It's not clear to me at all. Number systems, formal as they are, were arrived at via human reasoning, which is not a formal system. Human reasoning is, I maintain, complete, in that all possible truths are in principle available to it, including the truth or falisity of formally undecidable Gödel statements. The price paid for this is inconsistency: some wrong statements will seem to be true, and vice-versa. I don't think I need to make a case that human reasoning is inconsistent.
Which has no bearing. The number system is formal and contains Godel statements. It is (assumption) the natural product of human reasoning which is (another assumption) a natural product of physical law. By composition, the number system is a natural product of physical law. Ergo, physical law contains Godel statements.
Human reasoning is, I maintain, complete, in that all possible truths are in principle available to it ...
Not so. Human reasoning has bounded algorithmic complexity. Any truth exceeding that complexity is not open to it. So far as I know, there's no way out.
... including the truth or falisity of formally undecidable Gödel statements
Again not so. The nature of Godel statements is that they are neither true nor false but may be assumed to be either true or false.
I don't think I need to make a case that human reasoning is inconsistent.
Actually I think "unreliable" is what you're after. Inconsistency would be believing X and not-X at the same time and not thinking that's a problem.