TITLE: What are large cardinals for?
QUESTION [12 upvotes]: I've heard large cardinals talked about, and I (think I) understand a little about how you define them, but I don't understand why you would bother.
What's the simplest proof or whatever that requires the use of large cardinals? Is there some branch of mathematics that makes particularly heavy use of them?

REPLY [10 votes]: G. Rodrigues's specific answer gets at the general issue: large cardinals are used to examine how much more one can proof in ZFC set theory.  The first time I discovered large cardinals (in Jech's 2000 book Set Theory), I was amazed.  A large cardinal is just a "very big" set, after all, but I did not realize that the existence of such a set changed the nature of what was mathematically provable.  For example, there is, according to Jech, the event that started it all: Ulam's work on the problem of measure.  It is well-known that Lebesgue measure over the reals is not defined for all sets, but it turns out to be undecidable in ZFC alone if any non-trivial measure on the reals exists at all.  In order to get such a measure, one must assume the existence of a large cardinal, which is now called a measurable cardinal.  So I think of large cardinals as things that change the very nature of the mathematical "plumbing".  Deep stuff.
On a more practical level, I think it was Dudley who said that large cardinals can be useful for seeing why a proof is failing: if a proof is not working, seeing if it fails at a large cardinal can provide insight.