Upward Lowenheim-Skolem theorem
= Upward Lowenheim-Skolem theorem
{c}
If an $L$-theory has an infinite model, then it has models of arbitrarily large cardinality. For a cardinal $\kappa$, add constants $c_\alpha$ for $\alpha<\kappa$ and all inequalities $c_\alpha\ne c_\beta$. Every finite fragment is satisfiable in the given infinite model, so compactness gives a model with at least $\kappa$ elements. Applying the downward theorem produces a model of exactly $\kappa$ when $\kappa\geq|L|+\aleph_0$.