Some theories can lie about their own consistency (for example, if $PA$ is consistent then the theory $PA + \lnot CON(PA)$ is consistent, although it proves its own inconsistency). 


Now working with ZFC, one can avoid problems like ZFC lying about its own consistency, by assuming ZFC is $\omega$ - consistent.


that leads me to two questions:


the first one is - when we proved inside the metatheory of ZFC that the $\omega$ - consistency of ZFC avoids the problem mentioned above, how can we be so sure that this proof isn't another example of ZFC lying about itself? in different words, how can we know that this proof is expressing truth? 


and the second question is - without assuming ZFC is $\omega$ - consistent, how can we rely on any conclusions of ZFC? is there a way to separate between theorems which can be examples of ZFC lying about itself and theorems which does express truth?



ZFC ability to express truth and $\omega$ - consistency
The metatheory involved in proving "if ZFC is $\omega$-consistent then it doesn't prove its own inconsistency" is some tiny fragment of PA. I'd expect primitive recursive arithmetic (PRA) to be more than enough, since the proof is essentially just inspection of definitions. So we needn't worry that the metatheory argument is a case of ZFC lying to us. The only worry that I see is that PRA might lie to us; if you worry about that, then you should probably also worry about a lot of more basic things, like whether $2^n$ exists for all natural numbers $n$.


