A lattice-valued set theory
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.
scientific article; zbMATH DE number 1348050