Consider a model of the empty unsorted signature. Equivalently, a model of the signature having a single sort, and no function or relation symbols. Intuitively, such a model should be called a "set."
However, the emphasis is all wrong. For example, in ZFC everything is a set, but I don't feel comfortable saying that everything in ZFC is a model of the empty signature. Firstly, because its just false. Secondly, because we should only talk about models up to isomorphism, however your average ZFC set is interesting beyond its cardinality; that is, beyond its structure up to isomorphism. We care about more than just the cardinality of your average ZFC set.
Is there a good word to mean "a model of the empty signature"?