TITLE: Why study cardinals, ordinals and the like?
QUESTION [19 upvotes]: 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.

REPLY [4 votes]: 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!