Why do I run into trouble if I try to define the set of all groups? I get that defining the set of all sets could lead to paradoxes. But how is it that defining the set of all groups somehow leads to the same kind of problems? If I define the set of all groups as all the ordered pairs (x,y) such that y is a closed operation on x that satisfies all the axioms of a group. How will that get me in trouble??