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MrMind comments on Things philosophers have debated - Less Wrong Discussion
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Because even if we assume the existence of every mathematical structure, we are still assuming that they are coherent. Mind you, there are consistent models of some paraconsistent logic (even in set theory), but there is no model of the theory of all sentences. This is pretty standar model theory: the class of models of the total theory is empty (viceversa: the theory of the class of all models is empty).
Anyway, assuming trivialism is uninteresting (as the name correctly imply ;)): we still can play a formal game that mimics the difference between truth and falsity.
I'm not sure why level IV would restrict itself to standard model theory. In a tri-valued logic (i.e., all propositions are either true, false, or both), there are non-trivial models of trivialism.
Trivialism would not respect Tegmark IV's subsections which comply with our model of logic.