TITLE: The set that only contains itself
QUESTION [12 upvotes]: Ignoring the axiom of regularity (and therefore the implication of "no set can contain itself"), would it be correct to state that the set that contains only itself is unique?
My argument is that if $x$ is said set, then
$$ x = \{x\} = \{\{x\}\} = \{\{\{\cdot\cdot\cdot\}\}\} $$
Ad infinitum, which seems to be unique.

REPLY [14 votes]: It need not be unique; in fact, you can weaken the axiom of foundation to allow either well-founded sets or sets of the form $x=\{x\}$. Sets of the latter form are called Quine atoms, and play the role of urelements. These are useful in set theory with atoms because they allow you to formulate a theory of sets-and-atoms without resorting to multi-sorted logic.
For more, see The Axiom of Choice by T. J. Jech.