Let $L$ be a first-order language and let $\mathcal{A}$ be an infinite$L$ -structure. Then if $\kappa$ is a cardinal with $\kappa\geq\operatorname{Max}(\card{\mathcal{A}}, \card{L})$ then there is an $L$ -structure $\mathcal{B}$ such that $\card{\mathcal{B}} = \kappa$ and $\mathcal{A} \preccurlyeq \mathcal{B}$ .
"upward Lowenheim-Skolem theorem" is owned by Evandar.
