*Sounds plausible that self-aware systems can manage this. I'd like to
see this done as a formal system though, as I have a natural mistrust
of handwaving arguments! *

I like it too :).
I think the computational view would help in construction.

I definitely don't think the two systems could be complete, since (handwavey
argument follows) if you have two theorem-proving algorithms A and B, it's
trivial to just create a new algorithm that prints out the theorems that
either A or B could print out, and incompleteness should apply to this too*
They're not independent systems.putting that aside, I can't find the
correspondence to my argument. It would be nice if you could clarify your
* *
* *Brent,*
*But doesn't that depend on having adopted rules of inference and values,
i.e. "the sentence is either true or false".  Why shouldn't we suppose that
self-referential  sentences can have other values or that your informal
reasoning above is a proof and hence there is contradiction*

Actually this is the main advantage of making such loop. you're right only
when we're talking about ONE system and that system has concluded the truth
of such statement. But it can be avoided by two or more systems. They
are reliable in each other's view, and have statements for that. maybe that
is the only (symmetric!) difference of those systems. Except for that,
both(all) are the same.

Mohsen Ravanbakhsh

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