The system has a binary notion of truth which satisfies the law of excluded model because it was constructed in this manner. Mathematical truth does not exist in its own right, in only exists within a system of logic. Geometry, arithmetic and set theory can all be modelled within the same set-theoretic logic which has the same rules related to truth. But this doesn’t mean that truth is a set-theoretic concept—set-theory is only one possible way of modelling these systems which then lets us combine objects from these different domains into the one proposition. Set-theory simply shows us being within the true or false class has similar effects across multiple systems. This explains why we believe that mathematical truth exists—leaving us with no reason to suppose that this kind of “truth” has an inherent meaning. These aren’t models of the truth, “truth” is really just a set of useful models with similar properties.
The problem with that approach is that you still need a meta-language and a notion of “meta-truth” to talk about these models, and then you’re right back where you started.
That’s a good point, but I don’t think that it invalidates the whole approach. Non-classical logic is normally formulated within classical logic. I believe that other formulations of set theory are usually analysed from within standard set theory (can someone else confirm?).
My point was that just as some notion of set theory is necessary to talk about the different kinds of set theory, some notion of truth is needed to talk about the different notions of truth.
The problem with that approach is that you still need a meta-language and a notion of “meta-truth” to talk about these models, and then you’re right back where you started.
That’s a good point, but I don’t think that it invalidates the whole approach. Non-classical logic is normally formulated within classical logic. I believe that other formulations of set theory are usually analysed from within standard set theory (can someone else confirm?).
The liars paradox is a paradox in “meta-logic”. Standard set theory already has ways of dealing with it (by disallowing use of the word “truth”).
My point was that just as some notion of set theory is necessary to talk about the different kinds of set theory, some notion of truth is needed to talk about the different notions of truth.