Strict polynomial hierarchy and reduction
$\newcommand{\PH}{\mathsf{PH}}$To expand the comments:

Suppose there did exist a $\PH$-Complete language $L \in \Sigma_i$ for some $i$. By definition of completeness $L' \leq L$ for all $L' \in \PH$. Therefore there exists some language $L' \in \Sigma_{i+1}$ that reduces to $L$. This is a contradiction with the strictness assumption. Indeed, we conclude that no such $\PH$-Complete language exists.