Proper translation

Abstract

We continue our work on weak diamonds [J. Appl. Anal. 15 (1009)]. We show that 2ω = א2 together with the weak diamond for covering by thin trees, the weak diamond for covering by meagre sets, the weak diamond for covering by null sets, and "all Aronszajn trees are special" is consistent relative to ZFC. We iterate alternately forcings specialising Aronszajn trees without adding reals (the NNR forcing from ["Proper and Improper Forcing", Ch. V]) and