Why study cardinals, ordinals and the like?

Why is the study of infinite cardinals, ordinals and the like so prevalent in set theory and logic? What's so interesting about infinite cardinals beyond $\aleph _0 $ and $\mathfrak{c} $? It seems like they're enough for all practical purposes and they don't seem to pop up in pure mathematics also.

Solution 1:

It is true, when you only work on measure theory, or algebraic number theory, or classical analysis, you are unlikely to run into anything larger than $\frak c$.

But if you start working in arbitrary fields, and arbitrary modules, or arbitrary rings. Not just finitely generated, or countably generated. Then you need to have a better understanding of how infinities behave.

Moreover there are questions in analysis whose answers are decided by existence of large cardinals (where by large cardinal I don't mean $\aleph_{2412}$, but rather a technical term in set theory which implies the existence of cardinals whose size dwarfs $\frak c$ to insignificance). Questions like Lebesgue measurability, determinacy, and so on.

You can ask whether or not every normal Moore space is metrizable. And the question is independent of $\sf ZFC$, and the proofs include a deep understanding of ordinals, very large cardinals, and combinatorial structure of their subsets.

Yes, it's true, you will never run into anything larger than $2^\frak c$ if all you care about is applied mathematics, or mathematical physics. But at the same time you will not run into any uncountable set if you just restrict yourself to definable sets of integers; and you will never run into an infinite set if you only restrict yourself to bounded sets of integers.

Mathematics develops organically. From one question we draw certain abstractions and we continue to walk towards general notions and abstract notions. That is the nature of modern mathematics, we don't want to limit ourselves where we don't have to. But the result is that if you haven't sat down to learn something, you may have a hard time to understand "why is it interesting, other than 'it sounds cool'?"

Solution 2:

It's true that large cardinals don't often appear explicitly in most of mathematics. However, off the top of the head, I can think of two ways in which they might be relevant for “ordinary” mathematics.

For one, when considering some class of mathematical objects, it is nice to have something like an universal object. These universal objects can sometimes be pretty large in cardinality, even if we initially consider only a class of objects which are not very large themselves. (This is related to the concepts of universal domain in universal algebra and monster model in model theory.) Furthermore, it is often easier to find those universal objects if we assume the existence of large strongly inaccessible cardinals.

The other one is, large cardinal axioms seem to be effective in expressing consistency strength, and some apparently rather esoteric set-theoretical statements can be closely related to some “concrete” statements in “ordinary” mathematics. As a consequence, many of those abstract principles are believed to be “intuitively true” for some mathematicians (like axiom of choice is for most of us, in spite of its counter-intuitive consequences). There are many statements which we know are true or not regardless of those additional axioms (and even axiom of choice), but that's far from all we would want to decide.