  We explore the relationships between Description Logics and Set Theory. The
study is carried on using, on the set-theoretic side, a very rudimentary
axiomatic set theory Omega, consisting of only four axioms characterizing
binary union, set difference, inclusion, and the power-set. An extension of
ALC, ALC^Omega, is then defined in which concepts are naturally interpreted as
sets living in Omega-models. In ALC^Omega not only membership between concepts
is allowed---even admitting circularity---but also the power-set construct is
exploited to add metamodeling capabilities. We investigate translations of
ALC^Omega into standard description logics as well as a set-theoretic
translation. A polynomial encoding of ALC^Omega in ALCIO proves the validity of
the finite model property as well as an ExpTime upper bound on the complexity
of concept satisfiability. We develop a set-theoretic translation of ALC^Omega
in the theory Omega, exploiting a technique originally proposed for translating
normal modal and polymodal logics into Omega. Finally, we show that the
fragment LC^Omega of ALC^Omega, which does not admit roles and individual
names, is as expressive as ALC^Omega.
