Most mathematicians are willing to use ZFC (or something fairly similar) as a foundation for their work. However, since Gödel, we know that these axioms are essentially incomplete – that is, any consistent, recursive extension of them will still leave various statements undecided. Some people seem willing to say that this just means that there is no fact of the matter about statements that go beyond ZFC, but I think this is just too hasty.
After all, if we are platonists, then we think that ZFC is true, and thus consistent. And if we are fictionalists, we at least think that it’s a good story, but part of being a good story seems to require consistency. In fact, on just about any reasonable view of mathematics, it seems that however much we can actually say to be the case should at least be consistent. Thus, whatever theory T we use, we should probably be willing to say “T is consistent” as well.
Now, “T is consistent” is not itself a mathematical statement, but there is a susbtitute, which I will call “Con(T)”, which states that no natural number has a certain property, which we intuitively understand as saying that it codes a proof from T of a contradiction. Whatever we might or might not be able to say about the natural numbers, I think we understand them well enough to say that this proof-coding mechanism is actually correct. (You can think of this coding as the way that a computer codes text, and can reason syntactically about this text to see which strings are correct proofs in our formal system – there’s another coding that’s easier to work with and is therefore more standard in the literature, involving powers of primes.) Thus, if T is an appropriate formalization of some part of the truths (or useful fictions, or whatever) of mathematics, then Con(T) should be as well.
Now, by Gödel’s Completeness theorem (which requires some fragment of ZFC to prove), we know that if Con(T) is the case, then there is actually some set and a collection of relations and functions on that set that can be used as the interpretation of the symbols in T to make it come out true. That is, any consistent theory (according to the numerical coding) has a model. Thus, if we think that ZFC+Con(ZFC) is a reasonable formulation of some part of mathematics, then there must in addition be some model of ZFC. If in addition, we assume Con(ZFC+Con(ZFC)), then there is a model of ZFC, which itself contains a model of ZFC.
Now, these models of ZFC may bear very little resemblance to the “real thing”. In particular, the symbol for the set membership relation may be interpreted as some relation that has no connection to the real thing. In fact, there is a model whose domain is just the natural numbers, and whose “set membership relation” is some relation among numbers. But if there is an inaccessible cardinal I, then there is in fact a set (called VI) such that using the actual set membership relation on this domain creates a model of ZFC. This set contains all members of its elements, so it actually “knows” what sets they are (unlike one of those models whose “sets” are natural numbers, and assigns them elements in some seemingly arbitrary way), and also contains the actual powersets of its elements (and thus all their subsets, not just some of them), and thus has a lot more properties in common with the whole of mathematical reality than the other sorts of models of ZFC. In fact, it contains all the natural numbers and knows which set is the set of all natural numbers, so it knows any true statement (or true according to the fiction of mathematics, or whatever) of the form Con(T). So if there is an inaccessible cardinal, then VI is a model of ZFC, so Con(ZFC) is true, so VI is also a model of ZFC+Con(ZFC), so Con(ZFC+Con(ZFC)) is true, so VI is also a model of Con(ZFC+Con(ZFC+Con(ZFC))), etc. So finding a model of this appropriate sort guarantees a lot of consistency statements.
As I said above, we have good reason to adopt all these statements. In addition, for any (even non-recursive) combination of consistency statements, a model of the form Vk satisfies all of them, so in some sense, saying that a model of this form exists just says that these non-recursive sets of axioms are consistent, just as the statement Con(T) said that a certain recursive set of axioms was consistent. Thus, we seem to have good reason to adopt the existence of an inaccessible as well as all the consistency statements.
Of course, once we have adopted ZFC+Inaccessible, we can start the whole process all over again, and eventually reach ZFC+”There are at least two inaccessibles”. Similar arguments get us arbitrarily large numbers of inaccessibles (even into infinite orderings of them).
I’m told that larger large cardinal axioms state similar properties about the universe and guarantee that we can bring them down to models of a certain form, extending Gödel’s Completeness Theorem.
Now, I’ll consider the endpoint of all this. The collection of all true (or fictional, or whatever) statements of set theory should be consistent. Thus, to extend Gödel’s Completeness Theorem, there should be some model of the form Vk that satisfies all true statements of set theory, ie, is elementarily equivalent to the universe as a whole. Now, the statement that such a model exists is certainly not consistent if it can be stated (because it would violate Gödel’s Second Incompleteness Theorem, saying that no theory can assert its own consistency, or equivalently the existence of a model of the entire theory). Perhaps more relevantly, I think it can’t even be stated, because Tarski showed that we can’t define the truth-predicate for a model inside that very model, so we can’t state that a model of “all true statements” exists. So this statement can’t be adopted as a “final large cardinal”.
However, even if it could be stated, or if it were consistent, we might still want to go farther. Since the universe contains an elementary submodel of this form, let us say, Va, then Va must also contain such a model, say Vb. Thus, there should be a and b such that Vb is an elementary submodel of Va. And this statement can in fact be expressed in the language of set theory.
I don’t know if this statement can be shown to be inconsistent (which would call into question this means of justifying large cardinals). It might also be equivalent to some already-known large cardinal axiom (my friend Adam Booth suggested to me that it sounds like it could be a consequence of the existence of an inaccessible that is the limit of a set of measurables). If anyone with a relevant background in set theory could tell me, that would be great.
Anyway, all this is just one means of justifying large cardinal axioms, but it seems to make sense to me. It also has the added benefit of not requiring a platonist view of mathematics, but works also on a fictionalist view, and probably a variety of other views as well. All that is needed is to assume that whatever set of axioms we should adopt should be consistent, and that the set of natural numbers that our axioms describe should satisfy Con(T) iff T is actually consistent. Of course, there will still be further statements that we won’t be able to decide with a principle like this, but it gives us a means of going far beyond ZFC. Recent work of Hugh Woodin and others suggests that it won’t be enough to settle the Continuum Hypothesis however.