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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?

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Wikipedia, is the best source for understanding these things. Did you visit it. Here is the link: en.wikipedia.org/wiki/Large_cardinal –  anonymous Aug 15 '10 at 9:29
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The wiki article says nothing about why you might want to stipulate the existence of large cardinals... –  Seamus Aug 15 '10 at 13:26
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Maybe this should be community wiki. –  Tom Stephens Aug 15 '10 at 14:03
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Here is one of the main results about the usefulness of large cardinals: They provide us with a clear "complete" first order theory of the reals. What I mean is this: Without the assumption of large cardinals, there are many first order statements about the reals whose truth value can be changed by forcing. This is impossible in the presence of large cardinals. Thus, they provide us a way to complete the theory, and one can further argue that the resulting theory is not arbitrary but rather the right version. –  Andres Caicedo Jul 13 '11 at 21:58
    
@Andres I think that is an interesting and important enough comment to be made into an answer... –  Seamus Jul 14 '11 at 9:33
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up vote 12 down vote accepted

I will leave the explaining of large cardinals to someone more knowleadgeable and explain one place where they are useful in tidying up things: category theory. In category theory, you are constantly facede with proper classes (the category of all sets, of all groups, etc.). To make things worse, you want to form functor categories but due to the sizes of the classes involved there is no way to do that in ZFC (and other set theories like NBG or MK soon hit a wall of their own). The Grothendieck axiom of universes is a large cardinal axiom (and I am told that it is rather mild compared to the large cardinals that set theorists routinely consider) that allows us to tidy up things for this and other constructions without paying attention to set sizes. Truth be told, this is mainly a convenience, as if one payed proper attention, and at the cost of circumlocutions (and vast tribulations) one could avoid them. But really, why go to all this trouble to settle what is a minor technicality not germane to the problem at hand, when you have this labor-savibng device at hand?

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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.

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"seeing if it fails at a large cardinal can provide insight" ... does this also apply if we speak about proofs in ZFC (were we don't even have large cardinals). I don't quite see it. –  NikolajK May 28 '13 at 14:48
    
@Nick Kidman: I believe Dudley's comment was in the context of looking at ZFC + large cardinal axiom. In other words, if your proof fails there, there might be insight into it failing in ZFC alone (or perhaps it's undecidable). He was making a rather technical point in a book on real analysis and probability, that was not otherwise about set theory, and so did not go into much detail. –  trb456 May 28 '13 at 16:32
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