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Vej_Kse comments on Walkthrough of the Tiling Agents for Self-Modifying AI paper - Less Wrong
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Coincidentally, a paper based on Yudkowsky and Herreshoff's paper has appeared a few days ago on the arXiv. It's Paradoxes of rational agency and formal systems that verify their own soundness by Nik Weaver. Here's the abstract:
I read the paper. It seems interesting. Apparently one key to the "assertibility" predicate is that it works on intuitionistic logic: the axioms that describe it don't include the law of the excluded middle: "(X is assertible) OR (It is not the case that X is assertable)" isn't assumed to be true for all X.
In their assertibility logic, you can indeed construct the statement "This sentence is not assertible", and you can even prove that it's false... but you also can't prove that you can't assert it - you can only prove that if you could assert it, you could also assert a contradiction.
Someone who's actually working on the real math should go look at it.