an effective way of doing something sheds more light on the problem and its solution. This is one of the main goals in research related to the Axiom of Choice: discover the limitations of what we can and cannot construct explicitly in the mathematical universe. Despite its non-constructive nature and a history rife with controversy, its many important consequences make the Axiom of Choice a staple of modern mathematics.Another family of set-theoretic axioms is formed of the so-called "large cardinal axioms". These are axioms asserting the existence of objects - aptly referred to as "large cardinals" in most cases - which generalise the set of the natural numbers in certain kind of ways. These large cardinals are much larger than the objects mathematicians normally take interest in (such as the real numbers and so on), but their existence affects them nonetheless. There are concrete statements about natural numbers which cannot be proved without assuming that large cardinal axioms are consistent with set theory.The characterisations of large cardinals are often given from several different directions. Some are combinatorial in their nature, others are more technical. But the proofs that these characterisations are equivalent utilise the Axiom of Choice in a very significant way. We know that, in the absence of the Axiom of Choice, small cardinals may satisfy some of the combinatorial properties characterising large cardinals. And so far there has been very little research into what sort of implications there are to the existence of large cardinals when characterised by seemingly stronger properties.This project aims to explore the consequences of large cardinal axioms without the Axiom of Choice, and improve our understanding of how these axioms impact the structure of the set-theoretic universe, and the mathematical universe as a whole. Specifically, we are concerned with the question of what sort of consequences of the Axiom of Choice must follow from the existence of these large cardinals. For this we need to develop new methods that will let us explore these questions, and many others.

纯数学中的许多研究都是关于从一组初始假设或公理中证明抽象数学对象的存在性。集合论是数理逻辑的一个分支，是数学的主流基础。更确切地说，集合论的公理是所有纯数学研究的“通用解释器”，其中一个标准的集合论公理是选择公理。这个公理涉及到从集合中选择一个对象，假设集合是非空的。我们可以很容易地从一个非空的对象集合中选择一个对象，并且归纳地我们可以从任何有限数量的非空集合中选择。然而，即使我们知道没有一个集合是空的，也不可能连贯地描述一种方法，可以同时从无限多个集合中进行选择。例如，如果我们有200对蚂蚁，我们可以一次只去一对，从每对中选择一只蚂蚁。如果我们有无限多对蚂蚁，那么就没有明显的辨别性质，让我们用有限的算法来指定从每对蚂蚁中选择一只蚂蚁的方法。然而，如果我们有无限多对蚂蚁，每对蚂蚁由一只蚂蚁和一只黄蜂组成，我们总是可以选择黄蜂，选择公理断言，总是有一种方法可以做出一个连贯的选择，但它没有为我们提供一个描述这种选择是什么。事实上，在证明存在的过程中，对象的确切选择往往并不重要。尽管如此，我们经常对是否能找到一种方法来构造我们证明了其存在的对象的问题感兴趣，因为有一种有效的方法来做某事会使问题及其解决方案更加清晰。这是与选择公理相关的研究的主要目标之一：发现我们在数学世界中可以和不能明确构建的限制。尽管它的非构造性和充满争议的历史，它的许多重要结果使选择公理成为现代数学的主要内容。另一个集合论公理家族由所谓的“大基数公理”组成。这些公理断言对象的存在-在大多数情况下被恰当地称为“大基数”-以某种方式推广自然数的集合。这些大基数比数学家通常感兴趣的对象（如真实的数等）要大得多，但它们的存在仍然影响着它们。有一些关于自然数的具体陈述，如果不假设大基数公理与集合论一致，就无法证明。有些是组合性质的，有些则是技术性质的。但是，证明这些特征是等价的，是以一种非常有意义的方式利用了选择公理。我们知道，在没有选择公理的情况下，小基数可能满足一些表征大基数的组合性质。到目前为止，还很少有人研究大基数的存在有什么样的影响时，其特点是似乎更强的性质。这个项目的目的是探索的后果大基数公理没有公理的选择，并提高我们的理解如何影响结构的集合论宇宙，数学宇宙作为一个整体。具体地说，我们关心的问题是，选择公理的什么样的后果必须遵循这些大基数的存在。为此，我们需要开发新的方法，让我们探索这些问题，以及许多其他问题。