Proper translation

Heike Mildenberger, and Saharon Shelah. "Proper translation." Fundamenta Mathematicae 215.1 (2011): 1-38. <http://eudml.org/doc/286424>.

@article{HeikeMildenberger2011,
abstract = {We continue our work on weak diamonds [J. Appl. Anal. 15 (1009)]. We show that $2^\{ω\} = ℵ₂$ 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 < ω₁-proper $^\{ω\} ω$-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.},
author = {Heike Mildenberger, Saharon Shelah},
journal = {Fundamenta Mathematicae},
keywords = {specialising Aronszajn trees; Borel computations},
language = {eng},
number = {1},
pages = {1-38},
title = {Proper translation},
url = {http://eudml.org/doc/286424},
volume = {215},
year = {2011},
}

TY - JOUR
AU - Heike Mildenberger
AU - Saharon Shelah
TI - Proper translation
JO - Fundamenta Mathematicae
PY - 2011
VL - 215
IS - 1
SP - 1
EP - 38
AB - We continue our work on weak diamonds [J. Appl. Anal. 15 (1009)]. We show that $2^{ω} = ℵ₂$ 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 < ω₁-proper $^{ω} ω$-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.
LA - eng
KW - specialising Aronszajn trees; Borel computations
UR - http://eudml.org/doc/286424
ER -