Is the word "empty" in set theory different from the word "empty" in ordinary language? I skimmed over this question Is the empty set a subset of itself?, and I'm currently under the impression that it's a widespread belief that the empty set contains itself. But a contradiction seems to arise: if the empty set contains itself, then it's NOT empty. After all, if we call a set non-empty just because it contains the empty set, then why should we treat the empty set itself differently?
Then it occurred to me that maybe mathematicians define "empty" differently in set theory. Maybe by "empty set" they mean a "set that contains only itself", instead of a "set that contains absolutely nothing".
Is my surmise correct?
 A: The problem here is you are using the word "contains" for two different things:


*

*When $x$ is an element of a set $A$, we write $x\in A$ and sometimes say "$A$ contains $x$".

*When $B$ is a subset of $A$, we write $B\subseteq A$ and also sometimes say "$A$ contains $B$".


This is a common problem and usually we can determine from the context whether which of the two types of containment is being referred to.
With the empty set, it is always false that $x\in\varnothing$, i.e., the empty set does not have elements, it is empty.
However, it has a subset: $\varnothing\subseteq\varnothing$ is true. Why is that? Well, $B\subseteq A$ means that whenever $x$ is an element of $B$ it also is an element of $A$, or in other words, $A$ contains (as elements) at least all elements of $B$. In particular $\varnothing\subseteq X$ is true for any set $X$, since there are no elements in $\varnothing$ that have to be contained in $X$ at all.
A: $A\subseteq B$ is not the same as $A\in B$. The former one says that all elements of $A$ are also in $B$. Whereas the latter one means the set $A$ is an element of $B$. So $B=\{\,A,\dots\,\}$.
Therefore it is correct to say $\emptyset\subseteq\emptyset$ but not $\emptyset\in\emptyset$ since the empty set has no elements.
A: I think you are confusing two possible meanings of "contains". Often we say $A$ contains $B$ to mean $B$ is an element of $A$ (but see below). In this sense, $\varnothing$ doesn't contain itself, because it has no elements. (Also, no set is an element of itself.)
However, $\varnothing$ is a subset of itself, because it doesn't contain any elements outside itself. By the same logic, every set is a subset of itself. 
To see the distinction in non-empty sets, $\{1\}$ is a subset of $\{1,2\}$ but $\{1,2\}$ does not contain $\{1\}$. It does contain $1$, but that is not the same thing. 

On the usage of "contains", wikipedia says:

The expressions "A includes x" and "A contains x" are also used to mean set membership, however some authors use them to mean instead "x is a subset of A". Logician George Boolos strongly urged that "contains" be used for membership only and "includes" for the subset relation only.

