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I was reading a pdf on set theory that explains that because of the axiom of regularity, there is no set that contains all sets, $\{x: x =x\}$, because it would have to contain itself.

I was wondering if instead a set containing all sets containing one set, $\{x:\exists y\ x=\{y\}\}$, could exist. How could you prove/disprove it's existence?

Asaf Karagila
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    Hint: Regularity tells us there can't be a set $x$ such that $x\in x$. But it also tells us that there can't be sets $x, y$ such that $x\in y\in x$. –  Sep 19 '15 at 19:05
  • Related: http://math.stackexchange.com/questions/3815/why-does-the-set-of-all-singleton-sets-not-exist?rq=1 – Noah Schweber Sep 19 '15 at 19:10
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    You can also consider the set $B={x\in A:\forall y\in x,x\notin y}$ (where $A$ is the set of singletons). Prove that this is equal to ${{X}:{X}\notin X}$. Then ask if ${B}\in B$. (This does not appeal to regularity.) – Akiva Weinberger Sep 20 '15 at 05:42

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No. For any set of sets $\mathcal A, \bigcup \mathcal A = \{x : \exists A \in \mathcal A \text{ with } x \in A\}$ exists (assuming the normal axioms of set theory). But for your supposed set, its union would be the universal set, so your condition does not define a set.

Paul Sinclair
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