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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.

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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.

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