It leaves us with a system that we strongly suspect is consistent, but which we cannot prove is such (at least not without resorting to a stronger system whose

It leaves us with a system that we strongly suspect is consistent, but which we cannot prove is such (at least not without resorting to a stronger system whose consistency is again unproven). Note that even were ZFC be proven to be inconsistent it would probably not be a huge issue practically speaking for most mathematicians. This Math Overflow answer describes it best: https://mathoverflow.net/a/41030 https://mathoverflow.net/a/41030