S-forcing IIa: Adding diamonds and more applications: Coding sets, Arhangel’skii’s problem and ℒ $$[Q_1^{< \omega } ,Q_2^1 ]$$

Abstract

We continue our earlier paper [20] by proving the equivalence, for regularκ>ω, of the existence of (κ, 1) morasses with built-in ♦ sequences and a strengthening, SK◊ , of the forcing principle, SK◊ of [20]. We obtain various applications of SK◊, to wit: the existence of a stationary subset of [K+]<K with sup as coding function, the existence of a counterexample to Arhangel’skii’s conjecture (κ=ℵ1) and compactness, axiomatizability and transfer properties for the Magidor-Malitz language ℒ\([Q_1^{< \omega } ,Q_2^1 ]\) (κ=ℵ1).