Specifically what I'm referring to is, "In every mathematical system there WILL BE certain Truths found which are not derivable from the Axioms." What Kurt showed was that either that (the system is Incomplete iow) OR the system is inconsistent. AFAIK (and it has been more than two decades since I last looked at this seriously) we pretend our mathematics is consistent, but there's no firm proof that it is.

We know that either (a) our mathematics is incomplete (your stated observation) or (b) our mathematics is inconsistent. Kurt proved we cannot have both consistency and completeness with his (in)famous Incompleteness Theorems. To make things "just work", we claim (without proof) that (a) holds because (b) holding is really a terrible nuisance.

The first time I understood this, I chuckled to myself how closely this "pretending we know that (a) is the case" was akin to other "beliefs." I think that was the source of the fire-in-the-belly I had on this topic when I was a serious student of mathematics. WE were not supposed to "believe" in anything. WE were supposed to PROVE things. Unlike scientists who can and do have "exceptions to the Rule", mathematicians don't. A single counter-example is sufficient to throw out the proposition entirely. But here, here at the base we don't do that because, well, it's disturbing.