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

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.

Set theory is a branch of mathematical logic, which itself is a branch of pure mathematics. Modern mathematics can be interpreted in set theory, and this way set theory is the de facto foundation for all pure mathematics. This project investigates the foundations of modern mathematics, and in particular the interactions between the Axiom of Choice and a family of axioms called "large cardinal axioms". Much of the research in set theory is aimed to develop and understand new axioms and their influence on the mathematical universe, and a large part of this includes understanding their limitations which in turn motivates the formulation of new axioms.

1. Impact within set theory

As is the case in most research in pure mathematics, the immediate impact is within the academic research community. The research objectives include developing new tools and techniques for proving independence results (i.e. what things could be true, but we cannot prove them without adding new axioms), and to advance the research into questions related to the Axiom of Choice. This translates to questions of constructing and defining mathematical objects explicitly in the abstract mathematical universe.

At the start of the project we will host a small workshop at UEA where a number of experts will be invited to share ideas related to research on the topics of the Axiom of Choice and large cardinals. This will consolidate future collaborations on these topics. At the end of the project we will hold a large scale conference at UEA which will promote the results from this project, and from other important advances in related topics that happened during this period. As part of our commitment to impact, we will record the lectures in both of these meetings (with the permission of the speakers) and have them publicly accessible.

2. Academic impact outside of set theory

In 2019 a paper was published establishing a deep connection between set theoretic independence results (i.e., things which can be true, but are not necessarily true) and the theoretical limitations of machine learning. While being very theoretical, it does further our understanding of machine learning as an abstract process and improves our ability to better implement machine learning techniques in other fields (from neuroscience to fluid dynamics). So even if the impact is not direct, our project ultimately connects to these types of questions which have importance in theoretical computer science, and affect many other fields of science.

The project will also deepen our understanding related to both the Axiom of Choice and large cardinals, and thus help and clarify the uses of these types of axioms within the mathematical community. This will impact the philosophy of mathematics, which in turn will influence the type of research that will be done in the future.

3. Public engagement

Apart from the academic impact, we have taken the first steps towards collaborating with YouTube content creators to produce videos on mathematical topics aimed at various levels (from hobbyists to mathematicians). The goal is to explain set theoretic objects and ideas in understandable ways without compromising mathematical integrity. We will talk about the notions of ordinals and cardinals and the difference between them, the Axiom of Choice and its many uses, and the broad ideas behind the foundations of mathematics. My goal is to engage with the public at large in order to promote pure mathematics, and demystify it as something that "only geniuses can do" since it is my firm belief that everyone can do excellent mathematics and science. This, I hope, will increase the interactions between non-scientists and non-mathematicians with pure mathematics, and develop interest in younger audiences that will lead them to STEM fields, and in particular pure mathematics.