Once, someone showed me a nice argument using elementary substructures for proving chain conditions about forcings. It was a basic example, maybe that Cohen forcing is ccc. I've been trying to look up this argument, but I've only found more combinatorial proofs. Can someone remind me how this argument goes or point me to a source that does? Thanks.



Substructure Argument for Chain Conditions
Say that $P=\{p \subset \kappa \times 2 : p \mbox{ is a finite function} \}$ is ordered by reverse inclusion (adding $\kappa$ Cohen reals). We show that $P$ is ccc:


Let $A \subseteq P$ be an antichain and let $M$ be a countable elementary submodel of (a sufficiently large initial segment of) the universe such that $A,P \in M$. It is enough to show that $A \subseteq M$. If not, fix $p \in A \setminus M$, let $a=dom(p) \cap M$ and let $q=p \upharpoonright a$. It is clear that both $a$ and $q$ are in $M$. The existence of an $x \in A$ such that $a \subseteq dom(x)$ and $x \upharpoonright a = q$ is true in the universe ($p$ is a witness) and therefore in $M$ , so let $p' \in A \cap M$ be a witness of this. Now $p$ and $p'$ are compatible (because $dom(p) \cap dom(p')=a$) and they are different (because one is in $M$ and the other is not), a contradiction.


