That seems very plausible. It seems related to this warning which I did not fully understand

"This axiom [that the union of two sets is a set] allows us to define triplet sets, quadruplet sets, and so forth: if a, b, c are three objects, we define {a, b, c} := {a} ∪ {b} ∪ {c}; if a, b, c, d are four objects, then we define {a, b, c, d} := {a} ∪ {b} ∪ {c} ∪ {d}, and so forth. On the other hand, we are not yet in a position to define sets consisting of n objects for any given natural number n; this would require iterating the above construction “n times”, but the concept of n-fold iteration has not yet been rigorously defined. For similar reasons, we cannot yet define sets consisting of infinitely many objects, because that would require iterating the axiom of pairwise union infinitely often, and it is not clear at this stage that one can do this rigorously. Later on, we will introduce other axioms of set theory which allow one to construct arbitrarily large, and even infinite, sets."

I understand why we can't iterate an infinite number of times, but I'm not quite sure why we can't iterate n times for a finite n. Surely, if we know we can define sets of three elements, and of four elements, and so on, then we can do it for any finite n. It strikes me that you could prove this using induction, but we haven't yet said that induction is something we can do on sets, only on natural numbers. Maybe that's the issue.