Constructible sets in lattice-valued models: A negative result

Abstract

We investigate different set-theoretic constructions in Residuated Logic based on Fitting's work on Intuitionistic Set Theory. We start by stating some results concerning constructible sets within valued models of Set Theory. We present two distinct constructions of the constructible universe: $\mathfrak{L}^{\mathbb{Q}}$ and $\mathbb{L}^{\mathbb{Q}}$, and show that they are isomorphic to V (the classical von Neumann universe) and L (the classical G\"odel constructible universe), respectively. Even though lattice-valued models are the natural way to study non-classical Set Theory (e.g., Intuitionistic, Residuated, Paraconsistent Set Theory),our results prove that the use of lattice-valued models is not suitable to study the notion of constructibility in logics weaker than classical logic.