Are there any sets that cannot be constructed using the symbols $\{$ and $\}$?

Question

It seems to me all sets in mathematics can be constructed via these two symbols only. For instance, the natural numbers are defined as $0 = \varnothing = \{\}, 1 = \{0\}$, $2 = \{0, 1\}, 3 = \{0, 1, 2\}$ and so on. From this it follows that $\mathbb{Q}$ and $\mathbb{R}$ and even $\mathbb{C}$ can be constructed using $\{$ and $\}$ only. Are there any sets that cannot be, in theory, constructed via just these two symbols?

Answer 1

No, the axiom of regularity essentially tells us that given a set $x$, there is a finite sequence $x_0=x,x_n=\varnothing$ such that $x_n\in\ldots\in x_0$.

A difference way to look at it would be to see that in $\sf ZF$ we can show that the universe is really constructed from the empty set by iterating the power set. So essentially the answer is negative. Although it should be exercised with caution since the set of all natural numbers would be one that has increasingly more "opening braces", but there is no infinite sequence of closing braces. This might coincide and might not with the notion of "construct" people hold in their heads.