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

纯数学中的许多研究都关注于从一组特定的初始假设或公理来证明抽象数学对象的存在。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”。集合论的标准公理之一是选择公理。这条公理与从集合中选择对象有关，假设集合是非空的。我们可以很容易地从对象的单个非空集合中选择一个对象，并且归纳地，我们可以从任意有限个非空集合中选择。然而，即使我们知道这些集合中没有一个是空的，也并不总是能够连贯地描述一次从无限多个集合中进行选择的方法。例如，如果我们有有限多对蚂蚁，我们可以一次走一对，并从每对蚂蚁中选择一只蚂蚁。如果我们有无限多个蚂蚁对，那么就没有明显的辨别性质，可以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，如果我们被给予无限多对，每对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做出连贯的选择，但它没有为我们提供关于这种选择是什么的描述。事实上，在证明存在的时候，物体的确切选择往往无关紧要。然而，我们经常感兴趣的问题是，我们是否可以找到一种方法来构造我们证明了其存在的对象，因为拥有一种有效的做某事的方法可以更好地揭示问题及其解决方案。这是与选择公理相关的研究的主要目标之一：发现我们可以和不能在数学宇宙中明确构建的东西的局限性。尽管它的非建构性和历史上充满争议，但它的许多重要结果使选择公理成为现代数学的主要内容。另一族集合论公理是由所谓的“大基数公理”形成的。这些公理断言了对象的存在--在大多数情况下被恰当地称为“大基数”--它们以某种方式概括了自然数的集合。这些大基数比数学家通常感兴趣的对象(如实数等)要大得多，但它们的存在仍然会影响它们。有一些关于自然数的具体陈述，如果不假设大型基数公理与集合论一致，就无法证明这些陈述。大型基数的特征通常是从几个不同的方向给出的。其中一些是组合性的，另一些则更具技术性。但证明这些特征等价的证据以一种非常重要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，小基数可能满足大基数的一些组合性质。到目前为止，对于大型基数的存在有什么影响的研究很少，当大型基数的特征是看起来更强的性质时。这个项目旨在探索没有选择公理的大型基数公理的后果，并提高我们对这些公理如何影响集合论宇宙的结构，以及整个数学宇宙的理解。具体地说，我们关心的问题是，这些大基数的存在必然会导致选择公理的什么样的后果。为此，我们需要开发新的方法，使我们能够探索这些问题和许多其他问题。

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.