I've been trying to understand in a more formal way what a set actually is, but I have some questions. According to the axiom of regularity for every non-empty set A there exists an element in the set that's disjoint from A. That would mean that such element is also a set, right?
I read here: Axiom of Regularity , that in axiomatic set theory everything is a set, I understand that natural numbers are constructed from the empty set, integers are constructed from the naturals, rationals from the integers, and reals from the rationals. I can see how every element in such sets is also a set. But, for example, in the set of all the letters of the alphabet, or the sample space of an experiment when the possible results are not numbers, or the set of my classmates; it's not clear to me how their elements are also sets. So, are they really sets? Is every element in a set also a set?