> It states more about provability in the context of a single axiomatic system for constructed self-referential statements rather than truth in general.

Thanks. And oops, I didn't know that the result was proven in a larger system. Did some searching and found this short answer[1] and a longer description[2] that I liked.

[1]:https://math.stackexchange.com/a/120018/19178 [2]:https://mathoverflow.net/a/24919

This statement in particular helped:

> The Incompleteness Theorem prevents ZFC from proving its own consistency and for that we need to have an additional axiom, giving us a stronger theory which can then prove ZFC is consistent