There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large cardinal axioms, like the existence of an infinite number of Grothendieck universes.
Are there any such heuristic arguments for the existence of Vopenka cardinals or huge cardinals? I'd very much like to believe them, mainly because they simplify a great deal of trouble one has to go through when working with accessible categories and localization (every localizer is accessible on a presheaf category, for instance).
For Vopenka's principle, the category-theoretic definition is that every full complete (cocomplete) subcategory of a locally presentable category is reflective (coreflective). This seems rather unintuitive to me (and I don't even understand the model-theoretic definition of Vopenka's principle).
What reason is there to believe that ZFC+VP (or ZFC+HC, which implies the consistency of VP) is consistent? Obviously, I am willing to accept heuristic or informal arguments (since a formal proof is impossible).
Best Answer
Most of the arguments previously presented take a set-theoretic/logical point of view and apply to large cardinal axioms in general. There's a lot of good stuff there, but I think there are additional things to be said about Vopěnka's principle specifically from a category-theoretic point of view.
One formulation of Vopěnka's principle (which is the one that I'm used to calling "the" category-theoretic definition, and the one used as the definition in Adamek&Rosicky's book, although there are many category-theoretic statements equivalent to VP) is that there does not exist a large (= proper-class-sized) full discrete (= having no nonidentity morphims between its objects) subcategory of any locally presentable category. I think there is a good argument to be made for the naturalness of this from a category-theoretic perspective.
To explain why, let me back up a bit. To a category theorist of a certain philosophical bent, one thing that category theory teaches us is to avoid talking about equalities between objects of a category, rather than isomorphism. For instance, in doing group theory, we never talk about when two groups are equal, only when they are isomorphic. Likewise in doing topology, we never talk about when two spaces are equal, only when they are homeomorphic. Once you get used to this, it starts to feel like an accident that it even makes sense to ask whether two groups are equal, rather than merely isomorphic. And in fact, it is an accident, or at least dependent on the particular choice of axioms for a set-theoretic foundation; one can give other axiomatizations of set theory, provably equivalent to ZFC, in which it doesn't make sense to ask whether two sets are equal, only whether two elements of a given ambient set are equal. These are sometimes called "categorial" set theories, since the first example was Lawvere's ETCS which axiomatizes the category of sets, but I prefer to call them structural set theories, since there are other versions, like SEAR, which don't require any category theory.
Now there do exist categories in which it does make sense to talk about "equality" of objects. For instance, any set X can be regarded as a discrete category $X_d$, whose objects are the elements of X and in which the only morphisms are identities. Moreover, a category is equivalent to one of the form $X_d$, for some set X, iff it is both a groupoid and a preorder, i.e. every morphism is invertible and any parallel pair of morphisms are equal. I call such a category a "discrete category," although some people use that only for the stricter notion of a category isomorphic to some $X_d$. So it becomes tempting to think that one might instead consider "category" to be a fundamental notion, and define "set" to mean a discrete category.
Unfortunately, however, what I wrote in the previous paragraph is false: a category is equivalent to one of the form $X_d$, for some set X, iff it is a groupoid and a preorder and small. We can just as well construct a category $X_d$ when X is a proper class, and it will of course still be discrete. In fact, just as a set is the same thing as a small discrete category, a proper class is the same thing as a large discrete category. However, this feels kind of bizarre, because the large categories that arise in practice are almost never of the sort that admit a meaningful notion of "equality" between their objects, and in particular they are almost never discrete. Consider the categories of groups, or rings, or topological spaces, or sets for that matter. Outside of set theory, proper classes usually only arise as the class of objects of some large category, which is almost never discrete. The world would make much more sense, from a category-theoretic point of view, if there were no such things as proper classes, a.k.a. discrete large categories --- then we could define "set" to mean "discrete category" and life would be beautiful.
Unfortunately, we can't have large categories without having large discrete categories, at least not without restricting the rest of mathematics fairly severly. This is obviously true if we found mathematics on ZFC or NBG or some other traditional "membership-based" or "material" set theory, since there we need a proper class of objects before we can even define a large category. But it's also true if we use a structural set theory, since there are a few naturally and structurally defined large categories that are discrete, such as the category of well-orderings and all isomorphisms between them (the core of the full subcategory of Poset on the well-orderings).
Thus Vopěnka's principle, as I stated it above, is a weakened version of the thesis that large discrete categories don't exist: it says that at least they can't exist as full subcategories of locally presentable categories. Since locally presentable categories are otherwise very well-behaved, this is at least reasonable to hope for. In fact, from this perspective, if Vopěnka's principle turns out to be inconsistent with ZFC, then maybe it is ZFC that is at fault! (-: