While I think I agree with Tim Chow and Joel Hamkins in some of their comments above regarding a single *formal* definition of what it is to be a large cardinal, I want to suggest that a large cardinal be considered such if it satisfies the following open-ended, semi-formal "definition" (which I draw from history, not a priori ideas about largeness or related notions about the universe of sets (whatever that could mean)). I.e., a set/proposition is a large cardinal notion if it satisfies an open-ended disjunction which includes among its disjuncts the three categories (not intended as the logical notion, just plain ol' English) I mention below.

Historically, large cardinal assumptions seem to fall into one of several categories:

1) Inaccessibility "from below" of some sort.

2) As critical points of elementary embeddings between certain set-theoretic structures.

3) Propositions formalizable in, say, ZFC or some second-order strengthening/alternative, which logically entail the existence of other large cardinal notions.

Per 1: This criterion will obviously capture the weakly and strongly inaccessible large cardinals as closure points under the usual set-theoretic operations. I take it that this informal notion will also give us Mahlo cardinals. I would include in this category indescribable cardinals of various degrees, hence also weakly compact cardinals (though these will fall under both categories 2 and 3 as well).

This inaccessibility requirement also encompasses proof-theoretic considerations. So, for example, statements which entail the existence of models of say ZFC, would fall under this category as well. This would include the iterated consistency hierarchy, the existence of a transitive (set?) model of ZFC, and worldly cardinals and their generalizations.

I think this category might also include $V=L$ as a large cardinal axiom, though there is some conflict with the other 2 categories. I hope to explain this away by proclaiming that a large cardinal assumption satisfies at least one of the three categories above while at the same time *not* satisfying a plausible negation of any of the three categories (or any further, since the list is intended to be open-ended). I'm not sure this is fair to do, since I'm implicitly importing a consistency constraint, but at this point, I believe it's at least reasonable.

per 2: This particular category encompasses, as far as I can tell, most of the large large cardinal axioms. Since this is pretty standard, I take it that there is no objection to this category as a criterion for "large cardinal". But please do offer any objections in the comments section.

per 3: This last category is intended to encompass assumptions relating to the existence of various kinds of sharps, indiscernibles for various set-theoretic structures, and axioms like $AD^{L(\mathbb{R})}$ and generalizations of these structures.

I think this category will also encompass the existence of certain types of indestructible large cardinals, since many such cardinals have been to shown to exist by assuming the existence of a particular large cardinal and then providing a certain forcing construction.

Again, this criterion may be a little unfair since the proposition $0=1$ implies every set-theoretic statement, including all so-called large cardinals. Thus, I intend to implicitly require a consistency constraint...

Some remarks about the above proposal:

1) In an ideal world, large cardinals would logically imply some regular/predictable structure on the universe below. In particular, I think a particularly desirable feature would be that the assumption in question imposes a linear order on the large cardinals strictly smaller. From an aesthetic point-of-view, this would require that the "identity crisis" phenomenon observed in say, strongly compact cardinals, be eradicated by the assumption in question.

To clarify, I would not reject, at this point, a specific large cardinal assumption that did not "tame" the large cardinals below it as a large cardinal. Rather, I'm more inclined to reject that a specific notion proposed as a large cardinal notion, *no longer be considered a large cardinal in the context of assuming another large cardinal* if the first is not "tamed" in the context of the second, but virtually everything else is. This is a vague idea, I know. I'm simply trying to isolate which features become forcing invariant under a specific large cardinal assumption.

2) Somewhat contrary to the first remark, I'm inclined to believe that preservation under small forcing (this is Levy-Solovay) is not a feature necessarily shared by all large cardinal assumptions. Although this feature is an empirical phenomenon known for most large cardinals, I'm not yet convinced that it is a characterizing feature of large cardinal assumptions. If the consensus of the community is that Levy-Solovay is somehow an intrinsic or desirable feature for a large cardinal assumption to satisfy, so be it. But I think there are many examples showing that large cardinal assumptions can in fact be susceptible to both large and small forcing notions.

3) Pen Maddy has a very thorough set of principles for proposing new axioms, expressed across two articles, that are at the very least worth reading. I'll add a link when I find it. I encountered these articles a long time ago and even found them compelling from a certain point-of-view. However, I didn't consult them for this answer and I also believe there has been research done since their publication that undermines some (or maybe even most) of those principles.

I invite any and all critiques (as well as additions) to this proposal.

examplesof large cardinal axioms, but rather a formal definition of what it means for an axiom to be a large cardinal axiom. This is a fairly slippery concept, and naive attempts to formalize this are typically inadequate. $\endgroup$ – Joel David Hamkins Jan 24 '14 at 14:32