# The meaning of singleton and existence of $\{\varnothing, \{ \varnothing \}\}$

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?

• 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. – Asaf Karagila May 17 '15 at 9:48

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.
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 $\{\varnothing\}$.