Constructible sets in lattice-valued models: A negative result

TL;DR

This paper explores constructible sets within lattice-valued models of set theory, demonstrating that such models are inadequate for studying constructibility in non-classical logics, despite their natural appeal.

Contribution

It introduces two constructions of the constructible universe in lattice-valued models and proves their isomorphism to classical universes, revealing limitations in non-classical set theory studies.

Findings

Constructible universes $rak{L}^{Q}$ and $L^{Q}$ are isomorphic to classical universes V and L.

Lattice-valued models are not suitable for studying constructibility in weaker logics.

Results highlight limitations of lattice-valued models in non-classical set theory research.

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.