# Can there be multiple sets $X$ satisfying $X = \{X\}$?

Suppose that our axioms are such that a set $X$ exists satisfying $X = \{X\}$. Might it be consistent that there are multiple sets satisfying this property? Are there (well-known) axioms that guarantee existence and uniqueness?
My motivating intuition here is that in most cases, enumerating the elements of a set uniquely defines the set. For example, I'm tempted to say that the set $\{1,2,3\}$ is unique "because we've enumerated all of its elements". Does this intuition completely fail for these stranger sets?