ZFC can't prove that strongly inaccessible cardinals exist, or else it would prove that a model of itself exists and hence $Con(ZFC)$. So this leaves us with two options:
- ZFC proves there are no strongly inaccessible cardinals
- The existence of strongly inaccessible cardinals is independent from ZFC
I've heard, however, that you can't actually prove that it's #2. That is, you can't prove the independence of large cardinal axioms from ZFC. Why is this?
At first, I thought it was because proving independence would mean that ZFC proves the consistency of a larger theory that contains ZFC as a model. However, I don't think my reasoning was right. That wouldn't prove that a model exists - just that a model could exist. And isn't it proven that Con(ZFC) is independent from ZFC, and hence that ZFC + Con(ZFC) is consistent iff ZFC is?
So how do you show that you can't show the independence of large cardinal axioms?
Well. In order to establish "true independence" you need to show that $\sf ZFC$ is consistent with both the statement and its negation, at least assuming that $\sf ZFC$ is consistent.
However just assuming that $\sf ZFC$ is consistent is not enough to establish that $\sf ZFC$+large cardinal axioms are consistent, because from large cardinal axioms we can prove the consistency of $\sf ZFC$; thus a proof of consistency from $\sf ZFC$ of large cardinal axioms will ultimately prove the consistency of $\sf ZFC$, and by the second incompleteness theorem this is impossible.
So you can only prove that $\sf ZFC$ does not prove the existence of large cardinals, not that it does not refute them.
And indeed every now and then you can find people claiming to have proofs that inaccessible cardinals are inconsistent with $\sf ZFC$.
(The situation is similar with $\sf ZFC+\operatorname{Con}(ZFC)$, which is also strictly stronger than just $\sf ZFC$ in terms of consistency. But I don't think anyone who seriously considers $\sf ZFC$ a reasonable theory thinks it is inconsistent, so this assumption gets questioned far less often than inaccessible cardinals.)
Caveat lector:
Relative consistency results about $\sf ZFC$ and assumptions stronger than $\sf ZFC$ itself will depend a lot on the meta-theory. If you work in a theory which assumes $\sf\operatorname{Con}(ZFC)+\lnot\operatorname{Con}(ZFC+LC)$, then $\sf ZFC$ will refute large cardinals in that meta-theory. Because there will be a "natural number" encoding a proof of contradiction in a way recognizable by $\sf ZFC$.
The point is that when you want to talk about the ability to refute something stronger than $\sf ZFC$ you need to ask yourself what sort of meta-theory you have. If inaccessible cardinals are refutable from $\sf ZFC$, that's great (or actually the other thing, sucky); if not, then the question if they are refutable becomes meta-theory dependent.