A lattice-valued set theory
type: Item
instance of: scholarly article
title: A lattice-valued set theory
zbMATH Open document ID: 0936.03048
author: Satoko Titani
published in: Archive for Mathematical Logic
publication date: 2000-05-15T00:00:00Z
review text: The paper continues an earlier one of the same author on intuitionistic set theory for a universe of sets which is built up like the Boolean-valued universes [J. Symb. Log. 62, No. 2, 506-528 (1997; Zbl 0907.03023)]. In the present setting the former reference to a Heyting algebra is replaced by a reference to a complete lattice. This lattice is enriched with an implication operation which mimics just the characteristic function of the lattice ordering, and an intuitionistic-like negation operation. For the resulting set theory of such lattice-valued universes a sequent calculus is given, and its completeness is proved within ZFC as metatheory. Reviewers remark: The approach is (loosely) related to more general considerations on lattice-valued fuzzy sets. In this area, however, lattices are enriched with an additional abelian semigroup operation and its related residuation operation. Unfortunately the author does not mention this relationship at all.
reviewed by: Siegfried J. Gottwald
Mathematics Subject Classification ID: 03E70
Mathematics Subject Classification ID: 03B20
zbMATH DE Number: 1348050
zbMATH Keywords: intuitionistic logic
zbMATH Keywords: lattice-valued sets
zbMATH Keywords: set theory
zbMATH Keywords: intuitionistic set theory
zbMATH Keywords: sequent calculus
MaRDI profile type: Publication
full work available at URL: s001530050134
OpenAlex ID: W2062362173
DOI: 10.1007/S001530050134
description: scientific article; zbMATH DE number 1348050