I have found two versions of the famous Lowenheim-Skolem Theorem (LST) which appear outwardly to be saying different things. In Enderton's classic logic manual he states
LST (Enderton): (a) Let $\Gamma$ be a satisfiable set of formulas in a language of cardinality $\lambda$. Then $\Gamma$ is satisfiable in some structure of size $\le \lambda$. (b) Let $\Sigma$ be a set of sentences in a language of cardinality $\lambda$. If $\Sigma$ has a model, then it has a model of cardinality $\le \lambda$.
Whereas Jech in his classic tome 'Set Theory' states
LST (Jech): Every infinite model for a countable language has a countable elementary submodel. [my italics].
Now the problem I have is with Jech's use of elementary submodel. It seems to me that Jech is saying something much stronger. Enderton's version says that if a countable $\Sigma$ has a model $V$ then it has a countable model $U$. Jech on the other hand not only asserts this, but in addition states that $U$ is an elementary submodel of $V$. Surely this is a much stronger statement?