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.
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.