No formal mathematical system that's worth its salt can prove itself consistent. But you can assume it's consistent – creating a new mathematical system with a new axiom, that itself cannot prove its own consistency and requires more and more axioms to this extent.
I think in practice, all known systems are likely to be consistent no matter how many layers you add there. Peano arithmetic seems like a complete slam-dunk, although, because of these incompleteness issues, this is rooted in speculation. But here's my question: is there a proof system s.t. ZFC thinks is consistent, and where you can add in at least one layer of "and that system is consistent", but adding more than N layers makes it inconsistent?
The notation for this math subfield is tremendously strange, and I've yet to get the type of mathematical grip on it that I'd like. Obvious ZFC + [4 layers of "that's consistent"] thinks this is true about ZFC, so this seems generally tractable. It'd be fascinating to see if there's some way of having a system break at 2 layers, though, instead of needing an infinite stack. Basically, is there a ZFC - [N layers] equivalent? I'm totally open to replacing ZFC with almost anything else, the idea is, we can construct [System] + [N] by adding axioms, can we find a [System] - [N] equivalence?
I've also not checked if there is a literature about this, any pointers would be welcome and the email is always there after every blog post. I'll post any advances with credit in a follow up post.