In set theory, is the axiom of pairing circular? The axiom of pairing uses "objects" $a$ and $b$ to form the set $\{a,b\}$, and the objects in this case can be either individuals or sets. But it seems to me that the entire point of the axioms is to establish what a set is, so how can we use the concept of a set in its definition?
 A: First, the purpose of the axioms of set theory is not to tell us what a set is. This is a fundamentally philosophical question. The axioms’ purpose is to tell us some facts about how sets relate to each other (and, if we allow things that aren’t sets in our theory, how sets relate to these other things).
Second, when studying set theory, one traditionally works in a theory where everything is a set, such as ZFC. This is certainly not required; people often study the set theory ZFA, which stipulates the existence of a set of “atoms”, where the atoms are not sets. But it is customary to study ZFC-like theories, for better or worse. This is why you will see people comment that your distinction between “individuals” and “sets” is not meaningful.
A: (A) The Axiom is Pairing is

(B) That Axiom can then give the Interpretation that we have a Set with 2 elements , hence that Axiom is building up the Set theory.
(C) Consider https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory : The 1st is the Axiom of extensionality , containing words like "two sets are equal ...." , the 2nd is the Axiom of regularity containing words like "Every non-empty set ...." , the 8th is Axiom of Power Set where the title itself is containing the word Set.
The Axiom of Pairing is listed 4th & there is nothing special about it in terms of Circularity. All Axioms contain the word Set.
(D) We can have no way to make the Axioms without using the word Set. That term is left undefined. The Circularity is inherent ....
We must have some Intuition about what a Set is & what a Set is not. Then the Axioms formalize that Intuition. With that formalization , we can move on to various Definitions & theorems & other higher-level theories like Polynomials , groups , rings , vectors , Etc.
(E) We might think that we can avoid using Set in the Axioms by using a new term , something else like "Collection" or "Class" or "Category" , but that is only pushing the Circularity or undefinedness to that new term , "Collection" or "Class" or "Category".
In Set theory , the term Set is left undefined , with all Axioms using that term freely.
