Proving the consistency of ZFC in some other system While we cannot prove the consistency of PA within PA, we can prove that PA is consistent within ZFC. Is there a similar way to prove ZFC is consistent within a (concrete example of a) larger/different axiom set?
I assume the answer is 'yes', but my main question is whether this is meaningful at all, and (since I am not hopeful) why not? My motivation—for example—would be to apply this in my new system to undecidable sentences in ZFC of the form '$\varphi$ $\Leftrightarrow$ ZFC is inconsistent'. 
Is there any meaningful way to convince ourselves that a system is consistent, or is this something one should just 'believe'?
My motivation is somewhat philosophical—what is truth?—but undecidability and the likes is something I cannot get my head around, and it is giving me sleepless nights! :-) Thanks
 A: Yes, of course we can prove the consistency of $\sf ZFC$ in some other system. You can have "silly systems" like $\sf PRA+\operatorname{Con}(ZFC)$ in which we have enough arithmetic to formalize the notions of consistency and such. And there are natural extensions of $\sf ZFC$ likes adding large cardinals into the mix.
The question is, of course, who can ensure that the extended systems are consistent? This is exactly why we have a hierarchy of large cardinal axioms, which is almost linear in terms of consistency. Namely, in most cases, we can tell given two large cardinal axioms (e.g. "There exists an inaccessible cardinal" and "There exists a proper class of Woodin cardinals") that one will prove that the other is consistent with $\sf ZFC$.
But, again, who watches the watchers? And who watches the watchers who do not watch themselves? At some point we have to take something for granted. And the question is how well your philosophical views are aligned with some axiomatic system. Most set theorists will probably agree that inaccessible cardinals or even weakly compact cardinals are probably consistent; whereas many will hesitate to claim the same about supercompact cardinals.
And if this troubles you, well, whoever said that the induction we use to prove basic things about propositional calculus, which are necessary for the soundness of the underlying logic (even before talking about $\sf ZFC$!), is consistent at all? This is also something that you have to take on belief.
Two possible ways out of this are: (1) We looked very hard, and we found no inconsistencies so far, so might as well assume this whole thing works; or (2) I just like pretty math, and formally speaking if $\sf ZFC$ is inconsistent, the finitary proofs are still probably fine, just vacuous.
