Definitions:
For a given first order sentence $\phi$ define $\text{spectrum}(\phi)$ to be the set of all cardinalities of the finite models of $\phi$. A set $S\subseteq\mathbb N_+$ is said to be a spectrum, if there is a first order sentence $\phi$ with $\text{spectrum}(\phi)=S$.
For a given first order sentence $\phi$ define $\infty\text{-spectrum}(\phi)$ to be the class of all cardinal numbers $\kappa$ for which there is a model $M$ of $\phi$ such that the cardinality of $M$ is $\kappa$. Let $\text{Card}$ be the class of all cardinal numbers. Set $\text{Card}_+:=\text{Card}\setminus \{0\}$. A subclass $T$ of $\text{Card}_+$ is said to be an $\infty$-spectrum if there is a first order sentence $\phi$ with $\infty\text{-spectrum}(\phi)=T$.
Question: Let $S$ be an arbitrary spectrum. Does it follow that $S$ is an $\infty$-spectrum?
Hint: If a sentence $\phi$ has models of arbitrarily large finite cardinality, use compactness to show it must also have infinite models.