Proper translation

by Mildenberger and Shelah. [MdSh:951]
Fundamenta Math, 2011
We continue our work on weak diamonds [MdSh:848]. 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 [Sh:f, Ch.~IV]) 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 c.s. 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 mentionedweak diamonds.

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