Statement of empty sets(using Axiom schema of specification) Wikipedia says that this definition derives from Axiom schema of specification(wikiZfc). 

$\emptyset = \{ u \in w | (u \in u) ∧$ ~$(u \in u) \}$

Probably $(u \in u) ∧$ ~$(u \in u)$ intend to if the first one is true or false, then second has opposite truth value, so it's a contradiction. But $(u \in u)$ is a contradiction itself. I think $(u \in w) ∧ $~$(u \in w)$ is natural rather than $(u \in u) ∧ $~$(u \in u)$. What's wrong with my thought?
 A: Actually, $u\in u$ is not a contradiction in itself, but a contradiction to the axiom of foundation. If you remove the axiom of foundation from ZFC, you get a theory containing the axiom schema of specification in which $u\in u$ can be true.
That is, in the context of ZFC indeed $\{u\in w|u\in u\}$ gives the empty set, but in set theories that allow non-well-founded sets, it won't.
In the form written in Wikipedia, the empty set is obtained in any set theory that includes the axiom schema of specification (either explicitly, or as derivable theorem), that is, in any set theory in which that notation actually makes sense.
Note that the important part of the condition is that it never can be true for any element of $w$. The easiest way to achieve that is to have a formula that yields false forany set. However, also the following works:
$$\emptyset = \{u\in w|u\notin w\}$$
While there are certainly sets with $u\notin w$, by definition none of them is in  $w$.
However to see that this formula works needs more thinking than to see that the formula given by Wikipedia works, as the formula in Wikipedia by itself is contradictory. But the formula in Wikipedia is itself unnecessarily complicated; A much simpler one that also always works is
$$\emptyset = \{u\in w|u\ne u\}$$
