An early approach toward graded identity and graded membership in set theory

The paper considers an early approach toward a (fuzzy) set theory with a graded membership predicate and a graded equality relation which had been developed by the German mathematician D. Klaua in 1965. In the context of the mathematical fuzzy logic MTL of left-continuous t-norms we discuss some properties of these graded relations. We compare the simultaneous recursive definitions of these relations with the very similar approach toward Boolean algebra valued interpretations of membership and equality, presented in 1967 by D. Scott and R. Solovay in the context of independence proofs for ZF set theory. Finally we speculate about possible reasons why Klaua soon abandoned this approach.