# Are $\{\}$ and $\{\{\}\}$ equal in set theory? [duplicate]

According to set theory, are $\{\}$ and $\{\{\}\}$ equal?

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• $P(A)="\{\}\in A"$ is true for one and false for the other. Therefore $\forall x(x\in\{\}\leftrightarrow x\in\{\{\}\})$ cannot be true. – Bettybel Jul 7 '17 at 15:11
• We have $\{\}=\emptyset$, and $\{\{\}\}=\{\emptyset\}$. – Dietrich Burde Jul 7 '17 at 15:12
No. We have that $\{\}$ is the empty set, whereas $\{\{\}\}$ is the set which contains one element: the empty set. So $\{\{\}\}$ is not empty.
No, in fact one definition of the natural numbers has $\{\}$ as 0, $\{\{\}\}=\{0\}$ as 1, $\{\{\},\{\{\}\}\}=\{0,1\}$ as 2, etc.