Are the objects associated with large cardinals still sets?

Question

When set theorists speak of large cardinals, are they still referring to the cardinality of some collection? If so, is this object a (hypothetical) set, a proper class, or something else?

Answer 1

First let me point out that there is no uniform and accepted definition for what is a large cardinal. In some contexts, even cardinals like $\aleph_\omega$ might be considered large. Some large properties are in fact asserting the existence of certain real numbers.

Large cardinals are generally sets. But large cardinals are not just sets, they are sets which carry some structure and some properties. Like ultrafilters, or the existence of a branch in a tree.

Some large cardinals properties require a proper class of "witnesses" for it to hold. For example, $\kappa$ is supercompact if for every $\lambda\geq\kappa$ there exists an ultrafilter on some collection of subsets of $\lambda$. While this is perfectly expressible using the language of set theory, this is not just one set which witnesses the large cardinal property, but rather a proper class of sets that do that.

Different large cardinal notions have different requirements from their witnesses, and so the question whether or not these are sets or classes can change. And this cannot be bounded, i.e. we can't say that if a large cardinal property is stronger than so and so, then it requires a proper class of witnesses.

The reason is almost trivial. Given a large cardinal property $P$, we can always define a stronger one by requiring the existence of an inaccessible cardinal $\kappa$ such that $V_\kappa\models\exists\lambda\text{ such that } P\text{ holds}$.