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On the Consistency of Naive Set Theories over Substructural and Fuzzy Logics

Author Kazushige Terui

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LIPIcs.FSCD.2026.30.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Proof theory
Keywords
  • substructural logics
  • mathematical fuzzy logics
  • fixed points
  • naive set theory

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Abstract

The purpose of this paper is to invite structural proof theorists to a challenging problem in substructural and fuzzy logics: the consistency of Cantor-Łukasiewicz naive set theory. To this end, we consider two logics: FLew (Full Lambek calculus with exchange and weakening) and its extension Ł (Lukasiewicz logic). The former is equivalent to !-free intuitionistic linear logic with weakening, while the latter is the most prominent system of mathematical fuzzy logic. For each of them, we consider two extensions: one with a fixed point operator (which is neither least nor greatest) and the other with a naive set theory with unrestricted comprehension, so that we end up with four systems: FLew_{fp}, FLew_{set}, Ł_{fp} and Ł_{set}. The first two admit an easy proof of consistency by cut elimination, while the third admits a proof by the Brouwer fixed point theorem. The last system Ł_{set} (known as Cantor-Łukasiewicz set theory) is our main target. Although there are some partial results, the consistency of the full system is still open. In this paper, we consider a restricted fragment of Ł_{set} and prove its consistency.

Cite As Get BibTex

Kazushige Terui. On the Consistency of Naive Set Theories over Substructural and Fuzzy Logics. In 11th International Conference on Formal Structures for Computation and Deduction (FSCD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 378, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FSCD.2026.30

Author Details

Kazushige Terui
  • Kyoto University, Research Institute for Mathematical Sciences, Japan

Funding

This work was supported by JSPS KAKENHI Grant Number 20K11679.

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