I have recently been slogging my way through Shelah's "*Large continuum, oracles*". Essentially from the start there has been a question needling me which I cannot seem to answer.

- In the paper, Shelah says that a forcing notion $\mathcal{P}$ is
*absolutely ccc*if it remains ccc after forcing with any ccc notion. - Elsewhere, I have seen it defined that a forcing notion $\mathcal{P}$ is
*absolutely ccc*if it remains ccc after any forcing. (This would be*indestructibly ccc*from Bartoszyński-Judah.)

Any forcing having the Knaster property is absolutely ccc (in the strong sense), and MA$_{\aleph_1}$ implies that all ccc forcings have the Knaster property. Thus, it is consistent that the two are equivalent.

Do these two versions of absolute ccc-ness provably coincide?

of size less than the continuumis Knaster. – saf Mar 24 '12 at 0:59e.g., Jech (3rd ed.), Theorem 16.21, p.277. The proof actually gives the slightly stronger result that MA$_{\aleph_1} $implies that all ccc posets have precalibre $\aleph_1$. – arjafi Mar 24 '12 at 8:01