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I can't understand difference between $\varnothing$ and $\{\varnothing\}$. What is difference between $x$ and set that has only $x$ as element? I think the concept of set is meaningful when it has more than 1 element. If "$\varnothing$ and $\{\varnothing \}$ is different" is right, then I have another question. When we see the set $\{\varnothing,\{\varnothing\}\}, \varnothing$ and $\{\varnothing\}$ exist simultaneously. It means two empty sets exist. One empty set exists as $\varnothing$ itself and one empty set exists as element of $\{\varnothing\}$. I can't understand this situation. What am i misunderstanding?

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  • $\begingroup$ There are probably an infinite list of questions like this on the site already. Here are a few quick links. One, two, three, four, five... and there are more. Many many many more. $\endgroup$ – Asaf Karagila May 17 '15 at 9:48
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Think of sets as bags. $\varnothing$ is an empty bag. $\{\varnothing\}$ is a bag with an empty bag within it. Therefore the outermost bag is not empty: It has a bag inside it.

Also, with this analogy, we have that $\{\varnothing,\{\varnothing\}\}$ a bag is containing an empty bag and a bag with an empty bag within it.

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The void set $\varnothing$ has no elements.

The set $\{\varnothing\}$ has one element: the void set $\varnothing$.

The set $\{\varnothing,\{\varnothing\}\}$ has two elements : the void set $\varnothing$ and the set $\{\varnothing\}$.

The key fact is that a set (void or not) is a different thing with respect to his elements and can be element of another set.

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