### 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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