Effective Set Membership in Computer Algebra and Beyond

In previous work, we showed the importance of distinguishing "I know that $X\ne Y$" from "I don't know that X= Y". In this paper we look at effective set membership, starting with Grobner bases, where the issues are well-expressed in algebra systems, and going on to integration and other questions of `computer calculus'. In particular, we claim that a better recognition of the role of set membership would clarify some features of computer algebra systems, such as `what does an integral mean as output'.