TITLE: How necessary is Godel's Condensation Lemma
QUESTION [12 upvotes]: It seems that the Godel's Condensation Lemma is typically used to show that certain constructible sets will appear by some stage of the construction of $L$.  For example in the proof that CH holds in $L$, GCL is used to show that if $X \subset \omega$ is contructible, then $X \in L_{\omega_{1}}$.  
Does anyone know if there are combinatorial arguments for these sorts of facts; i.e. arguments that don't use GCL or results from model theory?  How would one prove $L \models$CH to someone who did not know the Lowenheim-Skolem theorem?

REPLY [5 votes]: It's been 25 years since I struggled through it, but I clearly remember that Godel's book Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory gets this result combinatorially --- exactly how, I have long forgotten. I only learned about the much, much easier Lowenheim-Skolem proof later on.
I've been told that Godel wrote his book that way, in machine language, essentially, because he didn't want any philosophical doubts people might have about logic to affect their reception of his theorem.