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


marked as duplicate by Asaf Karagila elementary-set-theory Jul 7 '17 at 15:23

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    $\begingroup$ NO; the first one is empty while the second one has one element: the empty set. $\endgroup$ – Mauro ALLEGRANZA Jul 7 '17 at 15:11
  • $\begingroup$ $P(A)="\{\}\in A"$ is true for one and false for the other. Therefore $\forall x(x\in\{\}\leftrightarrow x\in\{\{\}\})$ cannot be true. $\endgroup$ – Bettybel Jul 7 '17 at 15:11
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    $\begingroup$ We have $\{\}=\emptyset$, and $\{\{\}\}=\{\emptyset\}$. $\endgroup$ – Dietrich Burde Jul 7 '17 at 15:12
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    $\begingroup$ A bag with an empty bag in it is able to be distinguished with an empty bag. $\endgroup$ – JMoravitz Jul 7 '17 at 15:13
  • $\begingroup$ @JMoravitz I'll have to use that one! $\endgroup$ – Theo Bendit Jul 7 '17 at 15:18

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.


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