With Hájek, we defined and explored L-valued universes for various
algebras L pertaining to fuzzy logics, among them, for {\L}ukasiewicz
logic. We then introduced an axiomatic set theory and showed that formulas
provable in the theory are valid in the universe constructed over L.
In this talk, I would like to explore the limitations of this method
for MV-algebras. In particular, I'm interested in two problems.
1) the role of the $\Delta$ connective in our construction; we did not
attempt to work without it, and
2) the role of the choice of a particular MV-algebra.
This is part of a larger investigation that seeks to find out various
ways of model construction for set theories over substructural logics,
and eventually determine limits of consistency for them.