TITLE: Why is the supremum of the empty set $-\infty$ and the infimum $\infty$?
QUESTION [14 upvotes]: 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?

REPLY [6 votes]: This is an example of "vacuous implication", and as such it's sometimes easier to turn the question around and ask "what if it weren't so?"  That is, what if $\sup \emptyset = s \in \mathbb{R}$ were not $-\infty$?  This would mean that no $t < s$ could be an upper bound for $\emptyset$, which in turn implies that:

for any $t < s$, there exists an element $x \in \emptyset$ such that $t < x$.  

But that property is impossible, as there is no element in $\emptyset$.  This means that, should we desire to assign a "numerical" value to $\sup \emptyset$, it must be less than every element of $\mathbb{R}$, and that's what $-\infty$ is intended to mean.  The same logic applies to $\inf \emptyset$, but with all ordering relations reversed so it must be larger than every real number, hence $\infty$.