When I was a graduate student Jack Silver was famous for trying to refute first, the existence of measurable cardinals, and second, the consistency of ZFC. His attempt at refuting measurable cardinals led directly to the theory of ordinal indiscernibles over $L$ and $0^\sharp$, which are now a core part of our understanding of the nature of the constructible universe in the presence of large cardinals. What an amazing and valuable contribution to the subject he had made that way, even though he had never achieved his initial goal of refuting measurable cardinals.
And although I have heard many people speak against Silver's inconsistency research program, my view has always been:
Main observation. We cannot show that Silver's ideas about inconsistency are incoherent.
Because of the incompleteness theorem, we know that even if ZFC is consistent, then it is consistent with ZFC to hold that they are inconsistent. Thus, Silver's world view will be as consistent as his opponent, and there will never be an argument against the view that ZFC is inconsistent that does not beg the question by assuming that this theory is consistent or something stronger.
Meanwhile, of course, many set theorists, including almost every large cardinal set theorist (but not quite all, like Silver), believe not only that ZFC is consistent, but that consistency rides much higher in the large cardinal hierarchy. There is no proof, for the reasons I gave above.
Another part of my view, however, is that it is precisely because we cannot prove that ZFC and large cardinals are consistent that we are interested in them. Namely, Gödel had identified with his theorem the incompleteness phenomenon, and from that we know of the existence of a tower of theories standing transfinitely above any foundational theory we might entertain. We know there is a tower of stronger theories above whatever theory $T$ we might have. One way of constructing such a tower of theories is to form the theories $T+\text{Con}(T)$ and $T+\text{Con}(T+\text{Con}(T))$ and so on.
But how remarkable that the large cardinal theories themselves instantiate the predicted tower of increasing consistency strength. These theories are not formed by some trivial closure under consistency statements, but rather express profound statements of infinite uncountable combinatorics. These axioms thus have independent interest, and yet fulfill the predicted tower of consistency strength.
So we are glad to have found in the large cardinal hierarchy the predicted tower of consistency strength, and it is not worrisome that we cannot prove consistency — rather, this was just the feature that we were seeking.
Disclaimer. Although I heard Silver talk about set theory at length, having taken several of his graduate courses, I never once heard him discuss his inconsistency research program. All my knowledge about it is second-hand, from other senior set theorists whom I have no reason to doubt. I suppose that in light of the extent of opposing views, it might have made some sense or was at least understandable for Silver to keep his research program relatively private. But actually I wish he had given a series of talks about his research program and his views on the matter — that would have been fascinating! (From my current perspective, I chalk this up to the differing cultural norms about the tolerable levels of disagreement in mathematics vs philosophy.)
