Kripke-like models of Set Theory in Modal Residuated Logic

TL;DR

This paper extends Kripke models of Set Theory within Modal Residuated Logic, generalizing the von Neumann hierarchy and proposing a constructible universe concept, bridging intuitionistic and residuated logic frameworks.

Contribution

It introduces a novel generalization of Kripke models for Set Theory using Modal Residuated Logic, including a new hierarchy and constructible universe concept.

Findings

Generalization of von Neumann hierarchy in Modal Residuated Logic

Translation between Modal Residuated models and Heyting valued models

Proposal of a universe of constructible sets in this logic framework

Abstract

We generalize Fitting's work on Intuitionistic Kripke models of Set Theory using Ono and Komori's Residuated Kripke models. Based on these models, we provide a generalization of the von Neumann hierarchy in the context of Modal Residuated Logic and prove a translation of formulas between it and a suited Heyting valued model. We also propose a notion of universe of constructible sets in Modal Residuated Logic and discuss some aspects of it.