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.

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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’?”

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.

“Wir müssen wissen. Wir werden wissen.” (D. Hilbert)

Gödel’s incompleteness theorems say that however we axiomatise mathematics, there will always be theorems that are true but not provable in any given axiomatisation. In order to prove such theorems, we need to add more powerful axioms, proof principles. You can think of these higher infinities as such more powerful proof principles.

It’s not easy to come up with more powerful proof principles because of the danger of inconsistency.

Harvey Friedman has done a lot of work on showing examples of statements in ordinary mathematics that are only provable with large cardianl axioms. See in particular his book on boolean relation theory.

The answer of Martin Berger is not clear, but I think there is a point to it, allow me to reformulate it here.
It does not contradict the other answer of bof, which I completely agree with.

Even if some abstractions seem very far from reality, they can actually help us figuring “practical” things out, by providing us with more powerful proof tools.
Indeed, some very abstract set theoretic assumptions can help to prove consistency of some conjectures, therefore showing that we won’t find a proof of the opposite.

For instance any proof obtained using the axiom of choice can be argued to be nonpractical, because the objects involved cannot be built explicitely. But at least we won’t try to find a proof in ZFC that Banach-Tarski is impossible, or to deny some other apparent paradoxes.

Another interesting example of the sort was recently accepted for publication at ICALP 2014 (but not yet published): the goal was to find an algorithm to decide satisfiability of a certain logic. It was shown that under some “exotic” (at least for computer scientists) set-theoretic assumptions like V=L, we can prove that such an algorithm does not exist. Therefore, we have the very practical knowledge that it is a waste of time to look for such an algorithm, and it is thanks to set theorists…

Other examples of axioms independent of ZFC applied to computer science can be found here: https://cstheory.stackexchange.com/questions/5934/results-in-theoretical-cs-independent-of-zfc

As Asaf pointed out, once you start working with objects of arbitrary sizes, then you need to understand how different infinities behave.

Besides this obvious reason to study cardinals, as has been already hinted above, it is not true that you do not run into cardinals bigger than continuum or they do not pop up in practice. You might be just asking the wrong questions for big cardinals to get involved. I personally think that in every field of mathematics if you ask the appropriate questions, then you will have to involve seemingly irrelevant set theoretic objects. At least, this has been my experience since I started learning about independence results.

Let me give you some famous examples. Pick a Borel subset of $\mathbb{R}^2$, project it and then take the complement. If the resulting set is uncountable, does it necessarily contain a perfect set?

The statement of this problem involves no sets of cardinality bigger than continuum and the question seems completely topological/measure theoretic. So, you might think that why on earth should the answer involve any infinities that are obviously unrelated to these?

On the other hand, if you want to have a positive answer to this question, the answer requires large cardinals (for example, you need this type of cardinals). A negative answer will also be implied by some axioms that go beyond ZFC (like this one for example).

The point I am trying to make is that existence of very big infinities can effect behavior of small infinities. You do run into big infinities in practice, they are just in disguise until you reveal the connections!