TITLE: Amalgamation of two ccc algebras may collapse the continuum
QUESTION [5 upvotes]: The claim that appears in the title of this question is mentioned in the paper "On Shelah's amalgamation" by Judah and Roslanowski. I'd really like to see a proof of this fact, but unfortunately I couldn't find any reference for this claim.
So my question is: Are there relatively simple examples of amalgamation of ccc algebras that collapse the continuum? If there aren't any simple examples, then a reference for a harder example would be welcome as well.

REPLY [6 votes]: Suppose CH holds and there is a Souslin  tree $T$.  (For example, suppose $V=L$.)  Then $T$ itself can be regarded as a forcing notion; it has ccc and it adjoins a generic path through $T$.  There is also a ccc forcing notion that specializes $T$; a condition here is a finite partial function from $T$ into $\omega$ such that any two comparable nodes in the domain of the condition are assigned distinct values; the union of the generic filter is a function from all of $T$ into $\omega$ that never takes the same value at two comparable nodes of $T$.  If you force with both of these, then the specializing function restricted to the generic path gives a one-to-one map of $\omega_1$ into $\omega$.  So you've collapsed $\aleph_1$; since I assumed CH, this means you've collapsed the cardinal of the continuum.