What are the motivations of large cardinal research?

Question

Why do set theorists research large cardinals? Is it about the consistency results, or is there a type of mathematical beauty to large cardinals? If so, are there any examples of beauty in large cardinals someone with a knowledge of some mathematical logic could understand? To clarify, this question isn't about the practical uses or applications of large cardinals, but rather why someone would want to go into large cardinal research.

Answer 1

I had to understand exactly that when I was a PhD student. Not because it was my research, it wasn't really my research. But because I saw a lot of this work as a kind of "bubble". You start defining notions that seem unnatural, you start tweaking them in what I like to call "a roll of the dice" where you change different parameters in the definition in what seems like an arbitrary way.

And it bothered me. It really did. At first I scoffed at it. But then I also saw what is happening around my research field. And then I was enlightened.

We start asking some obvious question. For example:

1. Can there be a $\kappa$ such that $V_\kappa\models\sf ZFC$?
2. What can we say about $2^{\aleph_0}$ if the Lebesgue measure has an extension to a total measure (not necessarily translation invariant; we are not looking to give up chocie)?
3. Is it really necessary to use an inaccessible cardinal in the Solovay construction of a model with many "All sets of reals have a regularity property"? For which ones?
4. What properties follow from certain compactness properties of $\cal L_{\kappa,\lambda}$ and other logics?

And so on. So, either someone already used large cardinals in a certain way (e.g., 3) or maybe there is a natural question arising from other fields of mathematics that turns out to involve large cardinals in subtle ways, or maybe it was just natural to raise a question.

So, we want to understand how large cardinals interact with consistency statement. After all, this is exactly what large cardinal axioms are: a stick against which we measure consistency of mathematical statements.

And you can go pretty far with just that, to be honest. You can say, "I only want to consider large cardinal axioms that are 'useful' in these measurements", and you'd still have a lot of motivation to study a lot of mathematics. From I0, to supercompactness, to Shelah, to Woodin cardinals in the investigation of the consistency of $\sf AD$, to papers such as "Can you take Solovay's inaccessible away".

But then you can start varying the parameters. And sometimes it yields weird statements, like ineffable cardinals, like subtle cardinals, even things like "Ord is Mahlo" is kind of an odd statement.

But really, all this is is us trying to refine the previous results and better understand these definitions and these measuring sticks.

For example, the first strongly compact can be the first measurable or the first supercompact; but the first supercompact has a lot of measurables below it (and I mean that in a technical sense, namely, it reflects being measurable, and much more). But are they equiconsistent? It's not clear. The difference between the definitions is "kinda minor", but apparently minor enough to make the identity crisis real.

So now we start asking, how far can we tweak this difference before we can no longer get this sort of identity crisis? This is both interesting on its own, it's a beautiful question; and it's relevant to the overall understanding of large cardinals, because it tells us something real not only about consistency, but also about their "placement in the universe", if you will.

Some of these things can lead to new understanding which in retrospect seem like it was the obvious place to start looking. For example, Vopěnka's principle is equivalent to "Ord is Woodin for supercompactness", and it turns out that the weak Vopěnka principle is exactly equivalent to "Ord is Woodin". Somewhere in the things that we knew already is that strong cardinals are the "weak" version of supercompact cardinals. But whereas the purely large cardinal theoretic result came in the early-mid 2010s, the application of it to understanding VP only came very recently.

Other questions, arising out of my own work, is the study of how the axiom of choice plays a role in the various definitions. Measurable cardinals can be defined as critical points of elementary embeddings, but moving from measures to embeddings, we heavily rely on the axiom of choice. Indeed, Jech proved that $\omega_1$ can be measurable in $\sf ZF$, but it is very easy to see that $\omega_1$ could never be the critical point of an internal elementary embedding.

Similarly with weakly compact cardinals which enjoy a plethora of definitions. Or inaccessible cardinals, or so on.

This gives us better understanding of what is "the real largeness" of the large cardinal, and these sort of identity crises can also shed light on understanding equiconsistency between various large cardinal axioms.

TL;DR

Yes, it is beautiful for its own sake. Yes, it is elegant for its own sake. Yes, it is useful, since we do use large cardinals as a measuring stick for consistency results, so we ought to be refining it when possible. No, I don't want milk in my tea. Yes, choiceless large cardinals are awesome.