Power (set) alc

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 Ω, consisting of only four axioms characterizing binary union, set difference, inclusion, and the power-set. The approach is then completed defining ALC^Ω, an extension of ALC in which concepts are naturally interpreted as sets living in Ω-models. In ALC^Ω not only membership between concepts is allowed-even admitting circularity-but also the power-set construct is exploited to add metamodeling capabilities. We conclude providing a polynomial translation of ALC^Ω in ALCOI and proving its basic traits, among which the validity of the finite model property.