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?

notprove its own inconsistency. It proves inconsistency of $PA$. $\endgroup$ – Andrej Bauer Jan 9 at 13:11