Why is the supremum of the empty set $-\infty$ and the infimum $\infty$? I read in a paper on set theory that the supremum and the infimum of the empty set are defined as $\sup(\{\})=-\infty$ and $\inf(\{\})=\infty$. But intuitively I can't figure out why that is the case. Is there a reason why the supremum and infimum of the empty set are defined this way?

As robjohn says, for sets $A$ and $B$, we want $\sup (A\cup B) = \max(\sup (A), \sup (B))$. If $B=\emptyset$, then we have
$$\sup (A) = \max(\sup (A), \sup (\emptyset)).$$
In other words, $\sup (\emptyset)$ must be the identity element for the binary operation $\max$, which is $-\infty$. Dually, $\inf (\emptyset)$ must be $\infty$.
The nice thing about this viewpoint is that it generalizes immediately to complete lattices! The empty join is the least element and the empty meet is the greatest element.
