A dichotomy for forcing notions
In an earlier paper, the author and Shelah showed that under PFA adding \(\omega_1\) Cohen reals is universal in the following sense: every separative partial order of size \(\omega_1\) that does not add \(\omega_1\) Cohen reals is small, in the sense that below some condition it is exactly the partial order adding one Cohen real. Let \(B\) be the partial order due to Baumgartner adding a closed unbounded subset of \(\omega_1\) by finite conditions, with \({\mathbb B}\) its completion. This paper shows that, under PFA, \(B\) is universal in the following sense: If the completion of \(B\) does not embed completely in the completion of a separative partial order \(P\) of size \(\omega_1\), then \(P\) is small, in the sense that below some condition \(P\) is ccc. The proof of the theorem in this paper uses the earlier theorem quite heavily, by noting that the partial order adding \(\omega_1\) many Cohen reals is isomorphic to the product of \(\omega_1\) many forcings adding a single Cohen real, and then coding these countable forcings as approximations to the Baumgartner order. In neither theorem is the full strength of PFA used; the consistency of the existence of large cardinals is not necessary.
scientific article; zbMATH DE number 1187595