Proper translation

Heike Mildenberger ¹   ; Saharon Shelah ²

¹ Abteilung für Mathematische Logik Mathematisches Institut Universität Freiburg Eckerstr. 1 79104 Freiburg im Breisgau, Germany
² Einstein Institute of Mathematics The Hebrew University of Jerusalem Edmond Safra Campus Givat Ram 91904 Jerusalem, Israel and Mathematics Department Rutgers University Piscataway, NJ 08854-8019, U.S.A.

Fundamenta Mathematicae, Tome 215 (2011) no. 1, pp. 1-38

Résumé

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.

Détail

DOI : 10.4064/fm215-1-1

Keywords: continue work weak diamonds appl anal omega aleph together weak diamond covering thin trees weak diamond covering meagre sets weak diamond covering null sets aronszajn trees special consistent relative zfc iterate alternately forcings specialising aronszajn trees without adding reals nnr forcing proper improper forcing nbsp omega proper omega omega bounding forcings adding reals tower elementary submodels there sort reduction proper translation iteration countable support iteration simpler iterands only sacks iterands nnr iterands allows guess values borel functions small trees derive above mentioned weak diamonds