Based on my limited understanding, the precise formulation of Lowenheim-Skolem (LS) theorem is stated as follows:
Let $\Gamma$ be a set of sentences from a countable language $\mathcal L$. Let $\mathcal M$ be an infinite model for $\Gamma$, i.e. $|\text{dom}(\mathcal M)|\ge\aleph_0$. Then,
(Upward part) For every cardinal number $\kappa\ge|\text{dom}(\mathcal M)|$, there exists an elementary extension $\mathcal N$ of $\mathcal M$ such that $|\text{dom}(\mathcal N)|=\kappa$.
(Downward part) For every infinite cardinal number $\kappa<|\text{dom}(\mathcal M)|$, there exists an elementary substructure $\mathcal N$ of $\mathcal M$ such that $|\text{dom}(\mathcal N)|=\kappa$.
According to Sets, Logic, Computation, one of the consequences of this theorem is that FOL cannot express that the size of a structure is uncountable. I don't understand what it means. What does it mean to "express" exactly? I interpreted this statement as for any infinite cardinal $\kappa>\aleph_0$ (not sure if this result holds for $\kappa=\aleph_0$), there is no set of sentences $\Gamma$ such that it satisfies $\mathcal M\models\Gamma$ iff $|\text{dom}(\mathcal M)|=\kappa$. Is that correct? For if there is such $\kappa$ and $\Gamma$, then certaintly, one direction "$\mathcal M\models\Gamma$ if $|\text{dom}(\mathcal M)|=\kappa$" holds but the other "$\mathcal M\models\Gamma$ only if $|\text{dom}(\mathcal M)|=\kappa$" is false because FOL cannot "control" the size of $\text{dom}(\mathcal M)$ according to LS theorem...?
Did I misunderstand anything?
I think I figured it out now. The LS theorem tells us that for any set of sentences $\Gamma$ in countable $\mathcal L$, FOL cannot "control" the cardinalities of its infinite models, i.e. it is inevitable that there are other infinite models for $\Gamma$ with cardinality $\ge\aleph_0$. So, it has two main consequences:
There is no consistent set of sentences $\Gamma$ in $\mathcal L$ such that all of its models are uncountably infinite. Otherwise, the downward LS theorem implies $\Gamma$ has a countably infinite model, which is a contradiction.
There is no consistent set of sentences $\Gamma$ in $\mathcal L$ such that all of its models are countably infinite. Otherwise, the upward LS theorem implies $\Gamma$ has an uncountably infinite model, which is a contradiction.