[Timon3]

>> Can you try to describe how you'd "drill down" on informal speech to transform it into such a system?

> As I said, just keep asking "why?" or "what does that mean?", then repeat that with the answer. Sooner or later you hit an assumption and a shrug.

I still don't understand your assumption. Do you think that any axiomatic system is a formal axiomatic theory? As I've said before, this term is well-defined, and I don't see how you could "drill down" on natural language to arrive at such a system. There are many axiomatic systems that are not formal axiomatic theories, and Gödels proof doesn't apply to those.

> I wouldn't understand Gödel's proof even if I tried to, I'm sure -- it "rings true" because it matches my own intellectual observations regardless where I turn.

This is why I've been asking about how you'd bridge the gap between Gödels proof and your assumption, because Gödels proof applies strictly to one thing, and you seem to apply it to everything, even if it doesn't meet the requirements of the proof. But I guess you're arguing from a philosophical standpoint, not a logical one.