The paper continues the investigation of Poincare and Russel’s Vicious Circle Principle (VCP) in the context of the design of logic programming languages with sets. We expand previously introduced language Alog with aggregates by allowing infinite sets and several additional set related constructs useful for knowledge representation and teaching. In addition, we propose an alternative formalization of the original VCP and incorporate it into the semantics of new language, Slog+, which allows more liberal construction of sets and their use in programming rules. We show that, for programs without disjunction and infinite sets, the formal semantics of aggregates in Slog+ coincides with that of several other known languages. Their intuitive and formal semantics, however, are based on quite different ideas and seem to be more involved than that of Slog+.