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I thought about this while answering this question. To restate my question:

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?

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Ben Grossmann
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Such sets are called Quine atoms - see https://mathoverflow.net/questions/33282/can-we-have-a-a. In ZFC, no Quine atoms exist; in Aczel's set theory, exactly one Quine atom exists; and in general we can have lots of distinct Quine atoms in NF-like set theories.

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Noah Schweber
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