Substructure Argument for Chain Conditions

Question

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.

Answer 1

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.

Answer 2

A typical argument for showing that a notion of forcing is ccc will invoke the $\Delta$-system lemma at some point, and this result has a slick proof using elementary submodels (see Lemma 24.24 of "Discovering Modern Set Theory II" by Just and Weese, or Example 2.1 of Dow's "An introduction to applications of elementary submodels to topology".

One can usually combine these two steps into a single argument to prove that a notion of forcing is ccc using elementary submodels, although I can't point to a specific published example of this at the moment. I'll see if I can find a reference for you where someone does this.

Edit: The best reference is Ramiro's answer!