On pg 190 of Jech's Set Theory, he proves V = L implies GCH. I understand it all except the following: 



> 
> Thus let X ⊂ $ω\_α$. There exists a limit ordinal δ>$ω\_α$ such that X ∈ $L\_δ$.
>  Let M be an elementary submodel of $L\_δ$ such that $ω\_α$ ⊂ M and X ∈ M, and $|M| = \aleph\_{\alpha}$. 
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> 
> 


I think this uses the Lowenheim Skolem theorem to get the model of size $\aleph\_{\alpha}$, but why should $\omega\_{\alpha} \subset M$ and why should $X \in M$, or how do we construct such an M? 


I know there are other proofs of this theorem which uses a version of the reflection principle which gives you this directly, but the version of the reflection principle I know is the more basic one which doesn't deal with the cardinality of the sets for which the formulas are absolute (the beginning reflection theorems in Kunen). 


Basically, I would prefer an explanation that uses the Lowenheim Skolem theorem instead. How is this done? 



Question about Jech's proof of V = L implies GCH
The (downward) Löwenheim-Skolem theorem has several formulations. The one I use says that, if $\mathfrak A$ is a structure with universe $A$ and vocabulary (also called language and signature) $L$, and if $S\subseteq A$, then $\mathfrak A$ has an elementary substructure $\mathfrak B$ whose universe $B$ satisfies both $S\subseteq B$ and $|B|\leq\max\{|S|,|L|,\aleph\_0\}$. Applying this theorem with $\mathfrak A=(L\_\delta,\in)$ and $S=\omega\_\alpha\cup\{X\}$ gives you the result that Jech needs.


Some people use other (often weaker) formulations of the Löwenheim-Skolem theorem. If you're one of those people, go back to the proof of that formulation and tweak it slightly to get the formulation I use.


