Proper translation
Volume 215 / 2011
Abstract
We continue our work on weak diamonds [J. Appl. Anal. 15 (1009)]. We show that $2^\omega = \aleph _2$ 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 ${<}\omega _1$-proper ${}^\omega \omega $-bounding forcings adding reals. We show that over a tower of elementary submodels there is a sort of a reduction (“proper translation”) of our iteration to the countable support iteration of simpler iterands. If we use only Sacks iterands and NNR iterands, this allows us to guess the values of Borel functions into small trees and thus derive the above mentioned weak diamonds.
