S-forcing. IIa: Adding diamonds and more applications: Coding sets, Arhangel'skii's problem and \({\mathcal L}[Q_ 1^{<\omega},Q^ 1_ 2]\). (With an appendix by John P. Burgess)
This paper is a continuation of the author's earlier S-Forcing I. A ``black-box'' theorem for morasses with applications to super-Souslin trees [Israel J. Math. 43, 185-224 (1982; Zbl 0513.03023)]. Notation and terminology from this previous work are carried over to the current one. \(S_{\kappa}(\diamond)\) is a strengthening of \(S_{\kappa}\), the previously presented forcing principle. The stated main result is: Theorem. For regular \(\kappa >\omega\), \(S_{\kappa}(\diamond)\) iff there exist (\(\kappa\),1)-morasses with built-in \(\diamond\) sequences. This leads to: in L for all regular \(\kappa >\omega\) there are (\(\kappa\),1) morasses with built-in \(\diamond\) sequences. This paper contains several applications of \(S_{\kappa}(\diamond)\). For example, \(S_{\aleph_ 1}(\diamond)\) implies the Magidor-Malitz language \({\mathcal L}[Q_ 1^{<\omega},Q^ 1_ 2]\) has the compactness, axiomatizability and transfer properties. Also for topological spaces \(S_{\aleph_ 1}(\diamond)\) implies there is a counterexample to Arhangel'skii's conjecture. This is a long and technically complex paper and it refers both positively and negatively (corrections to the earlier literature are made) to a large body of work.
scientific article; zbMATH DE number 4025431
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