I think that the way that humans predict other humans is the wrong way to look at this, and instead consider how humans would reason about the behavior of an AI that they build. I'm not proposing simply "don't use formal systems", or even "don't limit yourself exclusively to a single formal system". I am actually alluding to a far more specific procedure:

• Come up with a small set of basic assumptions (axioms)
• Convince yourself that these assumptions accurately describe the system at hand
• Try to prove that the axioms would imply the desired behavior
• If you cannot do this return for the first step and see if additional assumptions are necessary

Now it turns out that for almost any mathematical problem that we are actually interested in, ZFC is going to be a sufficient set of assumptions, so the first few steps here are somewhat invisible, but they are still there. Somebody need need to come up with these axioms for the first time, and each individual who wants to use them should convince themselves that they are reasonable before relying on them.

A good AI should already do this to some degree. It needs to come up with models of a system that it is interacting with before determining its course of action. It is obvious that it might need to update what assumptions it's using the model physical laws, why shouldn't it just do the same thing for logical ones?