Internally club and approachable (Q2370614)

A set \(N\) of cardinality \(\aleph_1\) is called internally approachable if there is an increasing and continuous sequence \((N_i : i < \omega_1 )\) of countable sets whose union is \(N\) such that all initial segments \((N_i : i < \alpha)\), \(\alpha < \omega_1\), are elements of \(N\). \(N\) is internally club if there is such a sequence such that all \(N_\alpha\), \(\alpha < \omega_1\), are elements of \(N\). Clearly every internally approachable \(N \prec H(\lambda)\), \(\lambda \geq \aleph_2\) regular, is internally club. The author shows that the proper forcing axiom PFA implies that for all regular cardinals \(\lambda \geq \aleph_2\), there are stationarily many sets \(N \) in \([H(\lambda)]^{\aleph_1}\) which are internally club but not internally approachable. This result is obtained by applying PFA to a three-step iteration the first step of which adds a Cohen real, while the second shoots an \(\omega_1\)-sequence through the countable subsets of \(H(\lambda)\) with countable conditions, and the third is the standard ccc forcing specializing a tree of height and size \(\aleph_1\) which arises in the second intermediate extension.