  In this paper we use results from Computable Set Theory as a means to
represent and reason about description logics and rule languages for the
semantic web.
  Specifically, we introduce the description logic $\mathcal{DL}\langle
4LQS^R\rangle(\D)$--admitting features such as min/max cardinality constructs
on the left-hand/right-hand side of inclusion axioms, role chain axioms, and
datatypes--which turns out to be quite expressive if compared with
$\mathcal{SROIQ}(\D)$, the description logic underpinning the Web Ontology
Language OWL. Then we show that the consistency problem for
$\mathcal{DL}\langle 4LQS^R\rangle(\D)$-knowledge bases is decidable by
reducing it, through a suitable translation process, to the satisfiability
problem of the stratified fragment $4LQS^R$ of set theory, involving variables
of four sorts and a restricted form of quantification. We prove also that,
under suitable not very restrictive constraints, the consistency problem for
$\mathcal{DL}\langle 4LQS^R\rangle(\D)$-knowledge bases is
\textbf{NP}-complete. Finally, we provide a $4LQS^R$-translation of rules
belonging to the Semantic Web Rule Language (SWRL).
