Heike Mildenberger: Proper translation

Week 1, Thursday June 18, 14:10-15:00

This is joint work with S. Shelah. We continue our work on weak diamonds. We show that 2^omega= aleph_2 together with the weak diamond for covering by slaloms, 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 the Proper Forcing book) and <omega_1-proper 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 mentioned weak diamonds.