I'm not either, but I think it depends on which system you're working from within. There's the ugly, beautiful thing from Godel's second incompleteness theorem saying a formal system cannot prove its own consistency from within.

With set theory, it looks like there was some reformulating of the fundamental terms to prevent paradox, but I'm not quite clear on whether that leaves us with a consistent formal system in the end: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...