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• # Artykuł - szczegóły

## Fundamenta Mathematicae

2011 | 215 | 1 | 1-38

## Proper translation

EN

### Abstrakty

EN
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.

1-38

wydano
2011

### Twórcy

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