Question about Jech's proof of V = L implies GCH

Question

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 $|M| = \aleph_{\alpha}$.

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?

Answer 1

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.

Answer 2

You assume $\omega_\alpha\subseteq M$ and $X\in M$ so that $X$ belongs to the transitive collapse of $M$ (because if $\pi$ is the collapsing map, $\pi(X)=\pi[X]=X$. You assume $|M|=\aleph_\alpha$ so that the transitive collapse of $M$ has size $\aleph_\alpha$. Since you also have that this transitive collapse is of the form $L_\beta$ for some $\beta$, it follows that $|\beta|=\aleph_\alpha$.

Building $M$ is a direct application of the downward Lowenheim-Skolem theorem (so $M\preceq L_\delta$). If you do not find this apparent, I suggest you read the section(s) in Chang and Keisler's classic model theory book that cover this result, which they do in more detail than in Jech's or Kunen's texts.