Many precipitous ideals

In the famous paper Martin`s Maximum, Foreman, Magidor and Shelah developed methods to make the nonstationary ideal precipitous (on certain cardinals). Goldring extended this construction to P kappa lambda. In this project we will try to find as big as possible a class of ideals which can be made precipitous (assuming sufficiently large cardinals, e.g., a class of supercompacts). A first goal is a theorem such as: All nicely definable normal ideals on P kappa lambda (kappa, lambda regular) are precipitous. Of course we will have to find out what "nicely definable" means (for example, the notion cannot include the completely ineffable ideal). We will also investigate the case of non-normal ideals and non-regular cardinals, as well as variants of precipitous, and try to contribute something to the old, unsolved questions of the field, such as: Does a precipitous ideal imply a normal precipitous one? Does a large cardinal imply a precipitous ideal on aleph1? I will hire a PhD student (Wolfgang Wohofsky is designated for this) and collaborate with Moti Gitik (Tel Aviv University, Israel) and Saharon Shelah (The Hebrew University of Jerusalem, Israel). The project will last 24 months. The main part of the cost will be the salary of the PhD student. Other costs are four longer research visits of the PhD student and myself, and invitations of potential collaborators to two conferences in Austria.

In course of the project we answered a question that has been open for quite a while: Consistently, the Borel Conjecture and the dual Borel Conjecture hold (simultaneously). A set A of reals is strong measure zero (smz), if for every sequence of rational numbers there is a cover of A by intervals which lenghts correspond to the given sequence. Equivalently, A is smz if for every comeager set B there is a real r such that A translated by r is subset of B. A is strongly meager (sm), if for every set B of measure zero there is a real r such that A translated by r is subset of B. The Borel Conjecture (BC) states that every smz set ist countable. The dual Borel Conjecture (dBC), that every sm set ist countable. It is easy to see that under the continuum hypothesis BC and dBC both fail. Laver and Carlson showed that BC and dBC are (seperably) consistent. It remaind open whether BC+dBC is consistent.