We define the empty set as the set that contains no elements, but is there more than one empty set? So is there "an" empty set as opposed to "the" empty set? May one be able to receive values, while another is truly empty, etc.? And how is it possible to define the empty set by the absence of members or by emptiness?
The empty set is indeed defined to be that set which contains no elements. Another definition we need is that of identity of sets: we say that set A and set B are identical just in case they contain exactly the same elements, i.e., whatever is in A is also in B, and vice versa . So, with these two definitions in hand, consider empty set E 1 and empty set E 2 . Well, they are equal since any element that is in the one is in the other (for the trivial reason that neither set contains any elements). So there really is only one empty set - which is what licenses our use of the definite article "the" in " the empty set". I'm not sure I understand your last question. In set theory, you've specified a set completely when you've specified its elements. And when we say that the empty set contains nothing, we have indeed specified exactly which elements it contains (namely, none).