Why is the intersection of the empty set the universe? I haven't found an answer yet that I can make complete sense of, so I'm asking again to try and clear it up if I have any misconceptions. 
If one defines $\cap_{i\in I}\alpha_i = \{x:\forall i\in I, x\in\alpha_i\}$ then apparently if $I = \emptyset$ this definition yields the absolute universe. This is just stated as if it is clear why, though I cannot see why. If $i\notin I \forall i$ then there is no set $\alpha_i$ for any $x$ to be a member of...?
I know I must be misreading this, but I can't see how by so much...
Edit: Let this intersection be $Z$, for convenience.

Given $\cap_{i\in I}\alpha_i = \{x:\forall i\in I, x\in\alpha_i\}$. If I, the family of sets is empty then any x would satisfy the condition $\forall i\in I, x\in\alpha_i$ vacuously because there is no $\alpha_i$. Therefore $\cap_{i\in I}\alpha_i=$Universe if $I=\phi$.
Unfortunately, there is no universal set in Zermelo Frankel set theory. We can overcome this problem by restricting the sets to a universal set U. In this case nullary intersection would be U.
