TITLE: Is every element on a set also a set?
QUESTION [7 upvotes]: 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?
Thank you

REPLY [5 votes]: That depends on which axioms or system you're using. You could of course make up a system which allows (distinct) atomic objects (they're called ur-elements) that can be elements of sets, this is how ZF set theory originally started.
However in (modern) ZF set theory the axiom of extensionality basically prohibits anything that's not a set. Things are considered equal if they have the same elements and since anything not being a set does not have any elements they would be considered equal to the empty set.
As you pointed out one constructs the natural numbers and so on by constructing set. However one normally does not use those properties of them (being sets), you almost never see things like $1 \cup 2$ or $0 \in 1$. One should note that the way natural numbers are constructed is not standardized, that is there is different ways to achieve the same (standardized) properties of the numbers (which makes using numbers as they are sets being non-standard and have no universally accepted meaning).